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Compact space

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of rational numbers $$\mathbb{Q}$$ is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers $$\mathbb{R}$$ is not compact either, because it excludes the two limiting values $$+\infty$$ and $$-\infty$$. However, the extended real number line would be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces.

One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval $A = (−∞, −2]$, some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence $1⁄2$, $4⁄5$, $1⁄3$, $5⁄6$, $1⁄4$, $6⁄7$, ... accumulate to 0 (while others accumulate to 1). Since neither 0 nor 1 are members of the open unit interval $C = (2, 4)$, those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded. For example, considering $$\mathbb{R}^1$$ (the real number line), the sequence of points 0,  1,  2,  3, ... has no subsequence that converges to any real number.

Compactness was formally introduced by Maurice Fréchet in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to spaces of functions. The Arzelà–Ascoli theorem and the Peano existence theorem exemplify applications of this notion of compactness to classical analysis. Following its initial introduction, various equivalent notions of compactness, including sequential compactness and limit point compactness, were developed in general metric spaces. In general topological spaces, however, these notions of compactness are not necessarily equivalent. The most useful notion — and the standard definition of the unqualified term compactness — is phrased in terms of the existence of finite families of open sets that "cover" the space in the sense that each point of the space lies in some set contained in the family. This more subtle notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, exhibits compact spaces as generalizations of finite sets. In spaces that are compact in this sense, it is often possible to patch together information that holds locally – that is, in a neighborhood of each point – into corresponding statements that hold throughout the space, and many theorems are of this character.

The term compact set is sometimes used as a synonym for compact space, but also often refers to a compact subspace of a topological space.

Historical development
In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano (1817) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a limit point. Bolzano's proof relied on the method of bisection: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts – until it closes down on the desired limit point. The full significance of Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.

In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points. The idea of regarding functions as themselves points of a generalized space dates back to the investigations of Giulio Ascoli and Cesare Arzelà. The culmination of their investigations, the Arzelà–Ascoli theorem, was a generalization of the Bolzano–Weierstrass theorem to families of continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergent sequence of functions from a suitable family of functions. The uniform limit of this sequence then played precisely the same role as Bolzano's "limit point". Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of integral equations, as investigated by David Hilbert and Erhard Schmidt. For a certain class of Green's functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà–Ascoli theorem held in the sense of mean convergence – or convergence in what would later be dubbed a Hilbert space. This ultimately led to the notion of a compact operator as an offshoot of the general notion of a compact space. It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property and coined the term compactness to refer to this general phenomenon (he used the term already in his 1904 paper which led to the famous 1906 thesis).

However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuum, which was seen as fundamental for the rigorous formulation of analysis. In 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it. The significance of this lemma was recognized by Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and Henri Lebesgue (1904). The Heine–Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers.

This property was significant because it allowed for the passage from local information about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function). This sentiment was expressed by, who also exploited it in the development of the integral now bearing his name. Ultimately, the Russian school of point-set topology, under the direction of Pavel Alexandrov and Pavel Urysohn, formulated Heine–Borel compactness in a way that could be applied to the modern notion of a topological space. showed that the earlier version of compactness due to Fréchet, now called (relative) sequential compactness, under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers. It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.

Basic examples
Any finite space is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it. A nontrivial example of a compact space is the (closed) unit interval $X$ of real numbers. If one chooses an infinite number of distinct points in the unit interval, then there must be some accumulation point among these points in that interval. For instance, the odd-numbered terms of the sequence 1, $A$, $X$, $x$, $x$, $A$, $x$, $a$, ... get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1. The given example sequence shows the importance of including the boundary points of the interval, since the limit points must be in the space itself — an open (or half-open) interval of the real numbers is not compact. It is also crucial that the interval be bounded, since in the interval $x$, one could choose the sequence of points 0, 1, 2, 3, ..., of which no sub-sequence ultimately gets arbitrarily close to any given real number.

In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, an open disk is not compact, because a sequence of points can tend to the boundary – without getting arbitrarily close to any point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can still tend to the missing point, thereby not getting arbitrarily close to any point within the space. Lines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point.

Definitions
Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set. Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.

In contrast, the different notions of compactness are not equivalent in general topological spaces, and the most useful notion of compactness – originally called bicompactness – is defined using covers consisting of open sets (see Open cover definition below). That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine–Borel theorem. Compactness, when defined in this manner, often allows one to take information that is known locally – in a neighbourhood of each point of the space – and to extend it to information that holds globally throughout the space. An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a local property of the function, and uniform continuity the corresponding global property.

Open cover definition
Formally, a topological space $A$ is called compact if every open cover of $A$ has a finite subcover. That is, $x$ is compact if for every collection $x$ of open subsets of $A$ such that

$$X = \bigcup_{S \in C}S\ ,$$

there is a finite subcollection $A$ ⊆ $[0,1]$ such that

$$X = \bigcup_{S \in F} S\ .$$

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact. A compact set is sometimes referred to as a compactum, plural compacta.

Compactness of subsets
A subset $1⁄2$ of a topological space $1⁄3$ is said to be compact if it is compact as a subspace (in the subspace topology). That is, $3⁄4$ is compact if for every arbitrary collection $1⁄5$ of open subsets of $5⁄6$ such that

$$K \subseteq \bigcup_{S \in C} S\ ,$$

there is a finite subcollection $1⁄7$ ⊆ $7⁄8$ such that

$$K \subseteq \bigcup_{S \in F} S\ .$$

Compactness is a topological property. That is, if $$K \subset Z \subset Y$$, with subset $[0,∞)$ equipped with the subspace topology, then $X$ is compact in $X$ if and only if $X$ is compact in $C$.

Characterization
If $X$ is a topological space then the following are equivalent:
 * 1) $F$ is compact; i.e., every open cover of $C$ has a finite subcover.
 * 2) $K$ has a sub-base such that every cover of the space, by members of the sub-base, has a finite subcover (Alexander's sub-base theorem).
 * 3) $X$ is Lindelöf and countably compact.
 * 4) Any collection of closed subsets of $K$ with the finite intersection property has nonempty intersection.
 * 5) Every net on $C$ has a convergent subnet (see the article on nets for a proof).
 * 6) Every filter on $X$ has a convergent refinement.
 * 7) Every net on $F$ has a cluster point.
 * 8) Every filter on $C$ has a cluster point.
 * 9) Every ultrafilter on $Z$ converges to at least one point.
 * 10) Every infinite subset of $K$ has a complete accumulation point.
 * 11) For every topological space $Z$, the projection $$X \times Y \to Y$$ is a closed mapping (see proper map).
 * 12) Every open cover linearly ordered by subset inclusion contains $K$.

Bourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above).

Euclidean space
For any subset $Y$ of Euclidean space, $X$ is compact if and only if it is closed and bounded; this is the Heine–Borel theorem.

As a Euclidean space is a metric space, the conditions in the next subsection also apply to all of its subsets. Of all of the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for a closed interval or closed $X$-ball.

Metric spaces
For any metric space $B = [0, 1]$, the following are equivalent (assuming countable choice):
 * 1) $[0, 1]$ is compact.
 * 2) $(0, 1)$ is complete and totally bounded (this is also equivalent to compactness for uniform spaces).
 * 3) $x ∈ X$ is sequentially compact; that is, every sequence in $X$ has a convergent subsequence whose limit is in $X$ (this is also equivalent to compactness for first-countable uniform spaces).
 * 4) $a ∈ A$ is limit point compact (also called weakly countably compact); that is, every infinite subset of $X$ has at least one limit point in $X$.
 * 5) $U(x)$ is countably compact; that is, every countable open cover of $X$ has a finite subcover.
 * 6) $V(a)$ is an image of a continuous function from the Cantor set.
 * 7) Every decreasing nested sequence of nonempty closed subsets $U(x)$ in $V(a)$ has a nonempty intersection.
 * 8) Every increasing nested sequence of proper open subsets ${V(a) : a ∈ A}$ in $U(x)$ fails to cover $X$.

A compact metric space $(X, d)$ also satisfies the following properties:
 * 1) Lebesgue's number lemma: For every open cover of $X$, there exists a number δ > 0 such that every subset of $X$ of diameter < $X$ is contained in some member of the cover.
 * 2) $(X, d)$ is second-countable, separable and Lindelöf – these three conditions are equivalent for metric spaces. The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact.
 * 3) $X$ is closed and bounded (as a subset of any metric space whose restricted metric is $Y$). The converse may fail for a non-Euclidean space; e.g. the real line equipped with the discrete metric is closed and bounded but not compact, as the collection of all singletons of the space is an open cover which admits no finite subcover. It is complete but not totally bounded.

Ordered spaces
For an ordered space $(X, d)$ (i.e. a totally ordered set equipped with the order topology), the following are equivalent:
 * 1) $(X, d)$ is compact.
 * 2) Every subset of $X$ has a supremum (i.e. a least upper bound) in $A$.
 * 3) Every subset of $A$ has an infimum (i.e. a greatest lower bound) in $n$.
 * 4) Every nonempty closed subset of $X$ has a maximum and a minimum element.

An ordered space satisfying (any one of) these conditions is called a complete lattice.

In addition, the following are equivalent for all ordered spaces $(X, d)$, and (assuming countable choice) are true whenever $(X, d)$ is compact. (The converse in general fails if $(X, d)$ is not also metrizable.):
 * 1) Every sequence in $S_{1} ⊇ S_{2} ⊇ ...$ has a subsequence that converges in $(X, d)$.
 * 2) Every monotone increasing sequence in $X$ converges to a unique limit in $X$.
 * 3) Every monotone decreasing sequence in $X$ converges to a unique limit in $X$.
 * 4) Every decreasing nested sequence of nonempty closed subsets $X$1 ⊇ $X$2 ⊇ ... in $S_{1} ⊆ S_{2} ⊆ ...$ has a nonempty intersection.
 * 5) Every increasing nested sequence of proper open subsets $X$1 ⊆ $δ$2 ⊆ ... in $(X, d)$ fails to cover $X$.

Characterization by continuous functions
Let $d$ be a topological space and $(X, d)$ the ring of real continuous functions on $X$. For each $(X, d)$, the evaluation map $$\operatorname{ev}_p\colon C(X)\to \mathbb{R}$$ given by $(X, <)$ is a ring homomorphism. The kernel of $(X, <)$ is a maximal ideal, since the residue field $(X, <)$ is the field of real numbers, by the first isomorphism theorem. A topological space $X$ is pseudocompact if and only if every maximal ideal in $(X, <)$ has residue field the real numbers. For completely regular spaces, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism. There are pseudocompact spaces that are not compact, though.

In general, for non-pseudocompact spaces there are always maximal ideals $X$ in $(X, <)$ such that the residue field $(X, <)$ is a (non-Archimedean) hyperreal field. The framework of non-standard analysis allows for the following alternative characterization of compactness: a topological space $X$ is compact if and only if every point $X$ of the natural extension $(X, <)$ is infinitely close to a point $(X, <)$ of $X$ (more precisely, $X$ is contained in the monad of $(X, <)$).

Hyperreal definition
A space $X$ is compact if its hyperreal extension $C(X)$ (constructed, for example, by the ultrapower construction) has the property that every point of $p ∈ X$ is infinitely close to some point of $ev_{p}(f) = f(p)$. For example, an open real interval $ev_{p}$ is not compact because its hyperreal extension $C(X)/ker ev_{p}$ contains infinitesimals, which are infinitely close to 0, which is not a point of $X$.

Sufficient conditions

 * A closed subset of a compact space is compact.
 * A finite union of compact sets is compact.
 * A continuous image of a compact space is compact.
 * The intersection of any non-empty collection of compact subsets of a Hausdorff space is compact (and closed);
 * If $S$ is not Hausdorff then the intersection of two compact subsets may fail to be compact (see footnote for example).
 * The product of any collection of compact spaces is compact. (This is Tychonoff's theorem, which is equivalent to the axiom of choice.)
 * In a metrizable space, a subset is compact if and only if it is sequentially compact (assuming countable choice)
 * A finite set endowed with any topology is compact.

Properties of compact spaces

 * A compact subset of a Hausdorff space $S$ is closed.
 * If $S$ is not Hausdorff then a compact subset of $S$ may fail to be a closed subset of $X$ (see footnote for example).
 * If $X$ is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).
 * In any topological vector space (TVS), a compact subset is complete. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are not closed.
 * If $X$ and $X$ are disjoint compact subsets of a Hausdorff space $m$, then there exist disjoint open sets $X$ and $x$ in $X$ such that $C(X)$ and $C(X)$.
 * A continuous bijection from a compact space into a Hausdorff space is a homeomorphism.
 * A compact Hausdorff space is normal and regular.
 * If a space $x$ is compact and Hausdorff, then no finer topology on $X$ is compact and no coarser topology on $X$ is Hausdorff.
 * If a subset of a metric space $C(X)/m$ is compact then it is $X$-bounded.

Functions and compact spaces
Since a continuous image of a compact space is compact, the extreme value theorem holds for such spaces: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum. (Slightly more generally, this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of a compact space under a proper map is compact.

Compactifications
Every topological space $a$ is an open dense subspace of a compact space having at most one point more than $X$, by the Alexandroff one-point compactification. By the same construction, every locally compact Hausdorff space $U$ is an open dense subspace of a compact Hausdorff space having at most one point more than $b$.

Ordered compact spaces
A nonempty compact subset of the real numbers has a greatest element and a least element.

Let $X$ be a simply ordered set endowed with the order topology. Then $V$ is compact if and only if $U$ is a complete lattice (i.e. all subsets have suprema and infima).

Examples

 * Any finite topological space, including the empty set, is compact. More generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology.
 * Any space carrying the cofinite topology is compact.
 * Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification. The one-point compactification of $$\mathbb{R}$$ is homeomorphic to the circle $
 * X$; the one-point compactification of $$\mathbb{R}^2$$ is homeomorphic to the sphere $x_{0}$. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.
 * The right order topology or left order topology on any bounded totally ordered set is compact. In particular, Sierpiński space is compact.
 * No discrete space with an infinite number of points is compact. The collection of all singletons of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact.
 * In $$\mathbb{R}$$ carrying the lower limit topology, no uncountable set is compact.
 * In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous example, the space as a whole is not locally compact but is still Lindelöf.
 * The closed unit interval $x_{0}$ is compact. This follows from the Heine–Borel theorem. The open interval $V$ is not compact: the open cover $\left( \frac{1}{n}, 1 - \frac{1}{n} \right)$ for $
 * X$ does not have a finite subcover. Similarly, the set of rational numbers in the closed interval $U$ is not compact: the sets of rational numbers in the intervals $\left[0, \frac{1}{\pi} - \frac{1}{n}\right]\text{ and }\left[\frac{1}{\pi} + \frac{1}{n}, 1\right]$  cover all the rationals in [0, 1] for $
 * X$ but this cover does not have a finite subcover. Here, the sets are open in the subspace topology even though they are not open as subsets of $$\mathbb{R}$$.
 * The set $$\mathbb{R}$$ of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover. For example, intervals $X ⊂ *X$, where $V$ takes all integer values in $X = (0, 1)$, cover $$\mathbb{R}$$ but there is no finite subcover.
 * On the other hand, the extended real number line carrying the analogous topology is compact; note that the cover described above would never reach the points at infinity and thus would not cover the extended real line. In fact, the set has the homeomorphism to [−1, 1] of mapping each infinity to its corresponding unit and every real number to its sign multiplied by the unique number in the positive part of interval that results in its absolute value when divided by one minus itself, and since homeomorphisms preserve covers, the Heine-Borel property can be inferred.
 * For every natural number $X$, the $X$-sphere is compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensional normed vector space is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.
 * On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology. (Alaoglu's theorem)
 * The Cantor set is compact. In fact, every compact metric space is a continuous image of the Cantor set.
 * Consider the set $X$ of all functions $
 * (0,1)$ from the real number line to the closed unit interval, and define a topology on $X$ so that a sequence $$\{f_n\}$$ in $X$ converges towards $X = {a, b} ∪ $\mathbb{N}$$ if and only if $$\{f_n(x)\}$$ converges towards $U = {a} ∪ $\mathbb{N}$$ for all real numbers $X$. There is only one such topology; it is called the topology of pointwise convergence or the product topology. Then $X$ is a compact topological space; this follows from the Tychonoff theorem.
 * A subset of the Banach space of real-valued continuous functions on a compact Hausdorff space is relatively compact if and only if it is equicontinuous and pointwise bounded (Arzelà–Ascoli theorem).
 * Consider the set $X$ of all functions $V = {b} ∪ $\mathbb{N}$$ satisfying the Lipschitz condition $X$ for all $X = {a, b}$. Consider on $S$ the metric induced by the uniform distance $$d(f, g) = \sup_{x \in [0, 1]} |f(x) - g(x)|.$$ Then by the Arzelà–Ascoli theorem the space $X$ is compact.
 * The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers $$\mathbb{C}$$. Conversely, any compact subset of $$\mathbb{C}$$ arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space $\ell^2$ may have any compact nonempty subset of $$\mathbb{C}$$ as spectrum.
 * The space of Borel probability measures on a compact Hausdorff space is compact for the vague topology, by the Alaoglu theorem.
 * A collection of probability measures on the Borel sets of Euclidean space is called tight if, for any positive epsilon, there exists a compact subset containing all but at most epsilon of the mass of each of the measures. Helly's theorem then asserts that a collection of probability measures is relatively compact for the vague topology if and only if it is tight.

Algebraic examples

 * Topological groups such as an orthogonal group are compact, while groups such as a general linear group are not.
 * Since the $X$-adic integers are homeomorphic to the Cantor set, they form a compact set.
 * Any global field K is a discrete additive subgroup of its adele ring, and the quotient space is compact. This was used in John Tate's thesis to allow harmonic analysis to be used in number theory.
 * The spectrum of any commutative ring with the Zariski topology (that is, the set of all prime ideals) is compact, but never Hausdorff (except in trivial cases). In algebraic geometry, such topological spaces are examples of quasi-compact schemes, "quasi" referring to the non-Hausdorff nature of the topology.
 * The spectrum of a Boolean algebra is compact, a fact which is part of the Stone representation theorem. Stone spaces, compact totally disconnected Hausdorff spaces, form the abstract framework in which these spectra are studied. Such spaces are also useful in the study of profinite groups.
 * The structure space of a commutative unital Banach algebra is a compact Hausdorff space.
 * The Hilbert cube is compact, again a consequence of Tychonoff's theorem.
 * A profinite group (e.g. Galois group) is compact.

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title: Compact space


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Clodius

Clodius is an alternate form of the Roman nomen Claudius, a patrician gens that was traditionally regarded as Sabine in origin. The alternation of o and au is characteristic of the Sabine dialect. The feminine form is Clodia.

Publius Clodius Pulcher
During the Late Republic, the spelling Clodius is most prominently associated with Publius Clodius Pulcher, a popularis politician who gave up his patrician status through an order in order to qualify for the office of tribune of the plebs. Clodius positioned himself as a champion of the urban plebs, supporting free grain for the poor and the right of association in guilds (collegia); because of this individual's ideology, Clodius has often been taken as a more "plebeian" spelling and a gesture of political solidarity. Clodius's two elder brothers, the Appius Claudius Pulcher who was consul in 54 BC and the C. Claudius Pulcher who was praetor in 56 BC, conducted more conventional political careers and are referred to in contemporary sources with the traditional spelling.

The view that Clodius represents a plebeian or politicized form has been questioned by Clodius's chief modern-era biographer. In The Patrician Tribune, W. Jeffrey Tatum points out that the spelling is also associated with Clodius's sisters and that "the political explanation … is almost certainly wrong." A plebeian branch of the gens, the Claudii Marcelli, retained the supposedly patrician spelling, while there is some inscriptional evidence that the -o- form may also have been used on occasion by close male relatives of the "patrician tribune" Clodius. Tatum argues that the use of -o- by the "chic" Clodia was a fashionable affectation, and that Clodius, whose perhaps inordinately loving relationship with his sister was the subject of much gossip and insinuation, was imitating his stylish sibling. The linguistic variation of o for au was characteristic of the Umbrian language, of which Sabine was a branch. Forms using o were considered archaic or rustic in the 50s BC, and the use of Clodius would have been either a whimsical gesture of pastoral fantasy, or a trendy assertion of antiquarian authenticity.

Other Clodii of the Republic
In addition to Clodius, Clodii from the Republican era include:

 * Gnaeus Cornelius Lentulus Clodianus, presumably a "Clodius" before his adoption
 * Clodius Aesopus, a tragic actor in the 50s BC who may have been a freedman of one of the Clodii Pulchri.
 * Claudia, daughter of Clodius Pulcher and Fulvia, the first wife of emperor Augustus.
 * Clodia, sister of Publius Clodius Pulcher, sometimes identified in Catullus' poems as "Lesbia".

Women of the Claudii Marcelli branch were often called "Clodia" in the late Republic.

Imperial era
People using the name Clodius during the period of the Roman Empire include:
 * Gaius Clodius Licinus, consul suffectus in AD 4.
 * Gaius Clodius Vestalis, possible builder of the Via Clodia
 * Publius Clodius Thrasea Paetus, senator and philosopher during the reign of Nero
 * Lucius Clodius Macer, a legatus who revolted against Nero
 * Publius Clodius Quirinalis, from Arelate in Gaul, teacher of rhetoric in time of Nero
 * Decimus Clodius Septimius Albinus, commonly known as Clodius Albinus, rival emperor 196-197
 * Marcus Clodius Pupienus Maximus, known as Pupienus, co-emperor 238
 * Titus Clodius Pupienus Pulcher Maximus, son of emperor Pupienus and suffect consul c. 235

Clodii Celsini
The Clodii Celsini continued to practice the traditional religions of antiquity in the face of Christian hegemony through at least the 4th century, when Clodius Celsinus Adelphius (see below) converted. Members of this branch include:
 * Quintus Fabius Clodius Agrippianus Celsinus, proconsul of Caria in 249 and the son of Clodius Celsinus (b. ca. 185); see for other members of the family.
 * Clodius Celsinus Adelphius, praefectus urbi in 351.
 * Quintus Clodius Hermogenianus Olybrius, consul 379

Selected bibliography

 * Tatum, W. Jeffrey. The Patrician Tribune: P. Clodius Pulcher. Studies in the History of Greece and Rome series. University of North Carolina Press, 1999. Limited preview online. Hardcover ISBN 0-8078-2480-1.

title: Clodius


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Cicero

Marcus Tullius Cicero (3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, writer and Academic skeptic, who tried to uphold optimate principles during the political crises that led to the establishment of the Roman Empire. His extensive writings include treatises on rhetoric, philosophy and politics. He is considered one of Rome's greatest orators and prose stylists and the innovator of what became known as "Ciceronian rhetoric". Cicero was educated in Rome and in Greece. He came from a wealthy municipal family of the Roman equestrian order, and served as consul in 63 BC.

He greatly influenced both ancient and modern reception of the Latin language. A substantial percentage of his work has survived, and he was admired by both ancient and modern authors alike. Cicero adapted the arguments of the chief schools of Hellenistic philosophy in Latin and created a large amount of Latin philosophical vocabulary via lexical innovation (e.g. neologisms such as evidentia, generator, humanitas, infinitio, qualitas, quantitas), almost 150 of which were the result of translating Greek philosophical terms.

Though he was an accomplished orator and successful lawyer, Cicero believed his political career was his most important achievement. It was during his consulship that the Catiline conspiracy attempted to overthrow the government through an attack on the city by outside forces, and Cicero (by his own account) suppressed the revolt by summarily and controversially executing five conspirators without trial, an act which would later lead to his exile. During the chaotic middle period of the first century BC, marked by civil wars and the dictatorship of Julius Caesar, Cicero was a supporter of the Optimates faction. Following Caesar's death, Cicero became an enemy of Mark Antony in the ensuing power struggle, attacking him in a series of speeches. He was proscribed as an enemy of the state by the Second Triumvirate and consequently executed by soldiers operating on their behalf in 43 BC, having been intercepted during an attempted flight from the Italian peninsula. His severed hands and head (taken by order of Antony and displayed representing the repercussions of his anti-Antonian actions as a writer and as an orator, respectively) were then displayed on the Rostra.

Petrarch's rediscovery of Cicero's letters is often credited for initiating the 14th-century Renaissance in public affairs, humanism, and classical Roman culture. According to Polish historian Tadeusz Zieliński, "the Renaissance was above all things a revival of Cicero, and only after him and through him of the rest of Classical antiquity." The peak of Cicero's authority and prestige came during the 18th-century Enlightenment, and his impact on leading Enlightenment thinkers and political theorists such as John Locke, David Hume, Montesquieu, and Edmund Burke was substantial. His works rank among the most influential in global culture, and today still constitute one of the most important bodies of primary material for the writing and revision of Roman history, especially the last days of the Roman Republic.

Early life

Marcus Tullius Cicero was born on 3 January 106 BC in Arpinum, a hill town 100 km southeast of Rome. He belonged to the tribus Cornelia. His father was a wealthy member of the equestrian order and possessed good connections in Rome. However, being a semi-invalid, he could not enter public life and studied extensively to compensate. Little is known about Cicero's mother Helvia, but Cicero's brother Quintus wrote in a letter that she was a thrifty housewife.

Cicero's cognomen, a hereditary nickname, comes from the Latin for chickpea, cicer. Plutarch explains that the name was originally given to one of Cicero's ancestors who had a cleft in the tip of his nose resembling a chickpea. The famous family names of Fabius, Lentulus, and Piso come from the Latin names of beans, lentils, and peas, respectively. Plutarch writes that Cicero was urged to change this deprecatory name when he entered politics, but refused, saying that he would make Cicero more glorious than Scaurus ("Swollen-ankled") and Catulus ("Puppy").

At the age of 15, in 90 BC, Cicero started serving under Pompey Strabo and later Sulla in the Social war between Rome and its Italian allies. When in Rome during the turbulent plebeian tribunate of Publius Sulpicius Rufus in 88 BC which saw a short bout of fighting between the Sulpicius and Sulla, who had been elected consul for that year, Cicero found himself greatly impressed by Sulpicius' oratory even if he disagreed with his politics. He continued his studies at Rome, writing a pamphlet titled On Invention relating to rhetorical argumentation and studying philosophy with Greek academics who had fled the ongoing First Mithridatic War.

Education
During this period in Roman history, Greek language and cultural studies were highly valued by the elite classes. Cicero was therefore educated in the teachings of the ancient Greek philosophers, poets and historians; as he obtained much of his understanding of the theory and practice of rhetoric from the Greek poet Archias. Cicero used his knowledge of Greek to translate many of the theoretical concepts of Greek philosophy into Latin, thus translating Greek philosophical works for a larger audience. It was precisely his broad education that tied him to the traditional Roman elite.

Cicero's interest in philosophy figured heavily in his later career and led to him providing a comprehensive account of Greek philosophy for a Roman audience, including creating a philosophical vocabulary in Latin. In 87 BC, Philo of Larissa, the head of the Platonic Academy that had been founded by Plato in Athens about 300 years earlier, arrived in Rome. Cicero, "inspired by an extraordinary zeal for philosophy", sat enthusiastically at his feet and absorbed Carneades' Academic Skeptic philosophy.

According to Plutarch, Cicero was an extremely talented student, whose learning attracted attention from all over Rome, affording him the opportunity to study Roman law under Quintus Mucius Scaevola. Cicero's fellow students were Gaius Marius Minor, Servius Sulpicius Rufus (who became a famous lawyer, one of the few whom Cicero considered superior to himself in legal matters), and Titus Pomponius. The latter two became Cicero's friends for life, and Pomponius (who later received the nickname "Atticus", and whose sister married Cicero's brother) would become, in Cicero's own words, "as a second brother", with both maintaining a lifelong correspondence.

In 79 BC, Cicero left for Greece, Asia Minor and Rhodes. This was perhaps to avoid the potential wrath of Sulla, as Plutarch claims, though Cicero himself says it was to hone his skills and improve his physical fitness. In Athens he studied philosophy with Antiochus of Ascalon, the 'Old Academic' and initiator of Middle Platonism. In Asia Minor, he met the leading orators of the region and continued to study with them. Cicero then journeyed to Rhodes to meet his former teacher, Apollonius Molon, who had taught him in Rome. Molon helped Cicero hone the excesses in his style, as well as train his body and lungs for the demands of public speaking. Charting a middle path between the competing Attic and Asiatic styles, Cicero would ultimately become considered second only to Demosthenes among history's orators.

Early legal activity
While Cicero had feared that the law courts would be closed forever, they were reopened in the aftermath of Sulla's civil war and the purging of Sulla's political opponents in the proscriptions. Many of the orators whom Cicero had admired in his youth were now dead from age or political violence. His first major appearance in the courts was in 81 BC at the age of 26 when he delivered Pro Quinctio, a speech defending certain commercial transactions which Cicero had recorded and disseminated.

His more famous speech defending Sextus Roscius of Ameria – Pro Roscio Amerino – on charges of parricide in 80 BC was his first appearance in criminal court. In this high-profile case, Cicero accused a freedman of the dictator Sulla, Chrysogonus, of fabricating Roscius' father's proscription to obtain Roscius' family's property. Successful in his defence, Cicero tactfully avoided incriminating Sulla of any wrongdoing and developed a positive oratorical reputation for himself.

While Plutarch claims that Cicero left Rome shortly thereafter out of fear of Sulla's response, according to Kathryn Tempest, "most scholars now dismiss this suggestion" because Cicero left Rome after Sulla resigned his dictatorship. Cicero, for his part, later claimed that he left Rome, headed for Asia, to develop his physique and develop his oratory. After marrying his wife, Terentia, in 80 BC, he eventually left for Asia Minor with his brother Quintus, his friend Titus Atticus, and others on a long trip spanning most of 79 through 77 BC. Returning to Rome in 77 BC, Cicero again busied himself with legal defence.

Early political career
In 76 BC, at the quaestorian elections, Cicero was elected at the minimum age required – 30 years – in the first returns from the comitia tributa, to the post of quaestor. Ex officio, he also became a member of the Senate. In the quaestorian lot, he was assigned to Sicily for 75 BC. The post, which was largely one related to financial administration in support of the state or provincial governors, proved for Cicero an important place where he could gain clients in the provinces. His time in Sicily saw him balance his duties – largely in terms of sending more grain back to Rome – with his support for the provincials, Roman businessmen in the area, and local potentates. Adeptly balancing those responsibilities, he won their gratitude. He was also appreciated by local Syracusans for the rediscovery of the lost tomb of Archimedes, which he personally financed.

Promising to lend the Sicilians his oratorical voice, he was called on a few years after his quaestorship to prosecute the Roman province's governor Gaius Verres, for abuse of power and corruption. In 70 BC, at the age of 36, Cicero launched his first high-profile prosecution against Verres, an emblem of the corrupt Sullan supporters who had risen in the chaos of the civil war.

The prosecution of Gaius Verres was a great forensic success for Cicero. While Verres hired the prominent lawyer, Quintus Hortensius, after a lengthy period in Sicily collecting testimonials and evidence and persuading witnesses to come forward, Cicero returned to Rome and won the case in a series of dramatic court battles. His unique style of oratory set him apart from the flamboyant Hortensius. On the conclusion of this case, Cicero came to be considered the greatest orator in Rome. The view that Cicero may have taken the case for reasons of his own is viable. Hortensius was, at this point, known as the best lawyer in Rome; to beat him would guarantee much success and the prestige that Cicero needed to start his career. Cicero's oratorical ability is shown in his character assassination of Verres and various other techniques of persuasion used on the jury. One such example is found in the speech In Verrem, where he states "with you on this bench, gentlemen, with Marcus Acilius Glabrio as your president, I do not understand what Verres can hope to achieve". Oratory was considered a great art in ancient Rome and an important tool for disseminating knowledge and promoting oneself in elections, in part because there were no regular newspapers or mass media. Cicero was neither a patrician nor a plebeian noble; his rise to political office despite his relatively humble origins has traditionally been attributed to his brilliance as an orator.

Cicero grew up in a time of civil unrest and war. Sulla's victory in the first of a series of civil wars led to a new constitutional framework that undermined libertas (liberty), the fundamental value of the Roman Republic. Nonetheless, Sulla's reforms strengthened the position of the equestrian class, contributing to that class's growing political power. Cicero was both an Italian eques and a novus homo, but more importantly he was a Roman constitutionalist. His social class and loyalty to the Republic ensured that he would "command the support and confidence of the people as well as the Italian middle classes". The optimates faction never truly accepted Cicero, and this undermined his efforts to reform the Republic while preserving the constitution. Nevertheless, he successfully ascended the cursus honorum, holding each magistracy at or near the youngest possible age: quaestor in 75 BC (age 30), aedile in 69 BC (age 36), and praetor in 66 BC (age 39), when he served as president of the "Reclamation" (or extortion) Court. He was then elected consul at age 42.

Consulship
Cicero, seizing the opportunity offered by optimate fear of reform, was elected consul for the year 63 BC; he was elected with the support of every unit of the centuriate assembly, rival members of the post-Sullan establishment, and the leaders of municipalities throughout post-Social War Italy. His co-consul for the year, Gaius Antonius Hybrida, played a minor role.

He began his consular year by opposing a land bill proposed by a plebeian tribune which would have appointed commissioners with semi-permanent authority over land reform. Cicero was also active in the courts, defending Gaius Rabirius from accusations of participating in the unlawful killing of plebeian tribune Lucius Appuleius Saturninus in 100 BC. The prosecution occurred before the comita centuriata and threatened to reopen conflict between the Marian and Sullan factions at Rome. Cicero defended the use of force as being authorised by a senatus consultum ultimum, which would prove similar to his own use of force under such conditions.

Catilinarian Conspiracy
Most famously – in part because of his own publicity – he thwarted a conspiracy led by Lucius Sergius Catilina to overthrow the Roman Republic with the help of foreign armed forces. Cicero procured a senatus consultum ultimum (a recommendation from the senate attempting to legitimise the use of force) and drove Catiline from the city with four vehement speeches (the Catilinarian orations), which remain outstanding examples of his rhetorical style. The Orations listed Catiline and his followers' debaucheries, and denounced Catiline's senatorial sympathizers as roguish and dissolute debtors clinging to Catiline as a final and desperate hope. Cicero demanded that Catiline and his followers leave the city. At the conclusion of Cicero's first speech (which was made in the Temple of Jupiter Stator), Catiline hurriedly left the Senate. In his following speeches, Cicero did not directly address Catiline. He delivered the second and third orations before the people, and the last one again before the Senate. By these speeches, Cicero wanted to prepare the Senate for the worst possible case; he also delivered more evidence, against Catiline.

Catiline fled and left behind his followers to start the revolution from within while he himself assaulted the city with an army of "moral and financial bankrupts, or of honest fanatics and adventurers". It is alleged that Catiline had attempted to involve the Allobroges, a tribe of Transalpine Gaul, in their plot, but Cicero, working with the Gauls, was able to seize letters that incriminated the five conspirators and forced them to confess in front of the Senate. The senate then deliberated upon the conspirators' punishment. As it was the dominant advisory body to the various legislative assemblies rather than a judicial body, there were limits to its power; however, martial law was in effect, and it was feared that simple house arrest or exile – the standard options – would not remove the threat to the state. At first Decimus Junius Silanus spoke for the "extreme penalty"; but during the debate many were swayed by Julius Caesar, who decried the precedent it would set and argued in favor of life imprisonment in various Italian towns. Cato the Younger then rose in defense of the death penalty and the Senate finally agreed on the matter, and came down in support of the death penalty. Cicero had the conspirators taken to the Tullianum, the notorious Roman prison, where they were strangled. Cicero himself accompanied the former consul Publius Cornelius Lentulus Sura, one of the conspirators, to the Tullianum.

Cicero received the honorific "pater patriae" for his efforts to suppress the conspiracy, but lived thereafter in fear of trial or exile for having put Roman citizens to death without trial. While the senatus consultum ultimum gave some legitimacy to the use of force against the conspirators, Cicero also argued that Catiline's conspiracy, by virtue of its treason, made the conspirators enemies of the state and forfeited the protections intrinsically possessed by Roman citizens. The consuls moved decisively. Antonius Hybrida was dispatched to defeat Catiline in battle that year, preventing Crassus or Pompey from exploiting the situation for their own political aims.

After the suppression of the conspiracy, Cicero was proud of his accomplishment. Some of his political enemies argued that though the act gained Cicero popularity, he exaggerated the extent of his success. He overestimated his popularity again several years later after being exiled from Italy and then allowed back from exile. At this time, he claimed that the republic would be restored along with him.

Shortly after completing his consulship, in late 62 BC, Cicero arranged the purchase of a large townhouse on the Palatine Hill previously owned by Rome's richest citizen, Marcus Licinius Crassus. To finance the purchase, Cicero borrowed some two million sesterces from Publius Cornelius Sulla, whom he had previously defended from court. Cicero boasted his house was "in conspectu prope totius urbis" ("in sight of nearly the whole city"), only a short walk from the Roman Forum.

Exile and return
In 60 BC, Julius Caesar invited Cicero to be the fourth member of his existing partnership with Pompey and Marcus Licinius Crassus, an assembly that would eventually be called the First Triumvirate. Cicero refused the invitation because he suspected it would undermine the Republic, and because he was strongly opposed to anything unconstitutional that limited the powers of the consuls and replaced them with non-elected officials.

During Caesar's consulship of 59 BC, the triumvirate had achieved many of their goals of land reform, publicani debt forgiveness, ratification of Pompeian conquests, etc. With Caesar leaving for his provinces, they wished to maintain their hold on politics. They engineered the adoption of patrician Publius Clodius Pulcher into a plebeian family and had him elected as one of the ten tribunes of the plebs for 58 BC. Clodius used the triumvirate's backing to push through legislation that benefited them. He introduced several laws (the leges Clodiae) that made him popular with the people, strengthening his power base, then he turned on Cicero. Clodius passed a law which made it illegal to offer "fire and water" (i.e shelter or food) to anyone who executed a Roman citizen without a trial. Cicero, having executed members of the Catiline conspiracy four years previously without formal trial, was clearly the intended target. Furthermore, many believed that Clodius acted in concert with the triumvirate who feared that Cicero would seek to abolish many of Caesar's accomplishments while consul the year before. Cicero argued that the senatus consultum ultimum indemnified him from punishment, and he attempted to gain the support of the senators and consuls, especially of Pompey.

Cicero grew out his hair, dressed in mourning and toured the streets. Clodius' gangs dogged him, hurling abuse, stones and even excrement. Hortensius, trying to rally to his old rival's support, was almost lynched. The Senate and the consuls were cowed. Caesar, who was still encamped near Rome, was apologetic but said he could do nothing when Cicero brought himself to grovel in the proconsul's tent. Everyone seemed to have abandoned Cicero.

After Clodius passed a law to deny to Cicero fire and water (i.e. shelter) within four hundred miles of Rome, Cicero went into exile. He arrived at Thessalonica, on 23 May 58 BC. In his absence, Clodius, who lived next door to Cicero on the Palatine, arranged for Cicero's house to be confiscated by the state, and was even able to purchase a part of the property in order to extend his own house. After demolishing Cicero's house, Clodius had the land consecrated and symbolically erected a temple of Liberty (aedes Libertatis) on the vacant land.

Cicero's exile caused him to fall into depression. He wrote to Atticus: "Your pleas have prevented me from committing suicide. But what is there to live for? Don't blame me for complaining. My afflictions surpass any you ever heard of earlier". After the intervention of recently elected tribune Titus Annius Milo, acting on the behalf of Pompey who wanted Cicero as a client, the Senate voted in favor of recalling Cicero from exile. Clodius cast the single vote against the decree. Cicero returned to Italy on 5 August 57 BC, landing at Brundisium. He was greeted by a cheering crowd, and, to his delight, his beloved daughter Tullia. In his Oratio De Domo Sua Ad Pontifices, Cicero convinced the College of Pontiffs to rule that the consecration of his land was invalid, thereby allowing him to regain his property and rebuild his house on the Palatine.

Cicero tried to re-enter politics as an independent operator, but his attempts to attack portions of Caesar's legislation were unsuccessful and encouraged Caesar to re-solidify his political alliance with Pompey and Crassus. The conference at Luca in 56 BC left the three-man alliance in domination of the republic's politics; this forced Cicero to recant and support the triumvirate out of fear from being entirely excluded from public life. After the conference Cicero lavishly praised Caesar's achievements, got the Senate to vote a thanksgiving for Caesar's victories and grant money to pay his troops. He also delivered a speech 'On the consular provinces' (de provinciis consularibus) which checked an attempt by Caesar's enemies to strip him of his provinces in Gaul. After this, a cowed Cicero concentrated on his literary works. It is uncertain whether he was directly involved in politics for the following few years.

Governorship of Cilicia
In 51 BC he reluctantly accepted a promagistracy (as proconsul) in Cilicia for the year; there were few other former consuls eligible as a result of a legislative requirement enacted by Pompey in 52 BC specifying an interval of five years between a consulship or praetorship and a provincial command. He served as proconsul of Cilicia from May 51 BC, arriving in the provinces three months later around August.

In 53 BC Marcus Licinius Crassus had been defeated by the Parthians at the Battle of Carrhae. This opened the Roman East for a Parthian invasion, causing unrest in Syria and Cilicia. Cicero restored calm by his mild system of government. He discovered that a great amount of public property had been embezzled by corrupt previous governors and members of their staff, and did his utmost to restore it. Thus he greatly improved the condition of the cities. He retained the civil rights of, and exempted from penalties, the men who gave the property back. Besides this, he was extremely frugal in his outlays for staff and private expenses during his governorship, and this made him highly popular among the natives.

Besides his activity in ameliorating the hard pecuniary situation of the province, Cicero was also creditably active in the military sphere. Early in his governorship he received information that prince Pacorus, son of Orodes II the king of the Parthians, had crossed the Euphrates, and was ravaging the Syrian countryside and had even besieged Cassius (the interim Roman commander in Syria) in Antioch. Cicero eventually marched with two understrength legions and a large contingent of auxiliary cavalry to Cassius's relief. Pacorus and his army had already given up on besieging Antioch and were heading south through Syria, ravaging the countryside again. Cassius and his legions followed them, harrying them wherever they went, eventually ambushing and defeating them near Antigonea.

Another large troop of Parthian horsemen was defeated by Cicero's cavalry who happened to run into them while scouting ahead of the main army. Cicero next defeated some robbers who were based on Mount Amanus and was hailed as imperator by his troops. Afterwards he led his army against the independent Cilician mountain tribes, besieging their fortress of Pindenissum. It took him 47 days to reduce the place, which fell in December. On 30 July 50 BC Cicero left the province to his brother Quintus, who had accompanied him on his governorship as his legate. On his way back to Rome he stopped in Rhodes and then went to Athens, where he caught up with his old friend Titus Pomponius Atticus and met men of great learning.

Julius Caesar's civil war
Cicero arrived in Rome on 4 January 49 BC. He stayed outside the pomerium, to retain his promagisterial powers: either in expectation of a triumph or to retain his independent command authority in the coming civil war. The struggle between Pompey and Julius Caesar grew more intense in 50 BC. Cicero favored Pompey, seeing him as a defender of the senate and Republican tradition, but at that time avoided openly alienating Caesar. When Caesar invaded Italy in 49 BC, Cicero fled Rome. Caesar, seeking an endorsement by a senior senator, courted Cicero's favor, but even so Cicero slipped out of Italy and traveled to Dyrrhachium where Pompey's staff was situated. Cicero traveled with the Pompeian forces to Pharsalus in Macedonia in 48 BC, though he was quickly losing faith in the competence and righteousness of the Pompeian side. Eventually, he provoked the hostility of his fellow senator Cato, who told him that he would have been of more use to the cause of the optimates if he had stayed in Rome. After Caesar's victory at the Battle of Pharsalus on 9 August, Cicero refused to take command of the Pompeian forces and continue the war. He returned to Rome, still as a promagistrate with his lictors, in 47 BC, and dismissed them upon his crossing the pomerium and renouncing his command.

In a letter to Varro on c. 20 April 46 BC, Cicero outlined his strategy under Caesar's dictatorship. Cicero, however, was taken by surprise when the Liberatores assassinated Caesar on the ides of March, 44 BC. Cicero was not included in the conspiracy, even though the conspirators were sure of his sympathy. Marcus Junius Brutus called out Cicero's name, asking him to restore the republic when he lifted his bloodstained dagger after the assassination. A letter Cicero wrote in February 43 BC to Trebonius, one of the conspirators, began, "How I could wish that you had invited me to that most glorious banquet on the Ides of March!" Cicero became a popular leader during the period of instability following the assassination. He had no respect for Mark Antony, who was scheming to take revenge upon Caesar's murderers. In exchange for amnesty for the assassins, he arranged for the Senate to agree not to declare Caesar to have been a tyrant, which allowed the Caesarians to have lawful support and kept Caesar's reforms and policies intact.

Opposition to Mark Antony and death
In April 43 BC, "diehard republicans" may have revived the ancient position of princeps senatus (leader of the senate) for Cicero. This position had been very prestigious until the constitutional reforms of Sulla in 82–80 BC, which removed most of its importance.

On the other side, Antony was consul and leader of the Caesarian faction, and unofficial executor of Caesar's public will. Relations between the two were never friendly and worsened after Cicero claimed that Antony was taking liberties in interpreting Caesar's wishes and intentions. Octavian was Caesar's adopted son and heir. After he returned to Italy, Cicero began to play him against Antony. He praised Octavian, declaring he would not make the same mistakes as his father. He attacked Antony in a series of speeches he called the Philippics, named after Demosthenes's denunciations of Philip II of Macedon. At the time, Cicero's popularity as a public figure was unrivalled.

Cicero supported Decimus Junius Brutus Albinus as governor of Cisalpine Gaul (Gallia Cisalpina) and urged the Senate to name Antony an enemy of the state. The speech of Lucius Piso, Caesar's father-in-law, delayed proceedings against Antony. Antony was later declared an enemy of the state when he refused to lift the siege of Mutina, which was in the hands of Decimus Brutus. Cicero's plan to drive out Antony failed. Antony and Octavian reconciled and allied with Lepidus to form the Second Triumvirate after the successive battles of Forum Gallorum and Mutina. The alliance came into official existence with the lex Titia, passed on 27 November 43 BC, which gave each triumvir a consular imperium for five years. The Triumvirate immediately began a proscription of their enemies, modeled after that of Sulla in 82 BC. Cicero and all of his contacts and supporters were numbered among the enemies of the state, even though Octavian argued for two days against Cicero being added to the list. Cicero was one of the most viciously and doggedly hunted among the proscribed. He was viewed with sympathy by a large segment of the public and many people refused to report that they had seen him. He was caught on 7 December 43 BC leaving his villa in Formiae in a litter heading to the seaside, where he hoped to embark on a ship destined for Macedonia. When his killers – Herennius (a Centurion) and Popilius (a Tribune) – arrived, Cicero's own slaves said they had not seen him, but he was given away by Philologus, a freedman of his brother Quintus Cicero.

As reported by Seneca the Elder, according to the historian Aufidius Bassus, Cicero's last words are said to have been: "Ego vero consisto. Accede, veterane, et, si hoc saltim potes recte facere, incide cervicem. I go no further: approach, veteran soldier, and, if you can at least do so much properly, sever this neck." He bowed to his captors, leaning his head out of the litter in a gladiatorial gesture to ease the task. By baring his neck and throat to the soldiers, he was indicating that he would not resist. According to Plutarch, Herennius first slew him, then cut off his head. On Antony's instructions his hands, which had penned the Philippics against Antony, were cut off as well; these were nailed along with his head on the Rostra in the Forum Romanum according to the tradition of Marius and Sulla, both of whom had displayed the heads of their enemies in the Forum. Cicero was the only victim of the proscriptions who was displayed in that manner. According to Cassius Dio, in a story often mistakenly attributed to Plutarch, Antony's wife Fulvia took Cicero's head, pulled out his tongue, and jabbed it repeatedly with her hairpin in final revenge against Cicero's power of speech.

Cicero's son, Marcus Tullius Cicero Minor, during his year as a consul in 30 BC, avenged his father's death, to a certain extent, when he announced to the Senate Mark Antony's naval defeat at Actium in 31 BC by Octavian.

Octavian is reported to have praised Cicero as a patriot and a scholar of meaning in later times, within the circle of his family. However, it was Octavian's acquiescence that had allowed Cicero to be killed, as Cicero was condemned by the new triumvirate.

Cicero's career as a statesman was marked by inconsistencies and a tendency to shift his position in response to changes in the political climate. His indecision may be attributed to his sensitive and impressionable personality; he was prone to overreaction in the face of political and private change.

"Would that he had been able to endure prosperity with greater self-control, and adversity with more fortitude!" wrote C. Asinius Pollio, a contemporary Roman statesman and historian.

Personal life and family
Cicero married Terentia probably at the age of 27, in 79 BC. According to the upper-class mores of the day it was a marriage of convenience but lasted harmoniously for nearly 30 years. Terentia's family was wealthy, probably the plebeian noble house of Terenti Varrones, thus meeting the needs of Cicero's political ambitions in both economic and social terms. She had a half-sister named Fabia, who as a child had become a Vestal Virgin, a great honour. Terentia was a strong-willed woman and (citing Plutarch) "took more interest in her husband's political career than she allowed him to take in household affairs".

In the 50s BC, Cicero's letters to Terentia became shorter and colder. He complained to his friends that Terentia had betrayed him but did not specify in which sense. Perhaps the marriage could not outlast the strain of the political upheaval in Rome, Cicero's involvement in it, and various other disputes between the two. The divorce appears to have taken place in 51 BC or shortly before. In 46 or 45 BC, Cicero married a young girl, Publilia, who had been his ward. It is thought that Cicero needed her money, particularly after having to repay the dowry of Terentia, who came from a wealthy family.

Although his marriage to Terentia was one of convenience, it is commonly known that Cicero held great love for his daughter Tullia. When she suddenly became ill in February 45 BC and died after having seemingly recovered from giving birth to a son in January, Cicero was stunned. "I have lost the one thing that bound me to life," he wrote to Atticus. Atticus told him to come for a visit during the first weeks of his bereavement, so that he could comfort him when his pain was at its greatest. In Atticus's large library, Cicero read everything that the Greek philosophers had written about overcoming grief, "but my sorrow defeats all consolation." Caesar and Brutus, as well as Servius Sulpicius Rufus, sent him letters of condolence.

Cicero hoped that his son Marcus would become a philosopher like him, but Marcus himself wished for a military career. He joined the army of Pompey in 49 BC, and after Pompey's defeat at Pharsalus 48 BC, he was pardoned by Caesar. Cicero sent him to Athens to study as a disciple of the peripatetic philosopher Kratippos in 48 BC, but he used this absence from "his father's vigilant eye" to "eat, drink, and be merry." After Cicero's death, he joined the army of the Liberatores but was later pardoned by Augustus. Augustus's bad conscience for having given in to Cicero's being put on the proscription list during the Second Triumvirate led him to aid considerably Marcus Minor's career. He became an augur and was nominated consul in 30 BC together with Augustus. As such, he was responsible for revoking the honors of Mark Antony, who was responsible for the proscription and could in this way take revenge. Later he was appointed proconsul of Syria and the province of Asia.

Legacy
Cicero has been traditionally considered the master of Latin prose, with Quintilian declaring that Cicero was "not the name of a man, but of eloquence itself." The English words Ciceronian (meaning "eloquent") and cicerone (meaning "local guide") derive from his name. He is credited with transforming Latin from a modest utilitarian language into a versatile literary medium capable of expressing abstract and complicated thoughts with clarity. Julius Caesar praised Cicero's achievement by saying "it is more important to have greatly extended the frontiers of the Roman spirit than the frontiers of the Roman empire". According to John William Mackail, "Cicero's unique and imperishable glory is that he created the language of the civilized world, and used that language to create a style which nineteen centuries have not replaced, and in some respects have hardly altered."

Cicero was also an energetic writer with an interest in a wide variety of subjects, in keeping with the Hellenistic philosophical and rhetorical traditions in which he was trained. The quality and ready accessibility of Ciceronian texts favored very wide distribution and inclusion in teaching curricula, as suggested by a graffito at Pompeii, admonishing: "You will like Cicero, or you will be whipped".

Cicero was greatly admired by influential Church Fathers such as Augustine of Hippo, who credited Cicero's lost Hortensius for his eventual conversion to Christianity, and St. Jerome, who had a feverish vision in which he was accused of being "follower of Cicero and not of Christ" before the judgment seat.

This influence further increased after the Early Middle Ages in Europe, where more of his writings survived than any other Latin author. Medieval philosophers were influenced by Cicero's writings on natural law and innate rights.

Petrarch's rediscovery of Cicero's letters provided the impetus for searches for ancient Greek and Latin writings scattered throughout European monasteries, and the subsequent rediscovery of classical antiquity led to the Renaissance. Subsequently, Cicero became synonymous with classical Latin to such an extent that a number of humanist scholars began to assert that no Latin word or phrase should be used unless it appeared in Cicero's works, a stance criticised by Erasmus.

His voluminous correspondence, much of it addressed to his friend Atticus, has been especially influential, introducing the art of refined letter writing to European culture. Cornelius Nepos, the first century BC biographer of Atticus, remarked that Cicero's letters contained such a wealth of detail "concerning the inclinations of leading men, the faults of the generals, and the revolutions in the government" that their reader had little need for a history of the period.

Among Cicero's admirers were Desiderius Erasmus, Martin Luther, and John Locke. Following the invention of Johannes Gutenberg's printing press, De Officiis was the second book printed in Europe, after the Gutenberg Bible. Scholars note Cicero's influence on the rebirth of religious toleration in the 17th century.

Cicero was especially popular with the Philosophes of the 18th century, including Edward Gibbon, Diderot, David Hume, Montesquieu, and Voltaire. Gibbon wrote of his first experience reading the author's collective works thus: "I tasted the beauty of the language; I breathed the spirit of freedom; and I imbibed from his precepts and examples the public and private sense of a man...after finishing the great author, a library of eloquence and reason, I formed a more extensive plan of reviewing the Latin classics..."

Voltaire called Cicero "the greatest as well as the most elegant of Roman philosophers" and even staged a play based on Cicero's role in the Catilinarian conspiracy, called Rome Sauvée, ou Catilina, to "make young people who go to the theatre acquainted with Cicero." Voltaire was spurred to pen the drama as a rebuff to his rival Claude Prosper Jolyot de Crébillon's own play Catilina, which had portrayed Cicero as a coward and villain who hypocritically married his own daughter to Catiline.

Montesquieu produced his "Discourse on Cicero" in 1717, in which he heaped praise on the author because he rescued "philosophy from the hands of scholars, and freed it from the confusion of a foreign language". Montesquieu went on to declare that Cicero was "of all the ancients, the one who had the most personal merit, and whom I would prefer to resemble."

Internationally, Cicero the republican inspired the Founding Fathers of the United States and the revolutionaries of the French Revolution. John Adams said, "As all the ages of the world have not produced a greater statesman and philosopher united than Cicero, his authority should have great weight." Thomas Jefferson names Cicero as one of a handful of major figures who contributed to a tradition "of public right" that informed his draft of the Declaration of Independence and shaped American understandings of "the common sense" basis for the right of revolution. Camille Desmoulins said of the French republicans in 1789 that they were "mostly young people who, nourished by the reading of Cicero at school, had become passionate enthusiasts for liberty".

Jim Powell starts his book on the history of liberty with the sentence: "Marcus Tullius Cicero expressed principles that became the bedrock of liberty in the modern world."

Likewise, no other ancient personality has inspired as much venomous dislike as Cicero, especially in more modern times. His commitment to the values of the Republic accommodated a hatred of the poor and persistent opposition to the advocates and mechanisms of popular representation. Friedrich Engels referred to him as "the most contemptible scoundrel in history" for upholding republican "democracy" while at the same time denouncing land and class reforms. Cicero has faced criticism for exaggerating the democratic qualities of republican Rome, and for defending the Roman oligarchy against the popular reforms of Caesar. Michael Parenti admits Cicero's abilities as an orator, but finds him a vain, pompous and hypocritical personality who, when it suited him, could show public support for popular causes that he privately despised. Parenti presents Cicero's prosecution of the Catiline conspiracy as legally flawed at least, and possibly unlawful.

Cicero also had an influence on modern astronomy. Nicolaus Copernicus, searching for ancient views on earth motion, said that he "first ... found in Cicero that Hicetas supposed the earth to move."

Notably, "Cicero" was the name attributed to size 12 font in typesetting table drawers. For ease of reference, type sizes 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, and 20 were all given different names.

Works

Cicero was declared a righteous pagan by the Early Church, and therefore many of his works were deemed worthy of preservation. Subsequent Roman and medieval Christian writers quoted liberally from his works De re publica (On the Commonwealth) and De Legibus (On the Laws), and much of his work has been recreated from these surviving fragments. Cicero also articulated an early, abstract conceptualization of rights, based on ancient law and custom. Of Cicero's books, six on rhetoric have survived, as well as parts of seven on philosophy. Of his speeches, 88 were recorded, but only 52 survive.

In archaeology
Cicero's great repute in Italy has led to numerous ruins being identified as having belonged to him, though none have been substantiated with absolute certainty. In Formia, two Roman-era ruins are popularly believed to be Cicero's mausoleum, the Tomba di Cicerone, and the villa where he was assassinated in 43 BC. The latter building is centered around a central hall with Doric columns and a coffered vault, with a separate nymphaeum, on five acres of land near Formia. A modern villa was built on the site after the Rubino family purchased the land from Ferdinand II of the Two Sicilies in 1868. Cicero's supposed tomb is a 24-meter (79 feet) tall tower on an opus quadratum base on the ancient Via Appia outside of Formia. Some suggest that it is not in fact Cicero's tomb, but a monument built on the spot where Cicero was intercepted and assassinated while trying to reach the sea.

In Pompeii, a large villa excavated in the mid 18th century just outside the Herculaneum Gate was widely believed to have been Cicero's, who was known to have owned a holiday villa in Pompeii he called his Pompeianum. The villa was stripped of its fine frescoes and mosaics and then re-buried after 1763 – it has yet to be re-excavated. However, contemporaneous descriptions of the building from the excavators combined with Cicero's own references to his Pompeianum differ, making it unlikely that it is Cicero's villa.

In Rome, the location of Cicero's house has been roughly identified from excavations of the Republican-era stratum on the northwestern slope of the Palatine Hill. Cicero's domus has long been known to have stood in the area, according to his own descriptions and those of later authors, but there is some debate about whether it stood near the base of the hill, very close to the Roman Forum, or nearer to the summit. During his life the area was the most desirable in Rome, densely occupied with Patrician houses including the Domus Publica of Julius Caesar and the home of Cicero's mortal enemy Clodius.

Notable fictional portrayals
In Dante's 1320 poem the Divine Comedy, the author encounters Cicero, among other philosophers, in Limbo. Ben Jonson dramatised the conspiracy of Catiline in his play Catiline His Conspiracy, featuring Cicero as a character. Cicero also appears as a minor character in William Shakespeare's play Julius Caesar.

Cicero was portrayed on the motion picture screen by British actor Alan Napier in the 1953 film Julius Caesar, based on Shakespeare's play. He has also been played by such noted actors as Michael Hordern (in Cleopatra), and André Morell (in the 1970 Julius Caesar). Most recently, Cicero was portrayed by David Bamber in the HBO series Rome (2005–2007) and appeared in both seasons.

In the historical novel series Masters of Rome, Colleen McCullough presents a not-so-flattering depiction of Cicero's career, showing him struggling with an inferiority complex and vanity, morally flexible and fatally indiscreet, while his rival Julius Caesar is shown in a more approving light. Cicero is portrayed as a hero in the novel A Pillar of Iron by Taylor Caldwell (1965). Robert Harris' novels Imperium, Lustrum (published under the name Conspirata in the United States) and Dictator comprise a three-part series based on the life of Cicero. In these novels Cicero's character is depicted in a more favorable way than in those of McCullough, with his positive traits equaling or outweighing his weaknesses (while conversely Caesar is depicted as more sinister than in McCullough). Cicero is a major recurring character in the Roma Sub Rosa series of mystery novels by Steven Saylor. He also appears several times as a peripheral character in John Maddox Roberts' SPQR series.

Samuel Barnett portrays Cicero in a 2017 audio drama series pilot produced by Big Finish Productions. A full series was released the following year. All episodes are written by David Llewellyn and directed and produced by Scott Handcock.

Giancarlo Esposito portrays Mayor Franklyn Cicero, a character from Francis Ford Coppola's 2024 sci-fi epic film Megalopolis, which is a modern futuristic interpretation of the Catiline Conspiracy. In the film, Esposito stars opposite Adam Driver, who portrays a character named Cesar Catilina.

Modern sources

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title: Cicero


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Consul

Consul (abbrev. cos.; Latin plural consules) was the title of one of the two chief magistrates of the Roman Republic, and subsequently also an important title under the Roman Empire. The title was used in other European city-states through antiquity and the Middle Ages, in particular in the Republics of Genoa and Pisa, then revived in modern states, notably in the First French Republic. The related adjective is consular, from the Latin consularis.

This usage contrasts with modern terminology, where a consul is a type of diplomat.

Roman consul
A consul held the highest elected political office of the Roman Republic (509 to 27 BC), and ancient Romans considered the consulship the highest level of the cursus honorum (an ascending sequence of public offices to which politicians aspired). Consuls were elected to office and held power for one year. There were always two consuls in power at any time.

Private sphere
It was not uncommon for an organization under Roman private law to copy the terminology of state and city institutions for its own statutory agents. The founding statute, or contract, of such an organisation was called lex, 'law'. The people elected each year were patricians, members of the upper class.

City-states
While many cities, including the Gallic states and the Carthaginian Republic, had a double-headed chief magistracy, another title was often used, such as the Punic sufet, Duumvir, or native styles like Meddix.

Republic of Genoa
The city-state of Genoa, unlike ancient Rome, bestowed the title of consul on various state officials, not necessarily restricted to the highest. Among these were Genoese officials stationed in various Mediterranean ports, whose role included helping Genoese merchants and sailors in difficulties with the local authorities. Great Britain reciprocated by appointing consuls to Genoa from 1722. This institution, with its name, was later emulated by other powers and is reflected in the modern usage of the word (see Consul (representative)).

Republic of Pisa
In addition to the Genoese Republic, the Republic of Pisa also took the form of "Consul" in the early stages of its government. The Consulate of the Republic of Pisa was the major government institution present in Pisa between the 11th and 12th centuries. Despite losing space within the government since 1190 in favor of the Podestà, for some periods of the 13th century some citizens were again elected as consuls.

Other uses in the Medieval period
Throughout most of southern France, a consul (consul or consule) was an office equivalent to the of the north and roughly similar with English aldermen. The most prominent were those of Bordeaux and Toulouse, which came to be known as jurats and capitouls, respectively. The capitouls of Toulouse were granted transmittable nobility. In many other smaller towns the first consul was the equivalent of a mayor today, assisted by a variable number of secondary consuls and jurats. His main task was to levy and collect tax.

The Dukes of Gaeta often used also the title of "consul" in its Greek form "Hypatos" (see List of Hypati and Dukes of Gaeta).

French Republic 1799–1804

After Napoleon Bonaparte staged a coup against the Directory government in November 1799, the French Republic adopted a constitution which conferred executive powers upon three consuls, elected for a period of ten years. In reality, the first consul, Bonaparte, dominated his two colleagues and held supreme power, soon making himself consul for life (1802) and eventually, in 1804, emperor.

The office was held by:

 * Napoleon Bonaparte, Emmanuel-Joseph Sieyès, Roger Ducos, provisional consuls (10 November – 12 December 1799)
 * Napoleon Bonaparte (first consul), Jean-Jacques Cambacérès (second consul), Charles-François Lebrun (third consul), consuls (12 December 1799 – 18 May 1804)

Bolognese Republic, 1796
The short-lived Bolognese Republic, proclaimed in 1796 as a French client republic in the Central Italian city of Bologna, had a government consisting of nine consuls and its head of state was the Presidente del Magistrato, i.e., chief magistrate, a presiding office held for four months by one of the consuls. Bologna already had consuls at some parts of its Medieval history.

Roman Republic, 1798–1800
The French-sponsored Roman Republic (15 February 1798 – 23 June 1800) was headed by multiple consuls:
 * Francesco Riganti, Carlo Luigi Costantini, Duke Bonelli-Crescenzi, Antonio Bassi, Gioacchino Pessuti, Angelo Stampa, Domenico Maggi, provisional consuls (15 February – 20 March 1798)
 * Liborio Angelucci, Giacomo De Mattheis, Panazzi, Reppi, Ennio Quirino Visconti, consuls (20 March – September 1798)
 * Brigi, Calisti, Francesco Pierelli, Giuseppe Rey, Federico Maria Domenico Michele, Zaccaleoni, consuls (September – 24 July 1799)

Consular rule was interrupted by the Neapolitan occupation (27 November – 12 December 1798), which installed a Provisional Government:
 * Prince Giambattista Borghese, Prince Paolo-Maria Aldobrandini, Prince Gibrielli, Marchese Camillo Massimo, Giovanni Ricci (29 November 1798 - 12 December 1798)

Rome was occupied by France (11 July – 28 September 1799) and again by Naples (30 September 1799 – 23 June 1800), bringing an end to the Roman Republic.

Revolutionary Greece, 1821
Among the many petty local republics that were formed during the first year of the Greek Revolution, prior to the creation of a unified Provisional Government at the First National Assembly at Epidaurus, were: Note: in Greek, the term for "consul" is "hypatos" (ὕπατος), which translates as "supreme one", and hence does not necessarily imply a joint office.
 * The Consulate of Argos (from 26 May 1821, under the Senate of the Peloponnese) had a single head of state, styled consul, 28 March 1821 – 26 May 1821: Stamatellos Antonopoulos
 * The Consulate of East Greece (Livadeia) (from 15 November 1821, under the Areopagus of East Greece) was headed 1 April 1821 – 15 November 1821 by three consuls: Lambros Nakos, Ioannis Logothetis & Ioannis Filon

Paraguay, 1813–1844
In between a series of juntas and various other short-lived regimes, the young republic was governed by "consuls of the republic", with two consuls alternating in power every 4 months:
 * 12 October 1813 – 12 February 1814, José Gaspar Rodríguez de Francia y Velasco
 * 12 February 1814 – 12 June 1814, Fulgencio Yegros y Franco de Torres
 * 12 June 1814 – 3 October 1814, José Gaspar Rodríguez de Francia y Velasco; he stayed on as "supreme dictator" 3 October 1814 – 20 September 1840 (from 6 June 1816 styled "perpetual supreme dictator")

After a few presidents of the Provisional Junta, there were again consuls of the republic, 14 March 1841 – 13 March 1844 (ruling jointly, but occasionally styled "first consul", "second consul"): Carlos Antonio López Ynsfrán (b. 1792 – d. 1862) + Mariano Roque Alonzo Romero (d. 1853) (the lasts of the aforementioned juntistas, Commandant-General of the Army) Thereafter all republican rulers were styled "president".

Modern uses of the term
In modern terminology, a consul is a type of diplomat. The American Heritage Dictionary defines consul as "an official appointed by a government to reside in a foreign country and represent its interests there." The Devil's Dictionary defines Consul as "in American politics, a person who having failed to secure an office from the people is given one by the Administration on condition that he leave the country".

In most governments, the consul is the head of the consular section of an embassy, and is responsible for all consular services such as immigrant and non-immigrant visas, passports, and citizen services for expatriates living or traveling in the host country.

A less common modern usage is when the consul of one country takes a governing role in the host country.

Sources and references

 * WorldStatesmen.org, see each present country

Specific

Konzul (antický Řím) Konzul Konsulo Konsûl Конзул Konsulado

title: Consul


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Talk:Classical mechanics

'Classical Mechanics' byline "F = m a" issues
Actually, F = dp/dt ( Newtonian Force Equation ) Then dp/dt = m * a, yet only when the mass is constant. The mass isn't constant when dealing with many systems, such as rockets when considering the rocket to be the material inside the hard boundary of the rocket, material traveling at approximately the same velocity.

Wikipedia Classical Mechanics articles have a CM tag box on the right, with "F = m a" in it. I think it's silly that the formula used to symbolize classical mechanics is not accurate in many cases. However, I can't come up with a better one, so I am not proposing any changes here. Perhaps someone else can think of a better formula or other byline.

I bring this up because I would hate for people to be misguided, especially if they are trying to understand rocket dynamics. Trying-bold (talk) 23:44, 6 April 2019 (UTC)

Reference to article with NPOV issues
The list of formulations makes a reference to an article with clear neutrality issues. That list should only point to notable formulations that are on par in their fundamental impact with the Lagrangian or Hamilton-Jacobi formulations (as some examples). I suggest the removal of the reference to the Udwadia-Kalaba equation for lack of notability. Reading the linked article, it looks like an attempted promotional push. As is well known, there is no unique formulation of analytical mechanics and the list should only point of especially notable formulations such as the Hamilton-Jacobi equation. The reference to the little-known Udwadia-Kalaba equation in the same vein as the other substantially more notable ones is inappropriate.

- V madhu (talk) 11:38, 2 December 2019 (UTC)

uoioioioioioioioioioioioioioioio.com.net.@gmail 47.17.255.96 (talk) 23:23, 17 May 2023 (UTC)

puzzling paragraph on inertial frames
the unreferenced paragraph starting with "A key concept of inertial frames is the method for identifying" makes at least two dubious claims. first that acceleration is involved in Einstein relativity. second that stellar fixed coordinates have some meaning in mechanics. absent citations I believe both claims are incorrect. I propose to delete or significantly modify the paragraph. Johnjbarton (talk) 21:53, 14 June 2023 (UTC)

 * I removed the incorrect relativity comment and added citations for Goldstein on inertial frames.
 * Johnjbarton (talk) 18:08, 15 June 2023 (UTC)

Proposal to shorten Description and focus this article on overview.
The current "Description of the theory" is really a description of Newtonian mechanics. As presented it distorts the over all topic.

I think the overall topic would be better presented by lifting "Branches" to the spot where Description lives and expanding the subheadings to summary sections linking Main. The current "Description" material would shrink to a paragraph, with possible movement to subarticles. Johnjbarton (talk) 19:02, 4 January 2024 (UTC)

 * This article does rather seem to suffer from not having been written with an overall plan in mind. Like so many broad-scope articles, it has become a repository for every little thing that somebody wanted to say. I like the idea of elevating the "Branches" section and expanding with main and/or further links. XOR'easter (talk) 02:02, 5 January 2024 (UTC)
 * OK, I made a stab at doing something like that. Maybe "Description of the theory" can be tightened now. XOR'easter (talk) 02:36, 5 January 2024 (UTC)
 * What do you think about moving the Lagrangian, Hamiltonian and and Hamilton-Jacobi sections of Newton's laws of motion in to this article? The Newton's law section would be just the current top paragraph and a main link to classical mechanics. This article would get a summary of Newtonian mechanics and the three sections on analytical mechanics. Johnjbarton (talk) 03:13, 5 January 2024 (UTC)
 * I prefer those where they are, since they're specifically about the guises in which Newton's laws present themselves in the alternate formalisms, rather than how to do anything in those formalisms other than re-derive the Newtonian approach. XOR'easter (talk) 03:18, 5 January 2024 (UTC)
 * I added sections on Hamiltonian and Lagrangian mechanics, because I think it makes sense to have them; this article calls for high-level overviews, while the other zooms in to the granular details of a specific aspect. What the page needs next is, I believe, a careful condensation of the long kinematics passages and the Newtonian section, with an eye to removing tangents and overly textbook-like material. XOR'easter (talk) 04:34, 5 January 2024 (UTC)
 * yes, I think we are at fork in the road: the Branches content is short and the rest is too long while also repeating somewhat. Moving more into branches will make it harder to navigate. So my suggestion is to take both forks
 * Leave Branches.
 * Change the name of "Description of the theory" to "Comparison among branches of classical mechanics"
 * Convert the remaining content up to Limitations into summary paragraphs matching the Branches organization. These summaries should focus on helping reader choose, not on explaining.
 * Johnjbarton (talk) 19:19, 5 January 2024 (UTC)
 * I changed the heading of "Description of the theory" and split it in two, since the first part was about kinematics and could apply more broadly than just the Newtonian formalism (e.g., we still use reference frames and coordinates in the Lagrangian method). XOR'easter (talk) 20:50, 5 January 2024 (UTC)

title: Talk:Classical mechanics


text:
List of equations in classical mechanics

Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known. The subject is based upon a three-dimensional Euclidean space with fixed axes, called a frame of reference. The point of concurrency of the three axes is known as the origin of the particular space.

Classical mechanics utilises many equations—as well as other mathematical concepts—which relate various physical quantities to one another. These include differential equations, manifolds, Lie groups, and ergodic theory. This article gives a summary of the most important of these.

This article lists equations from Newtonian mechanics, see analytical mechanics for the more general formulation of classical mechanics (which includes Lagrangian and Hamiltonian mechanics).

General energy definitions
Every conservative force has a potential energy. By following two principles one can consistently assign a non-relative value to U:

 * Wherever the force is zero, its potential energy is defined to be zero as well.
 * Whenever the force does work, potential energy is lost.

Kinematics
In the following rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to use θ, but this does not have to be the polar angle used in polar coordinate systems. The unit axial vector

$$\mathbf{\hat{n}} = \mathbf{\hat{e}}_r\times\mathbf{\hat{e}}_\theta $$

defines the axis of rotation, $$ \scriptstyle \mathbf{\hat{e}}_r $$ = unit vector in direction of $r$, $$ \scriptstyle \mathbf{\hat{e}}_\theta $$ = unit vector tangential to the angle.

Precession
The precession angular speed of a spinning top is given by:

$$ \boldsymbol{\Omega} = \frac{wr}{I\boldsymbol{\omega}} $$

where w is the weight of the spinning flywheel.

Energy
The mechanical work done by an external agent on a system is equal to the change in kinetic energy of the system:

General work-energy theorem (translation and rotation)
The work done W by an external agent which exerts a force F (at r) and torque τ on an object along a curved path C is:

$$ W = \Delta T = \int_C \left ( \mathbf{F} \cdot \mathrm{d} \mathbf{r} + \boldsymbol{\tau} \cdot \mathbf{n} \, {\mathrm{d} \theta} \right ) $$

where θ is the angle of rotation about an axis defined by a unit vector n.

Kinetic energy
The change in kinetic energy for an object initially traveling at speed $$v_0$$ and later at speed $$v$$ is: $$ \Delta E_k = W = \frac{1}{2} m(v^2 - {v_0}^2) $$

Elastic potential energy
For a stretched spring fixed at one end obeying Hooke's law, the elastic potential energy is

$$ \Delta E_p = \frac{1}{2} k(r_2-r_1)^2 $$

where r2 and r1 are collinear coordinates of the free end of the spring, in the direction of the extension/compression, and k is the spring constant.

Euler's equations for rigid body dynamics
Euler also worked out analogous laws of motion to those of Newton, see Euler's laws of motion. These extend the scope of Newton's laws to rigid bodies, but are essentially the same as above. A new equation Euler formulated is:

$$ \mathbf{I} \cdot \boldsymbol{\alpha} + \boldsymbol{\omega} \times \left ( \mathbf{I} \cdot \boldsymbol{\omega} \right ) = \boldsymbol{\tau} $$

where I is the moment of inertia tensor.

General planar motion
The previous equations for planar motion can be used here: corollaries of momentum, angular momentum etc. can immediately follow by applying the above definitions. For any object moving in any path in a plane,

$$ \mathbf{r} = \mathbf{r}(t) = r\hat\mathbf r $$

the following general results apply to the particle.

Central force motion
For a massive body moving in a central potential due to another object, which depends only on the radial separation between the centers of masses of the two objects, the equation of motion is:

$$\frac{d^2}{d\theta^2}\left(\frac{1}{\mathbf{r}}\right) + \frac{1}{\mathbf{r}} = -\frac{\mu\mathbf{r}^2}{\mathbf{l}^2}\mathbf{F}(\mathbf{r})$$

Equations of motion (constant acceleration)
These equations can be used only when acceleration is constant. If acceleration is not constant then the general calculus equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).

Galilean frame transforms
For classical (Galileo-Newtonian) mechanics, the transformation law from one inertial or accelerating (including rotation) frame (reference frame traveling at constant velocity - including zero) to another is the Galilean transform.

Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative accelerations.

Mechanical oscillators
SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively.