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Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates $(q, p) → (Q, P)$ that preserves the form of Hamilton's equations. This is sometimes known as form invariance. Although Hamilton's equations are preserved, it need not preserve the explicit form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).

Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates $q → Q$ do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into $$ P_i = \frac{ \partial L }{ \partial \dot{Q}_i }\ ,$$ where $$\left\{\ (P_1, Q_1),\ (P_2, Q_2),\ (P_3, Q_3),\ \ldots\ \right\} $$ are the new co‑ordinates, grouped in canonical conjugate pairs of momenta $$P_i $$ and corresponding positions $$Q_i,$$ for $$i = 1, 2, \ldots\ N,$$ with $$N $$ being the number of degrees of freedom in both co‑ordinate systems.

Therefore, coordinate transformations (also called point transformations) are a type of canonical transformation. However, the class of canonical transformations is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta. Canonical transformations that do not include the time explicitly are called restricted canonical transformations (many textbooks consider only this type).

Modern mathematical descriptions of canonical transformations are considered under the broader topic of symplectomorphism which covers the subject with advanced mathematical prerequisites such as cotangent bundles, exterior derivatives and symplectic manifolds.

Notation
Boldface variables such as $q$ represent a list of $N$ generalized coordinates that need not transform like a vector under rotation and similarly $p$ represents the corresponding generalized momentum, e.g., $$\begin{align} \mathbf{q} &\equiv \left (q_{1}, q_{2}, \ldots, q_{N-1}, q_{N} \right )\\ \mathbf{p} &\equiv \left (p_{1}, p_{2}, \ldots, p_{N-1}, p_{N} \right ). \end{align}$$

A dot over a variable or list signifies the time derivative, e.g., $$\dot{\mathbf{q}} \equiv \frac{d\mathbf{q}}{dt}$$and the equalities are read to be satisfied for all coordinates, for example:$$\dot{\mathbf{p}} = -\frac{\partial f}{\partial \mathbf{q}}\quad \Longleftrightarrow \quad  \dot = -\frac{\partial f}{\partial {q_i}}\quad (\forall  i). $$

The dot product notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, e.g., $$\mathbf{p} \cdot \mathbf{q} \equiv \sum_{k=1}^{N} p_{k} q_{k}.$$

The dot product (also known as an "inner product") maps the two coordinate lists into one variable representing a single numerical value. The coordinates after transformation are similarly labelled with $Q$ for transformed generalized coordinates and $P$ for transformed generalized momentum.

Conditions for restricted canonical transformation
Restricted canonical transformations are coordinate transformations where transformed coordinates $Q$ and $P$ do not have explicit time dependence, ie. $\mathbf Q=\mathbf Q(\mathbf q,\mathbf p)$  and $\mathbf  P=\mathbf P(\mathbf q,\mathbf p) $. The functional form of Hamilton's equations is$$\begin{align} \dot{\mathbf{p}} &= -\frac{\partial H}{\partial \mathbf{q}} \\ \dot{\mathbf{q}} &= \frac{\partial H}{\partial \mathbf{p}} \end{align}$$ In general, a transformation $(q, p) → (Q, P)$ does not preserve the form of Hamilton's equations but in the absence of time dependence in transformation, some simplifications are possible. Following the formal definition for a canonical transformation, it can be shown that for this type of transformation, the new Hamiltonian (sometimes called the Kamiltonian ) can be expressed as:$$K(\mathbf Q, \mathbf P, t)= H(q(\mathbf Q,\mathbf P),p(\mathbf Q,\mathbf P),t) + \frac{\partial G}{\partial t}(t)$$where it differs by a partial time derivative of a function known as generator, which reduces to being only a function of time for restricted canonical transformations.

In addition to leaving the form of the Hamiltonian unchanged, it is also permits the use of the unchanged Hamiltonian in the Hamilton's equations of motion due to the above form as:

$$\begin{alignat}{3} \dot{\mathbf{P}} &= -\frac{\partial K}{\partial \mathbf{Q}} &&= -\left(\frac{\partial H}{\partial \mathbf{Q}}\right)_{\mathbf Q,\mathbf P,t}\\ \dot{\mathbf{Q}} &= \,\,\,\, \frac{\partial K}{\partial \mathbf{P}} &&= \,\,\,\, \, \left(\frac{\partial H}{\partial \mathbf{P}}\right)_{\mathbf Q,\mathbf P ,t}\\ \end{alignat}$$

Although canonical transformations refers to a more general set of transformations of phase space corresponding with less permissive transformations of the Hamiltonian, it provides simpler conditions to obtain results that can be further generalized. All of the following conditions, with the exception of bilinear invariance condition, can be generalized for canonical transformations, including time dependance.

Indirect conditions
Since restricted transformations have no explicit time dependence (by definition), the time derivative of a new generalized coordinate $Q_{m}$ is

$$\begin{align} \dot{Q}_{m} &= \frac{\partial Q_{m}}{\partial \mathbf{q}} \cdot \dot{\mathbf{q}} + \frac{\partial Q_{m}}{\partial \mathbf{p}} \cdot \dot{\mathbf{p}} \\ &= \frac{\partial Q_{m}}{\partial \mathbf{q}} \cdot \frac{\partial H}{\partial \mathbf{p}} - \frac{\partial Q_{m}}{\partial \mathbf{p}} \cdot \frac{\partial H}{\partial \mathbf{q}} \\ &= \lbrace Q_m, H \rbrace \end{align}$$

where ${⋅, ⋅}$ is the Poisson bracket.

Similarly for the identity for the conjugate momentum, Pm using the form of the Kamiltonian it follows that:

$$\begin{align} &\frac{\partial K(\mathbf{Q}, \mathbf{P}, t)}{\partial P_{m}} \\ &= \frac{\partial K(\mathbf{Q}(\mathbf{q}, \mathbf{p}), \mathbf{P}(\mathbf{q}, \mathbf{p}), t)}{\partial \mathbf{q}} \cdot \frac{\partial \mathbf{q}}{\partial P_{m}} + \frac{\partial K(\mathbf{Q}(\mathbf{q}, \mathbf{p}), \mathbf{P}(\mathbf{q}, \mathbf{p}), t)}{\partial \mathbf{p}} \cdot \frac{\partial \mathbf{p}}{\partial P_{m}} \\ &= \frac{\partial H(\mathbf{q}, \mathbf{p}, t)}{\partial \mathbf{q}} \cdot \frac{\partial \mathbf{q}}{\partial P_{m}} + \frac{\partial H(\mathbf{q}, \mathbf{p}, t)}{\partial \mathbf{p}} \cdot \frac{\partial \mathbf{p}}{\partial P_{m}} \\ &= \frac{\partial H}{\partial \mathbf{q}} \cdot \frac{\partial \mathbf{q}}{\partial P_{m}} + \frac{\partial H}{\partial \mathbf{p}} \cdot \frac{\partial \mathbf{p}}{\partial P_{m}} \end{align}$$

Due to the form of the Hamiltonian equations of motion,

$$\begin{align} \dot{\mathbf{P}} &= -\frac{\partial K}{\partial \mathbf{Q}}\\ \dot{\mathbf{Q}} &= \,\,\,\, \frac{\partial K}{\partial \mathbf{P}} \end{align}$$

if the transformation is canonical, the two derived results must be equal, resulting in the equations: $$\begin{align} \left( \frac{\partial Q_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p}} &= -\left( \frac{\partial q_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P}} \\ \left( \frac{\partial Q_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p}} &= \left( \frac{\partial p_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P}} \end{align}$$

The analogous argument for the generalized momenta Pm leads to two other sets of equations: $$\begin{align} \left( \frac{\partial P_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p}} &= \left( \frac{\partial q_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P}} \\ \left( \frac{\partial P_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p}} &= -\left( \frac{\partial p_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P}} \end{align}$$

These are the indirect conditions to check whether a given transformation is canonical.

Symplectic condition
Sometimes the Hamiltonian relations are represented as:

$$\dot{\eta}= J \nabla_\eta H $$

Where $J := \begin{pmatrix} 0 & I_n \\ -I_n & 0 \\ \end{pmatrix},$

and $\mathbf{\eta} = \begin{bmatrix} q_1\\ \vdots \\ q_n\\ p_1\\ \vdots\\ p_n\\ \end{bmatrix} $ . Similarly, let $\mathbf{\varepsilon} = \begin{bmatrix} Q_1\\ \vdots \\ Q_n\\ P_1\\ \vdots\\ P_n\\ \end{bmatrix} $.

From the relation of partial derivatives, converting the $$\dot{\eta}= J \nabla_\eta H $$ relation in terms of partial derivatives with new variables gives $$\dot{\eta}=J ( M^T \nabla_\varepsilon H) $$ where $M := \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{p})}$. Similarly for $\dot{\varepsilon} $ ,

$$\dot{\varepsilon}=M\dot{\eta} =M J M^T \nabla_\varepsilon H $$

Due to form of the Hamiltonian equations for $\dot{\varepsilon} $ ,

$$\dot{\varepsilon}=J \nabla_\varepsilon K =  J \nabla_\varepsilon H $$

where $\nabla_\varepsilon K =  \nabla_\varepsilon H $  can be used due to the form of Kamiltonian. Equating the two equations gives the symplectic condition as:

$$M J M^T = J $$The left hand side of the above is called the Poisson matrix of $$\varepsilon $$, denoted as $\mathcal P(\varepsilon) = MJM^T $ . Similarly, a Lagrange matrix of $$\eta $$ can be constructed as $\mathcal L(\eta) = M^TJM $ . It can be shown that the symplectic condition is also equivalent to $M^T J M = J $  by using the $J^{-1}=-J $ property. The set of all matrices $M $ which satisfy symplectic conditions form a symplectic group. The symplectic conditions are equivalent with indirect conditions as they both lead to the equation $\dot{\varepsilon}= J \nabla_\varepsilon H $, which is used in both of the derivations.

Invariance of Poisson Bracket
The Poisson bracket which is defined as:$$\{u, v\}_\eta := \sum_{i=1}^{n} \left( \frac{\partial u}{\partial q_{i}} \frac{\partial v}{\partial p_{i}} - \frac{\partial u}{\partial p_i} \frac{\partial v}{\partial q_i}\right)$$can be represented in matrix form as:$$\{u, v\}_\eta := (\nabla_\eta u)^T J (\nabla_\eta v)$$Hence using partial derivative relations and symplectic condition gives: $$\{u, v\}_\eta = (\nabla_\eta u)^T J (\nabla_\eta v) = (M^T \nabla_\varepsilon u)^T J (M^T \nabla_\varepsilon v) = (\nabla_\varepsilon u)^T M J M^T (\nabla_\varepsilon v) = (\nabla_\varepsilon u)^T J (\nabla_\varepsilon v) = \{u, v\}_\varepsilon$$

The symplectic condition can also be recovered by taking $u=\varepsilon_i $ and $v=\varepsilon_j $ which shows that $(M J M^T )_{ij}= J_{i j} $. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that $\mathcal P_{ij}(\varepsilon) = \{ \varepsilon_i,\varepsilon_j\}_\eta =(M J M^T )_{ij} $, which is also the result of explicitly calculating the matrix element by expanding it.

Invariance of Lagrange Bracket
The Lagrange bracket which is defined as:

$$ [ u, v ]_{\eta} := \sum_{i=1}^n \left(\frac{\partial q_i}{\partial u} \frac{\partial p_i}{\partial v} - \frac{\partial p_i}{\partial u} \frac{\partial q_i}{\partial v }  \right) $$

can be represented in matrix form as:

$$ [ u, v ]_{\eta} := \left(\frac {\partial \eta}{\partial u}\right)^T J \left(\frac {\partial \eta}{\partial v}\right) $$

Using similar derivation, gives:

$$[u, v]_\varepsilon = (\partial_u \varepsilon )^T \,J\, (\partial_v \varepsilon) = (M \, \partial_u  \eta )^T \,J \, ( M \,\partial_v \eta) = (\partial_u \eta )^T\, M^TJ M\, (\partial_v \eta) = (\partial_u \eta )^T\, J\,(\partial_v \eta) = [u, v]_\eta$$The symplectic condition can also be recovered by taking $u=\eta_i $ and $v=\eta_j $ which shows that $(M^T J M )_{ij}= J_{i j} $. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that $\mathcal L_{ij}(\eta) =[\eta_i,\eta_j]_\varepsilon=(M^T J M )_{ij} $, which is also the result of explicitly calculating the matrix element by expanding it.

Bilinear invariance conditions
These set of conditions only apply to restricted canonical transformations or canonical transformations that are independent of time variable.

Consider arbitrary variations of two kinds, in a single pair of generalized coordinate and the corresponding momentum:

$ d \varepsilon=( dq_1, dp_{1},0,0,\ldots),\quad\delta \varepsilon=(\delta q_{1},\delta p_{1},0,0,\ldots). $

The area of the infinitesimal parallelogram is given by:

$ \delta a(12)=d q_{1}\delta p_{1}-\delta q_{1} d p_{1}={(\delta\varepsilon)}^T\,J \, d \varepsilon. $

It follows from the $M^T J M = J $  symplectic condition that the infinitesimal area is conserved under canonical transformation:

$ \delta a(12)={(\delta\varepsilon)}^T\,J \,d \varepsilon={(M\delta\eta)}^T\,J \,Md \eta= {(\delta\eta)}^T\,M^TJM \,d \eta = {(\delta\eta)}^T\,J \,d\eta = \delta A(12). $

Note that the new coordinates need not be completely oriented in one coordinate momentum plane.

Hence, the condition is more generally stated as an invariance of the form $ {(d\varepsilon)}^T\,J \, \delta \varepsilon $ under canonical transformation, expanded as:$$ \sum \delta q \cdot dp - \delta p \cdot dq = \sum \delta Q \cdot dP - \delta P \cdot dQ $$If the above is obeyed for any arbitrary variations, it would be only possible if the indirect conditions are met. The form of the equation, $ {v}^T\,J \, w $ is also known as a symplectic product of the vectors $ {v} $  and $ w $  and the bilinear invariance condition can be stated as a local conservation of the symplectic product.

Liouville's theorem
The indirect conditions allow us to prove Liouville's theorem, which states that the volume in phase space is conserved under canonical transformations, i.e., $$ \int \mathrm{d}\mathbf{q}\, \mathrm{d}\mathbf{p} = \int \mathrm{d}\mathbf{Q}\, \mathrm{d}\mathbf{P}$$

By calculus, the latter integral must equal the former times the determinant of Jacobian $M$$$\int \mathrm{d}\mathbf{Q}\, \mathrm{d}\mathbf{P} = \int \det (M) \, \mathrm{d}\mathbf{q}\, \mathrm{d}\mathbf{p}$$Where $M := \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{p})}$

Exploiting the "division" property of Jacobians yields$$ M \equiv \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{P})} \left/ \frac{\partial (\mathbf{q}, \mathbf{p})}{\partial (\mathbf{q}, \mathbf{P})} \right. $$

Eliminating the repeated variables gives$$M \equiv \frac{\partial (\mathbf{Q})}{\partial (\mathbf{q})} \left/ \frac{\partial (\mathbf{p})}{\partial (\mathbf{P})} \right.$$

Application of the indirect conditions above yields $$\operatorname{det}(M)=1$$.

Generating function approach
To guarantee a valid transformation between $(q, p, H)$ and $(Q, P, K)$, we may resort to a direct generating function approach. Both sets of variables must obey Hamilton's principle. That is the action integral over the Lagrangians $$\mathcal{L}_{qp}=\mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t)$$ and $$\mathcal{L}_{QP}=\mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t)$$, obtained from the respective Hamiltonian via an "inverse" Legendre transformation, must be stationary in both cases (so that one can use the Euler–Lagrange equations to arrive at Hamiltonian equations of motion of the designated form; as it is shown for example here): $$\begin{align} \delta \int_{t_{1}}^{t_{2}} \left[ \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) \right] dt &= 0 \\ \delta \int_{t_{1}}^{t_{2}} \left[ \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) \right] dt &= 0 \end{align}$$

One way for both variational integral equalities to be satisfied is to have $$\lambda \left[ \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) \right] = \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{dG}{dt} $$

Lagrangians are not unique: one can always multiply by a constant $λ$ and add a total time derivative $dG⁄dt$ and yield the same equations of motion (as discussed on Wikibooks). In general, the scaling factor $λ$ is set equal to one; canonical transformations for which $λ ≠ 1$ are called extended canonical transformations. $dG⁄dt$ is kept, otherwise the problem would be rendered trivial and there would be not much freedom for the new canonical variables to differ from the old ones.

Here $G$ is a generating function of one old canonical coordinate ($q$ or $p$), one new canonical coordinate ($Q$ or $P$) and (possibly) the time $t$. Thus, there are four basic types of generating functions (although mixtures of these four types can exist), depending on the choice of variables. As will be shown below, the generating function will define a transformation from old to new canonical coordinates, and any such transformation $(q, p) → (Q, P)$ is guaranteed to be canonical.

The various generating functions and its properties tabulated below is discussed in detail:

Type 1 generating function
The type 1 generating function $G_{1}$ depends only on the old and new generalized coordinates $$G \equiv G_{1}(\mathbf{q}, \mathbf{Q}, t)$$ To derive the implicit transformation, we expand the defining equation above $$ \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) = \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{\partial G_{1}}{\partial t} + \frac{\partial G_{1}}{\partial \mathbf{q}} \cdot \dot{\mathbf{q}} + \frac{\partial G_{1}}{\partial \mathbf{Q}} \cdot \dot{\mathbf{Q}}$$

Since the new and old coordinates are each independent, the following $2N + 1$ equations must hold $$\begin{align} \mathbf{p} &= \frac{\partial G_{1}}{\partial \mathbf{q}} \\ \mathbf{P} &= -\frac{\partial G_{1}}{\partial \mathbf{Q}} \\ K &= H + \frac{\partial G_{1}}{\partial t} \end{align}$$

These equations define the transformation $(q, p) → (Q, P)$ as follows: The first set of $N$ equations $$\ \mathbf{p} = \frac{\ \partial G_{1}\ }{ \partial \mathbf{q} }\ $$ define relations between the new generalized coordinates $Q$ and the old canonical coordinates $(q, p)$. Ideally, one can invert these relations to obtain formulae for each $Q_{k}$ as a function of the old canonical coordinates. Substitution of these formulae for the $Q$ coordinates into the second set of $N$ equations $$\mathbf{P} = -\frac{\partial G_{1}}{\partial \mathbf{Q}}$$ yields analogous formulae for the new generalized momenta $P$ in terms of the old canonical coordinates $(q, p)$. We then invert both sets of formulae to obtain the old canonical coordinates $(q, p)$ as functions of the new canonical coordinates $(Q, P)$. Substitution of the inverted formulae into the final equation $$K = H + \frac{\partial G_{1}}{\partial t}$$ yields a formula for $K$ as a function of the new canonical coordinates $(Q, P)$.

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let $$G_{1} \equiv \mathbf{q} \cdot \mathbf{Q}$$ This results in swapping the generalized coordinates for the momenta and vice versa $$\begin{align} \mathbf{p} &= \frac{\partial G_{1}}{\partial \mathbf{q}} = \mathbf{Q} \\ \mathbf{P} &= -\frac{\partial G_{1}}{\partial \mathbf{Q}} = -\mathbf{q} \end{align}$$ and $K = H$. This example illustrates how independent the coordinates and momenta are in the Hamiltonian formulation; they are equivalent variables.

Type 2 generating function
The type 2 generating function $$G_{2}(\mathbf{q}, \mathbf{P}, t)$$ depends only on the old generalized coordinates and the new generalized momenta $$G \equiv G_{2}(\mathbf{q}, \mathbf{P}, t)-\mathbf{Q} \cdot \mathbf{P}$$ where the $$-\mathbf{Q} \cdot \mathbf{P}$$ terms represent a Legendre transformation to change the right-hand side of the equation below. To derive the implicit transformation, we expand the defining equation above $$ \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) = -\mathbf{Q} \cdot \dot{\mathbf{P}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{\partial G_{2}}{\partial t} + \frac{\partial G_{2}}{\partial \mathbf{q}} \cdot \dot{\mathbf{q}} + \frac{\partial G_{2}}{\partial \mathbf{P}} \cdot \dot{\mathbf{P}}$$

Since the old coordinates and new momenta are each independent, the following $2N + 1$ equations must hold $$\begin{align} \mathbf{p} &= \frac{\partial G_{2}}{\partial \mathbf{q}} \\ \mathbf{Q} &= \frac{\partial G_{2}}{\partial \mathbf{P}} \\ K &= H + \frac{\partial G_{2}}{\partial t} \end{align}$$

These equations define the transformation $(q, p) → (Q, P)$ as follows: The first set of $N$ equations $$\mathbf{p} = \frac{\partial G_{2}}{\partial \mathbf{q}}$$ define relations between the new generalized momenta $P$ and the old canonical coordinates $(q, p)$. Ideally, one can invert these relations to obtain formulae for each $P_{k}$ as a function of the old canonical coordinates. Substitution of these formulae for the $P$ coordinates into the second set of $N$ equations $$\mathbf{Q} = \frac{\partial G_{2}}{\partial \mathbf{P}}$$ yields analogous formulae for the new generalized coordinates $Q$ in terms of the old canonical coordinates $(q, p)$. We then invert both sets of formulae to obtain the old canonical coordinates $(q, p)$ as functions of the new canonical coordinates $(Q, P)$. Substitution of the inverted formulae into the final equation $$K = H + \frac{\partial G_{2}}{\partial t}$$ yields a formula for $K$ as a function of the new canonical coordinates $(Q, P)$.

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let $$G_{2} \equiv \mathbf{g}(\mathbf{q}; t) \cdot \mathbf{P}$$ where $g$ is a set of $N$ functions. This results in a point transformation of the generalized coordinates $$\mathbf{Q} = \frac{\partial G_{2}}{\partial \mathbf{P}} = \mathbf{g}(\mathbf{q}; t)$$

Type 3 generating function
The type 3 generating function $$G_{3}(\mathbf{p}, \mathbf{Q}, t)$$ depends only on the old generalized momenta and the new generalized coordinates $$G \equiv G_{3}(\mathbf{p}, \mathbf{Q}, t)+ \mathbf{q} \cdot \mathbf{p}$$ where the $$\mathbf{q} \cdot \mathbf{p}$$ terms represent a Legendre transformation to change the left-hand side of the equation below. To derive the implicit transformation, we expand the defining equation above $$-\mathbf{q} \cdot \dot{\mathbf{p}} - H(\mathbf{q}, \mathbf{p}, t) = \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{\partial G_{3}}{\partial t} + \frac{\partial G_{3}}{\partial \mathbf{p}} \cdot \dot{\mathbf{p}} + \frac{\partial G_{3}}{\partial \mathbf{Q}} \cdot \dot{\mathbf{Q}}$$

Since the new and old coordinates are each independent, the following $2N + 1$ equations must hold $$\begin{align} \mathbf{q} &= -\frac{\partial G_{3}}{\partial \mathbf{p}} \\ \mathbf{P} &= -\frac{\partial G_{3}}{\partial \mathbf{Q}} \\ K &= H + \frac{\partial G_{3}}{\partial t} \end{align}$$

These equations define the transformation $(q, p) → (Q, P)$ as follows: The first set of $N$ equations $$ \mathbf{q} = -\frac{\partial G_{3}}{\partial \mathbf{p}}$$ define relations between the new generalized coordinates $Q$ and the old canonical coordinates $(q, p)$. Ideally, one can invert these relations to obtain formulae for each $Q_{k}$ as a function of the old canonical coordinates. Substitution of these formulae for the $Q$ coordinates into the second set of $N$ equations $$\mathbf{P} = -\frac{\partial G_{3}}{\partial \mathbf{Q}}$$ yields analogous formulae for the new generalized momenta $P$ in terms of the old canonical coordinates $(q, p)$. We then invert both sets of formulae to obtain the old canonical coordinates $(q, p)$ as functions of the new canonical coordinates $(Q, P)$. Substitution of the inverted formulae into the final equation $$K = H + \frac{\partial G_{3}}{\partial t}$$ yields a formula for $K$ as a function of the new canonical coordinates $(Q, P)$.

In practice, this procedure is easier than it sounds, because the generating function is usually simple.

Type 4 generating function
The type 4 generating function $$G_{4}(\mathbf{p}, \mathbf{P}, t)$$ depends only on the old and new generalized momenta $$G \equiv G_{4}(\mathbf{p}, \mathbf{P}, t) +\mathbf{q} \cdot \mathbf{p} - \mathbf{Q} \cdot \mathbf{P} $$ where the $$\mathbf{q} \cdot \mathbf{p} - \mathbf{Q} \cdot \mathbf{P}$$ terms represent a Legendre transformation to change both sides of the equation below. To derive the implicit transformation, we expand the defining equation above $$-\mathbf{q} \cdot \dot{\mathbf{p}} - H(\mathbf{q}, \mathbf{p}, t) = -\mathbf{Q} \cdot \dot{\mathbf{P}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{\partial G_{4}}{\partial t} + \frac{\partial G_{4}}{\partial \mathbf{p}} \cdot \dot{\mathbf{p}} + \frac{\partial G_{4}}{\partial \mathbf{P}} \cdot \dot{\mathbf{P}} $$

Since the new and old coordinates are each independent, the following $2N + 1$ equations must hold $$\begin{align} \mathbf{q} &= -\frac{\partial G_{4}}{\partial \mathbf{p}} \\ \mathbf{Q} &= \frac{\partial G_{4}}{\partial \mathbf{P}} \\ K &= H + \frac{\partial G_{4}}{\partial t} \end{align}$$

These equations define the transformation $(q, p) → (Q, P)$ as follows: The first set of $N$ equations $$\mathbf{q} = -\frac{\partial G_{4}}{\partial \mathbf{p}}$$ define relations between the new generalized momenta $P$ and the old canonical coordinates $(q, p)$. Ideally, one can invert these relations to obtain formulae for each $P_{k}$ as a function of the old canonical coordinates. Substitution of these formulae for the $P$ coordinates into the second set of $N$ equations $$\mathbf{Q} = \frac{\partial G_{4}}{\partial \mathbf{P}} $$ yields analogous formulae for the new generalized coordinates $Q$ in terms of the old canonical coordinates $(q, p)$. We then invert both sets of formulae to obtain the old canonical coordinates $(q, p)$ as functions of the new canonical coordinates $(Q, P)$. Substitution of the inverted formulae into the final equation $$K = H + \frac{\partial G_{4}}{\partial t}$$ yields a formula for $K$ as a function of the new canonical coordinates $(Q, P)$.

Restrictions on generating functions
For example, using generating function of second kind: ${p}_i = \frac{\partial G_{2}}{\partial {q}_i} $ and ${Q}_i = \frac{\partial G_{2}}{\partial {P}_i} $, the first set of equations consisting of variables $\mathbf{p} $ , $\mathbf{q} $  and $\mathbf{P} $  has to be inverted to get $\mathbf{P}(\mathbf q, \mathbf p) $. This process is possible when the matrix defined by $a_{ij}=\frac{\partial {p}_i(\mathbf q,\mathbf P)}{\partial P_j} $ is non-singular.

$$\left|\begin{array}{l l l}&&\\ {\quad \vdots} & {\ddots}&{\quad \vdots}\\&&\end{array}\right|{\neq0}$$

Hence, restrictions are placed on generating functions to have the matrices: $\left[\frac{\partial^2 G_1}{\partial Q_j\partial q_i} \right] $, $\left[\frac{\partial^2 G_2}{\partial P_j\partial q_i} \right] $ , $\left[\frac{\partial^2 G_3}{\partial p_j\partial Q_i} \right]  $  and $\left[\frac{\partial^2 G_4}{\partial p_j\partial P_i} \right]  $ , being non-singular.

Limitations of generating functions
Since $\left[\frac{\partial^2 G_2}{\partial P_j\partial q_i} \right] $  is non-singular, it implies that $\left[\frac{\partial {p}_i(\mathbf q,\mathbf P)}{\partial P_j}\right]  $  is also non-singular. Since the matrix $\left[\frac{\partial P_i(\mathbf q,\mathbf p)}{\partial p_j} \right] $  is inverse of $\left[\frac{\partial {p}_i(\mathbf q,\mathbf P)}{\partial P_j}\right]  $, the transformations of type 2 generating functions always have a non-singular $\left[\frac{\partial P_i(\mathbf q,\mathbf p)}{\partial p_j} \right]  $  matrix. Similarly, it can be stated that type 1 and type 4 generating functions always have a non-singular $\left[\frac{\partial Q_i(\mathbf q,\mathbf p)}{\partial p_j} \right] $  matrix whereas type 2 and type 3 generating functions always have a non-singular $\left[\frac{\partial P_i(\mathbf q,\mathbf p)}{\partial p_j} \right]  $  matrix. Hence, the canonical transformations resulting from these generating functions are not completely general.

In other words, since $(Q, P)$ and $(q, p)$ are each $2N$ independent functions, it follows that to have generating function of the form $G_{1}(\mathbf{q}, \mathbf{Q}, t) $ and $$G_{4}(\mathbf{p}, \mathbf{P}, t)$$ or $$G_{2}(\mathbf{q}, \mathbf{P}, t)$$ and $$G_{3}(\mathbf{p}, \mathbf{Q}, t)$$, the corresponding Jacobian matrices $\left[\frac{\partial Q_i}{\partial p_j} \right]  $  and $\left[\frac{\partial P_i}{\partial p_j} \right]  $  are restricted to be non singular, ensuring that the generating function is a function of $2N + 1$ independent variables. However, as a feature of canonical transformations, it is always possible to choose $2N$ such independent functions from sets $(q, p)$ or $(Q, P)$, to form a generating function representation of canonical transformations, including the time variable. Hence, it can be proved that every finite canonical transformation can be given as a closed but implicit form that is a variant of the given four simple forms.

Canonical transformation relations
From: $$K = H + \frac{\partial G}{\partial t} $$, calculate $\frac{\partial (K-H)}{\partial P} $ :

$$\begin{align} \left( \frac{\partial (K-H)}{\partial P}\right)_{Q,P,t}= \frac{\partial K}{\partial P} - \frac{\partial H}{\partial p}\frac{\partial p}{\partial P} - \frac{\partial H}{\partial q}\frac{\partial q}{\partial P} - \frac{\partial H}{\partial t}\left( \frac{\partial t}{\partial P}\right)_{Q,P,t} = \dot{Q} + \dot{p} \frac{\partial q}{\partial P} - \dot{q}\frac{\partial p}{\partial P} \\ = \frac{\partial Q}{\partial t} + \frac{\partial Q}{\partial q} \cdot \dot{q} + \frac{\partial Q}{\partial p} \cdot \dot{p} + \dot{p} \frac{\partial q}{\partial P} - \dot{q}\frac{\partial p}{\partial P} \\ =\dot{q}\left(\frac{\partial Q}{\partial q} - \frac{\partial p}{\partial P}\right)+\dot{p}\left(\frac{\partial q}{\partial P} +\frac{\partial Q}{\partial p} \right) + \frac{\partial Q}{\partial t} \end{align}       $$Since the left hand side is $\frac{\partial (K-H)}{\partial P} = \frac \partial {\partial P}\left( \frac{\partial G}{\partial t} \right) \bigg |_{Q,P,t} $   which is independent of dynamics of the particles, equating coefficients of $\dot q $  and $\dot p $  to zero, canonical transformation rules are obtained. This step is equivalent to equating the left hand side as $\frac{\partial (K-H)}{\partial P} = \frac{\partial Q}{\partial t}  $.

Similarly:

$$\begin{align} \left(\frac{\partial (K-H)}{\partial Q}\right)_{Q,P,t}= \frac{\partial K}{\partial Q} - \frac{\partial H}{\partial p}\frac{\partial p}{\partial Q} - \frac{\partial H}{\partial q}\frac{\partial q}{\partial Q} - \frac{\partial H}{\partial t}\left(\frac{\partial t}{\partial Q}\right)_{Q,P,t} = -\dot{P} + \dot{p} \frac{\partial q}{\partial Q} - \dot{q}\frac{\partial p}{\partial Q} \\ = -\frac{\partial P}{\partial t} -\frac{\partial P}{\partial q} \cdot \dot{q} - \frac{\partial P}{\partial p} \cdot \dot{p} + \dot{p} \frac{\partial q}{\partial Q} - \dot{q}\frac{\partial p}{\partial Q} \\ =-\left(\dot{q}\left(\frac{\partial P}{\partial q} + \frac{\partial p}{\partial Q}\right)+\dot{p}\left(\frac{\partial P}{\partial p} -\frac{\partial q}{\partial Q} \right) + \frac{\partial P}{\partial t} \right) \end{align}       $$Similarly the canonical transformation rules are obtained by equating the left hand side as $\frac{\partial (K-H)}{\partial Q} = - \frac{\partial P}{\partial t}   $.

The above two relations can be combined in matrix form as: $J \left(\nabla_\varepsilon \frac{\partial G}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t} $  (which will also retain same form for extended canonical transformation) where the result $\frac{\partial G}{\partial t} = K-H $, has been used. The canonical transformation relations are hence said to be equivalent to $J \left(\nabla_\varepsilon \frac{\partial G}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t} $  in this context.

The canonical transformation relations can now be restated to include time dependance:$$\begin{align} \left( \frac{\partial Q_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= - \left( \frac{\partial q_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \\ \left( \frac{\partial Q_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= \left( \frac{\partial p_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \end{align} $$$$\begin{align} \left( \frac{\partial P_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= \left( \frac{\partial q_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \\ \left( \frac{\partial P_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= - \left( \frac{\partial p_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \end{align}$$Since $\frac{\partial (K-H)}{\partial P} = \frac{\partial Q}{\partial t}  $  and $\frac{\partial (K-H)}{\partial Q} = - \frac{\partial P}{\partial t}   $, if $Q$ and $P$ do not explicitly depend on time, $K= H + \frac{\partial G}{\partial t}(t)$  can be taken. The analysis of restricted canonical transformations is hence consistent with this generalization.

Symplectic Condition
Applying transformation of co-ordinates formula for $$ \nabla_\eta H = M^T \nabla_\varepsilon H  $$, in Hamiltonian's equations gives:$$\dot{\eta}=J\nabla_\eta H =J ( M^T \nabla_\varepsilon H) $$

Similarly for $\dot{\varepsilon} $ :$$\dot{\varepsilon}=M\dot{\eta} + \frac{\partial \varepsilon}{\partial t} =M J M^T \nabla_\varepsilon H + \frac{\partial \varepsilon}{\partial t} $$or:$$\dot{\varepsilon}=J \nabla_\varepsilon K =  J \nabla_\varepsilon H + J \nabla_\varepsilon \left( \frac{\partial G}{\partial t}\right) $$Where the last terms of each equation cancel due to $J \left(\nabla_\varepsilon \frac{\partial G}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t} $  condition from canonical transformations. Hence leaving the symplectic relation: $M J M^T = J $ which is also equivalent with the condition $M^T J M = J $. It follows from the above two equations that the symplectic condition implies the equation $J \left(\nabla_\varepsilon \frac{\partial G}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t} $, from which the indirect conditions can be recovered. Thus, symplectic conditions and indirect conditions can be said to be equivalent in the context of using generating functions.

Invariance of Poisson and Lagrange Bracket
Since $\mathcal P_{ij}(\varepsilon) = \{ \varepsilon_i,\varepsilon_j\}_\eta =(M J M^T )_{ij} = J_{ij} $ and $\mathcal L_{ij}(\eta) =[\eta_i,\eta_j]_\varepsilon=(M^T J M )_{ij} = J_{ij} $ where the symplectic condition is used in the last equalities. Using $\{\varepsilon_i,\varepsilon_j\}_\varepsilon=[\eta_i,\eta_j]_\eta = J_{ij} $, the equalities $\{ \varepsilon_i,\varepsilon_j\}_\eta= \{ \varepsilon_i,\varepsilon_j\}_\varepsilon $ and $[\eta_i,\eta_j]_\varepsilon= [\eta_i,\eta_j]_\eta $ are obtained which imply the invariance of Poisson and Lagrange brackets.

Canonical transformation relations
By solving for:$$\lambda \left[ \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) \right] = \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{dG}{dt} $$with various forms of generating function, the relation between K and H goes as $\frac{\partial G}{\partial t} = K-\lambda H $ instead, which also applies for $\lambda = 1 $  case.

All results presented below can also be obtained by replacing $q \rightarrow \sqrt{\lambda}q $, $p \rightarrow \sqrt{\lambda}p $ and $H \rightarrow {\lambda}H $  from known solutions, since it retains the form of Hamilton's equations. The extended canonical transformations are hence said to be result of a canonical transformation ($\lambda = 1 $ ) and a trivial canonical transformation ($\lambda \neq 1 $ ) which has $M J M^T = \lambda J $ (for the given example, $M = \sqrt{\lambda} I $  which satisfies the condition).

Using same steps previously used in previous generalization, with $\frac{\partial G}{\partial t} = K-\lambda H $ in the general case, and retaining the equation $J \left(\nabla_\varepsilon \frac{\partial g}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t}  $, extended canonical transformation partial differential relations are obtained as:$$\begin{align} \left( \frac{\partial Q_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= -\lambda \left( \frac{\partial q_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \\ \left( \frac{\partial Q_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= \lambda \left( \frac{\partial p_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \end{align}$$$$\begin{align} \left( \frac{\partial P_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= \lambda \left( \frac{\partial q_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \\ \left( \frac{\partial P_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= -\lambda \left( \frac{\partial p_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \end{align}$$

Symplectic condition
Following the same steps to derive the symplectic conditions, as: $$\dot{\eta}=J\nabla_\eta H =J ( M^T \nabla_\varepsilon H) $$and $$\dot{\varepsilon}=M\dot{\eta} + \frac{\partial \varepsilon}{\partial t} =M J M^T \nabla_\varepsilon H + \frac{\partial \varepsilon}{\partial t} $$

where using $\frac{\partial G}{\partial t} = K-\lambda H $ instead gives:$$\dot{\varepsilon}=J \nabla_\varepsilon K  = \lambda J \nabla_\varepsilon H + J \nabla_\varepsilon \left( \frac{\partial G}{\partial t}\right) $$The second part of each equation cancel. Hence the condition for extended canonical transformation instead becomes: $M J M^T = \lambda J $.

Poisson and Lagrange Brackets
The Poisson brackets are changed as follows:$$\{u, v\}_\eta = (\nabla_\eta u)^T J (\nabla_\eta v) = (M^T \nabla_\varepsilon u)^T J (M^T \nabla_\varepsilon v) = (\nabla_\varepsilon u)^T M J M^T (\nabla_\varepsilon v) = \lambda (\nabla_\varepsilon u)^T J (\nabla_\varepsilon v) = \lambda \{u, v\}_\varepsilon$$whereas, the Lagrange brackets are changed as:

$$[u, v]_\varepsilon = (\partial_u \varepsilon )^T \,J\, (\partial_v \varepsilon) = (M \, \partial_u  \eta )^T \,J \, ( M \,\partial_v \eta) = (\partial_u \eta )^T\, M^TJ M\, (\partial_v \eta) = \lambda (\partial_u  \eta )^T\, J\,(\partial_v \eta) = \lambda [u, v]_\eta$$ Hence, the Poisson bracket scales by the inverse of $\lambda $ whereas the Lagrange bracket scales by a factor of $\lambda $.

Infinitesimal canonical transformation
Consider the canonical transformation that depends on a continuous parameter $$\alpha $$, as follows:

$$\begin{align} & Q(q,p,t;\alpha) \quad \quad \quad & Q(q,p,t;0)=q \\ & P(q,p,t;\alpha) \quad \quad \text{with} \quad & P(q,p,t;0)=p   \\ \end{align} $$

For infinitesimal values of $$\alpha $$, the corresponding transformations are called as infinitesimal canonical transformations which are also known as differential canonical transformations.

Consider the following generating function:

$$G_2(q,P,t)= qP + \alpha G(q,P,t) $$

Since for $$\alpha=0 $$, $$G_2 = qP $$ has the resulting canonical transformation, $$Q = q $$ and $$P = p $$, this type of generating function can be used for infinitesimal canonical transformation by restricting $$\alpha $$ to an infinitesimal value. From the conditions of generators of second type:$$\begin{align} {p} &= \frac{\partial G_{2}}{\partial {q}} = P + \alpha \frac{\partial G}{\partial {q}} (q,P,t) \\ {Q} &= \frac{\partial G_{2}}{\partial {P}} = q + \alpha \frac{\partial G}{\partial {P}}  (q,P,t) \\ \end{align}$$Since $$P = P(q,p,t;\alpha) $$, changing the variables of the function $$G $$ to $$G(q,p,t) $$ and neglecting terms of higher order of $$\alpha $$, gives: $$\begin{align} {p} &= P + \alpha \frac{\partial G}{\partial {q}} (q,p,t) \\ {Q} &= q + \alpha \frac{\partial G}{\partial p}  (q,p,t) \\ \end{align}$$Infinitesimal canonical transformations can also be derived using the matrix form of the symplectic condition.

Active canonical transformations
In the passive view of transformations, the coordinate system is changed without the physical system changing, whereas in the active view of transformation, the coordinate system is retained and the physical system is said to undergo transformations. Thus, using the relations from infinitesimal canonical transformations, the change in the system states under active view of the canonical transformation is said to be:

$$\begin{align} & \delta q = \alpha \frac{\partial G}{\partial p} (q,p,t)   \quad \text{and} \quad \delta p = - \alpha  \frac{\partial G}{\partial q}  (q,p,t), \\ \end{align} $$

or as $$\delta \eta = \alpha J \nabla_\eta G $$ in matrix form.

For any function $$u(\eta)  $$, it changes under active view of the transformation according to:

$$\delta u = u(\eta +\delta \eta)-u(\eta) = (\nabla_\eta u)^T\delta\eta=\alpha (\nabla_\eta u)^T J (\nabla_\eta G) = \alpha \{ u,G \}. $$

Considering the change of Hamiltonians in the active view, ie. for a fixed point,$$K(Q=q_0,P=p_0,t) - H(q=q_0,p=p_0,t) = \left(H(q_0',p_0',t) + \frac{\partial G_{2}}{\partial t}\right) - H(q_0,p_0,t) = - \delta H +\alpha \frac{\partial G}{\partial t} = \alpha\left(\{ G,H\}+\frac{\partial G}{\partial t} \right)=\alpha\frac{dG}{dt}   $$where $(q=q_0',p=p_0') $  are mapped to the point, $(Q=q_0,P=p_0) $  by the infinitesimal canonical transformation, and similar change of variables for $$G(q,P,t) $$ to $$G(q,p,t) $$ is considered up-to first order of $$\alpha $$. Hence, if the Hamiltonian is invariant for infinitesimal canonical transformations, its generator is a constant of motion.

Time evolution
Taking $$G(q,p,t)=H(q,p,t) $$ and $$\alpha = dt $$, then $$\delta \eta = (J \nabla_\eta H) dt = \dot{\eta} dt = d\eta $$. Thus the continuous application of such a transformation maps the coordinates $$\eta(\tau) $$ to $$\eta(\tau+t)  $$. Hence if the Hamiltonian is time translation invariant, i.e. does not have explicit time dependance, its value is conserved for the motion.

Translation
Taking $$G(q,p,t)=p_k $$, $$ \delta p_i = 0 $$ and $$ \delta q_i = \alpha \delta_{ik} $$. Hence, the canonical momentum generates a shift in the corresponding generalized coordinate and if the Hamiltonian is invariant of translation, the momentum is a constant of motion.

Rotation
Consider an orthogonal system for an N-particle system:

$$\begin{array}{l}\\ \end{array}$$

Choosing the generator to be: $$G=L_{z}=\sum_{i=1}^{n}\left(x_{i}p_{i y}-y_{i}p_{i x}\right) $$ and the infinitesimal value of $$ \alpha = \delta \phi $$, then the change in the coordinates is given for x by:

$$\begin{array}{c} {\delta x_{i}=\{x_{i},G\}\delta\phi=\displaystyle\sum_{j} \{x_{i},x_{j}p_{j y}-y_{j}p_{j x}\}\delta\phi=\displaystyle\sum_{j}(\underbrace{\{x_{i},x_{j}p_{j y}\}}_{=0} -{ \{x_{i},y_{j}p_{j x}\}}})\delta\phi\\ \end{array} $$

and similarly for y:

$$\begin{array}{c}

\delta y_{i}=\{y_{i},G\}\delta\phi=\displaystyle\sum_{j}\{y_{i},x_{j}p_{j y}-y_{j}p_{j x}\}\delta\phi=\displaystyle\sum_{j}(\{y_{i},x_{j}p_{j y}\}-\underbrace{ \{y_{i},y_{j}p_{j x}\}}_{=0})\delta \phi\\ {=\displaystyle\sum_{j}x_{j}\underbrace{\{y_i,p_{jy}\}}_{=\delta_{ij}} \delta\phi=x_{i}\delta\phi\,,} \end{array} $$

whereas the z component of all particles is unchanged: $$ \delta z_{ i }=\left\{z_{i},G\right\}\delta\phi=\sum_{j}\left\{z_{i},x_{j}p_{j y}-y_{j}p_{j x}\right\}\delta \phi =0$$.

These transformations correspond to rotation about the z axis by angle $$\delta \phi $$ in its first order approximation. Hence, repeated application of the infinitesimal canonical transformation generates a rotation of system of particles about the z axis. If the Hamiltonian is invariant under rotation about the z axis, the generator, the component of angular momentum along the axis of rotation, is an invariant of motion.

Motion as canonical transformation
Motion itself (or, equivalently, a shift in the time origin) is a canonical transformation. If $$\mathbf{Q}(t) \equiv \mathbf{q}(t+\tau)$$ and $$\mathbf{P}(t) \equiv \mathbf{p}(t+\tau)$$, then Hamilton's principle is automatically satisfied$$ \delta \int_{t_1}^{t_2} \left[ \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) \right] dt = \delta \int_{t_1 + \tau}^{t_2 + \tau} \left[ \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t+\tau) \right] dt = 0 $$since a valid trajectory $$(\mathbf{q}(t), \mathbf{p}(t))$$ should always satisfy Hamilton's principle, regardless of the endpoints.

Examples

 * The translation $$\mathbf{Q}(\mathbf{q}, \mathbf{p})= \mathbf{q} + \mathbf{a}, \mathbf{P}(\mathbf{q}, \mathbf{p})= \mathbf{p} + \mathbf{b}$$ where $$\mathbf{a}, \mathbf{b}$$ are two constant vectors is a canonical transformation. Indeed, the Jacobian matrix is the identity, which is symplectic: $$I^\text{T}JI=J$$.
 * Set $$\mathbf{x}=(q,p)$$ and $$\mathbf{X}=(Q,P)$$, the transformation $$\mathbf{X}(\mathbf{x})=R \mathbf{x}$$ where $$R \in SO(2)$$ is a rotation matrix of order 2 is canonical. Keeping in mind that special orthogonal matrices obey $$R^\text{T}R=I$$ it's easy to see that the Jacobian is symplectic. However, this example only works in dimension 2: $$SO(2)$$ is the only special orthogonal group in which every matrix is symplectic. Note that the rotation here acts on $$(q,p)$$ and not on $$q$$ and $$p$$ independently, so these are not the same as a physical rotation of an orthogonal spatial coordinate system.
 * The transformation $$(Q(q,p), P(q,p))=(q+f(p), p)$$, where $$f(p)$$ is an arbitrary function of $$p$$, is canonical. Jacobian matrix is indeed given by $$\frac{\partial X}{\partial x} = \begin{bmatrix} 1 & f'(p) \\ 0 & 1 \end{bmatrix}$$ which is symplectic.

Modern mathematical description
In mathematical terms, canonical coordinates are any coordinates on the phase space (cotangent bundle) of the system that allow the canonical one-form to be written as $$\sum_i p_i\,dq^i$$ up to a total differential (exact form). The change of variable between one set of canonical coordinates and another is a canonical transformation. The index of the generalized coordinates $q$ is written here as a superscript ($$q^{i}$$), not as a subscript as done above ($$q_{i}$$). The superscript conveys the contravariant transformation properties of the generalized coordinates, and does not mean that the coordinate is being raised to a power. Further details may be found at the symplectomorphism article.

History
The first major application of the canonical transformation was in 1846, by Charles Delaunay, in the study of the Earth-Moon-Sun system. This work resulted in the publication of a pair of large volumes as Mémoires by the French Academy of Sciences, in 1860 and 1867.

title: Canonical transformation


text:
Huell Howser

Huell Burnley Howser (October 18, 1945 – January 7, 2013) was an American television personality, actor, producer, writer, singer, and voice artist, best known for hosting, producing, and writing California's Gold and his human interest show Visiting... with Huell Howser, produced by KCET in Los Angeles for California PBS stations. The archive of his video chronicles offers an enhanced understanding of the history, culture, and people of California. He also voiced the Backson in Winnie the Pooh (2011).

Early life
Howser was born in Gallatin, Tennessee, on October 18, 1945, to Harold Chamberlain and Jewell Havens (Burnley) Howser. Howser's first name is a portmanteau of his parents' given names, Harold and Jewell, as Howser explained in the California's Gold episode "Smartsville."

Howser graduated from the University School of Nashville in 1963, then studied history and political science at the University of Tennessee, where he served as student body president.

Career
After serving in the U.S. Marine Corps Reserve and on the staff of U.S. Senator Howard Baker, Howser began his television career at WSMV-TV in Nashville, Tennessee, where he produced shows focused on human interest stories, such as Happy Features and The Happy World of Huell Howser. Howser was also a television personality working for the University of Tennessee.

After working in New York City as the host of WCBS-TV's Real Life show, Howser moved to Los Angeles, California, in 1981 to work as a reporter for KCBS-TV. During 1982 and 1983, he served as weekend host and correspondent for Entertainment Tonight. In 1983, he joined KCET (then a PBS affiliate) as host and producer of Videolog, a series of brief human-interest segments running less than 10 minutes each, that aired in between the station's shorter programs to fill up air time. "Videolog" eventually became one of the more popular programs on KCET, and in 1990, the show was expanded to half hour-long episodes. Included in Videolog was lint artist Slater Barron among other topics relevant to Los Angeles and adjacent communities.

California's Gold
In 1991, after spending his vacation driving across the Golden State and visiting with all 13 PBS stations in California, California's Gold premiered in April of that year. California's Gold highlights small towns, landmarks, events, or places of interest throughout California that are not well known to the general public. Howser conducted informal, often impromptu, interviews with locals involved with the sites he visited. He also produced California's Communities, California's Golden Fairs, Downtown, California's Water, California's Green, California's Golden Coast, California's Golden Parks, Road Trip, Visiting... with Huell Howser, California Missions, Palm Springs, Our Neighborhoods, The Bench, and various specials.

Visiting... with Huell Howser
Visiting... with Huell Howser was a weekly PBS series hosted by Howser that originally aired from 1991 until his retirement in 2012 that focused on "the diverse people, places, and events that make southern California such a unique community." , KCET, and other Southern California PBS continue to rerun this popular series. Episodes can be viewed on demand on the Chapman University and PBS affiliates' websites.

Other work
Articles written by Howser appeared in Westways, the magazine of the Automobile Club of Southern California.

In 1997, he featured prominently as himself alongside Tracey Ullman in character as Ruby Romaine in the Tracey Takes On... episode "Hollywood."

Howser spearheaded an unsuccessful effort to stop the demolition of buildings designed by Paul Williams at the Long Beach Naval Shipyard.

He appeared in Who Killed the Electric Car? (2006) in his capacity as a reporter, witnessing the demolition and shredding of a Honda EV Plus.

In 2011, Howser voiced the Backson in the post-credits scene of Walt Disney Animation Studios' feature film Winnie the Pooh.

Personal life
Howser lived in the historic El Royale apartments in Los Angeles, California, in an apartment which had previously been home to character actor William Frawley. Howser also had homes in Palm Springs and Twentynine Palms.

On June 29, 2015, Howser's Twentynine Palms home became available for rentals and weddings.

Howser mentioned that he was a Methodist during his episode covering the Nevada County Fair on California's Golden Fairs.

Volcano House
In 2003, Howser purchased the 1800 sqft Volcano House, situated on a volcanic cinder cone just outside Barstow in Newberry Springs, along with 60 acre of desert and a man-made lake. In 2010, Howser put the unusual residence on the market for $650,000.

In June 2012, The Panther, a student-run newspaper for Chapman University, announced that Howser had donated the Volcano House to the school. On September 3, 2015, Chapman University sold the Volcano House for $750,000.

Retirement and death
On November 27, 2012, The Sacramento Bee reported that Howser was retiring from making new shows, amid speculation in the television community that he was seriously ill.

On January 7, 2013, Howser died at his Palm Springs home, at the age of 67. He had been battling cancer for several years and his death certificate listed metastatic prostate cancer as the cause. Howser's body was cremated and his ashes were scattered at sea off the coast of Los Angeles County.

On January 15, 2013, a memorial was held for Howser, who said before his death that he did not want a funeral as he did not want attention.

Legacy
Howser donated his videotaped collection of California's Gold episodes, as well as those of his other series, to Chapman University in 2011. He also donated his personal papers, and a large collection of books on California history to the university. The school established the Huell Howser Archives, which, when completed, will offer the public free access to the entire digitized collection of his life's work. The archives can be accessed at Chapman University as well as on the internet. He also gave his extensive art collection, which consists mostly of "found-object" art collected during his travels, to the university, and endowed the California's Gold Scholarship Fund. Upon his death he bequeathed his remaining two homes to the university, the proceeds from the sale of which will be added to the scholarship fund.

Testimonials to Howser's unique contribution to the celebration of California history and culture were acknowledged in numerous media sources upon word of his death. Gustavo Arellano, then-editor of the OC Weekly, called Howser "the greatest Californian since Hiram Johnson," noting that for Howser, "California was the ultimate temple of the American dream." This reflects the high regard in which many Californians hold Howser's unabashedly enthusiastic promotion of their state's heritage.

In 2015, a Golden Palm Star on the Palm Springs, California, Walk of Stars was dedicated to him.

In popular culture
Howser's enthusiastic style as host of his various travel shows led to him being impersonated and lampooned by comedians and radio personalities, such as Adam Carolla, Dana Gould, Ralph Garman, and James Adomian.

Matt Groening has stated he is a fan of Howser and featured him in two episodes of The Simpsons: "There's Something About Marrying," in which a character named Howell Huser (voiced by Dan Castellaneta) falls off a turnip truck, and "O Brother, Where Bart Thou?," in which the real Howser presents a program similar to Marc Summers' Unwrapped television program titled Under the Wrapper. Howser received a voice credit for the episode. The Simpsons episode "A Test Before Trying" commemorated his death with a quick cartoon memorial shot at the end of the episode, stating: "In Memory of Huell Howser, Friend of the Simpsons and a friend of California."

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Huell Howser


text:
Talk:Symplectomorphisms

Wiktionary definition, if this is all

title: Talk:Symplectomorphisms


text:
Champion (supermarket)

Champion was a supermarket chain that operated in France and owned by the international retailer Carrefour. There were also Champion supermarkets in Belgium, Spain, Poland, Greece, Turkey, Argentina and Brazil.

History
In 1993, Champion became the main sponsor of the polka dot jersey in the Tour de France until 2008. From 2009, Carrefour took over the sponsorship.

In 1999, Champion is taken over by the Carrefour Group after the fusion of Promodès and Champion grows by absorbing the Stoc chain.

In 2006, Carrefour closed most Brazilian Champion stores and rebranded the remainder to the Carrefour Bairro. Also, all Spanish and Turkish Champion supermarkets were rebranded as Carrefour Express in 2006; some of them became Maxi DIA.

In 2008, Champion was the second largest supermarket chain in France, its annual turnover was €13.5bn. There were over 1,000 stores and 45,000 employees.

In 2008, it was announced that most stores would be rebranded to Carrefour Market between September 2008 and October 2009.

In 2009, there were 135 supermarkets, in 2011 there were less than 100 stores.

By 2013, only two Champion stores were left in European France (excluding French overseas islands), in Paris (18th arrondissement) and in Pouzac. In 2018, after a legal battle filed against Carrefour was concluded, the last remaining (Paris) store was also rebranded, marking the end of the "Champion" supermarket brand in Europe.

Champion in Belgium
In Belgium, the Champion stores are owned by Carrefour and Groupe Mestdagh. In 2009, there were 67 Champion stores. From 2013, it was decided by Carrefour that most Champion stores in Belgium would be rebranded as Carrefour Market-Groupe Mestdagh.

As of July 2013, there are 21 stores were left ; 7 in Hainaut, 5 in Namur, 4 in Liège, 3 in Walloon Brabant, 1 in Limburg and Luxembourg. The rebranding was complete the following year (2014) when the last single remaining Champion store in Wallonia was renamed Carrefour.

title: Champion (supermarket)


text:
The Eleventh Hour (Canadian TV series)

The Eleventh Hour is a Canadian television drama series which aired weekly on CTV from 2002 to 2005.

The show revolves around the reporters and producers at a fictional television news magazine series, The Eleventh Hour. Unhappy with the newsmagazine's shrinking audience, the network has brought in a new executive producer, Kennedy Marsh, to reorient the show in a more ratings-driven tabloid journalism direction.

The tension between the ratings imperative and the more traditional journalistic ethics of the show's senior staff is the primary conflict that drives the show, but storylines also include the team's efforts to get the stories that will make it to air each week.

The Eleventh Hour was produced by Alliance Atlantis, Canada's largest film and television production house. It aired in the U.S. on Sleuth, under the title Bury the Lead, to distinguish it from a CBS series with a similar name.

Ratings
Although the show started off poorly in the Canadian television ratings, attracting an average of just 400,000 viewers each week, its audience during the start of its 2004 season was sufficiently high that CTV ordered a third season. However, ratings did not improve further. The third season, already ordered, aired irregularly on Saturday nights with very little promotion, and proved to be its last. The final season also received a Best Dramatic Series Gemini Award despite the cancellation.

The Eleventh Hour was the latest in a lengthy line of drama series at CTV which had high-profile launches but were quickly marginalized to little-watched weekend timeslots and ultimately cancelled. However, it outlasted several other recent CTV dramas, including Power Play, The City, and The Associates, which each lasted two seasons.

The show's theme song for season 3 is "Weapon", by Matthew Good.

Cast

 * Sonja Smits as correspondent Megan Redner
 * Shawn Doyle as producer Dennis Langley
 * Waneta Storms as producer Isobel Lambert
 * Tanya Reid as executive producer Kennedy Marsh
 * Jeff Seymour as correspondent Kamal Azizi
 * Peter MacNeill as network news head Warren Donohue
 * Inga Cadranel as researcher Brooke Fairburn
 * Scott McCord as researcher James Joy
 * Matt Gordon as legal counsel Murray Dann
 * Jonas Chernick as editor Gavin Kowalchuk
 * John Neville as interviewer Deaton Hill

Main Crew

 * Creators: Semi Chellas, Ilana Frank
 * Executive Producers: Ilana Frank, Semi Chellas, David Wellington
 * Directors: David Wellington, Kelly Makin, Philip Earnshaw, T.W. Peacocke, Graeme Campbell, Milan Cheylov, Stephen Reynolds
 * Writers: Semi Chellas, Esta Spalding, Tassie Cameron, Frank Borg, Sean Reycraft, Peter Wellington, Karen Walton, Morwyn Brebner

Awards and nominations
The show was nominated for 14 Gemini Awards in 2003, and won for Best Dramatic Series, Best Actor in a Leading Role (Seymour) and Best Supporting Actor (MacNeill).

The show was nominated for Best Dramatic Series Gemini Awards again in 2004. In 2005, the show won Gemini Awards for Best Dramatic Series and for Best Writing for Semi Chellas and Tassie Cameron.

Season 1

 * 1) Mad As Hatters
 * 2) I'm Mad As Hell
 * 3) The Source
 * 4) A Low, Dishonest Decade
 * 5) Tree Hugger
 * 6) A Modern Mata Hari
 * 7) Not Without My Reefer
 * 8) The 37-Year Itch
 * 9) Don't Have a Cow
 * 10) Shelter
 * 11) Cell Phone Slaves
 * 12) The RASH Troops of Error
 * 13) Hall of Mirrors

Season 2

 * 1) Cowboy
 * 2) Stormy Petrel
 * 3) Hard Seven
 * 4) Swimmers
 * 5) Wonderland
 * 6) Gone Baby Gone
 * 7) Nadir
 * 8) Rather Be Wrong
 * 9) Georgia
 * 10) I'll Build Me An Island
 * 11) Strange Bedfellows
 * 12) The Revenge Specialist
 * 13) The Missionary Position

Season 3

 * 1) Eden
 * 2) In Spite of All the Damage
 * 3) Megan Ice Cream
 * 4) Bedfellas
 * 5) A Virgin Walks into a Bar
 * 6) ZUGZWANG
 * 7) Hit Delete
 * 8) Kettle Black
 * 9) In Another Life
 * 10) The Miracle Worker
 * 11) Special Delivery
 * 12) Das Bootcamp
 * 13) Bumpy Cover

title: The Eleventh Hour (Canadian TV series)


text:
MNA

Mna or MNA may refer to:

 * Mna, a Latin and Greek form of the Mina (unit), a unit of weight and money

Acronyms

 * Ma'an News Agency, Palestinian Territories
 * Magyar Nemzeti Arcvonal (Hungarian National Front)
 * Mehr News Agency, Iran
 * Member of the National Assembly (Quebec)
 * Member of the National Assembly of Pakistan
 * Merpati Nusantara Airlines, ICAO code
 * Midland News Association, UK
 * Minnesota Nurses Association, labor union, US
 * Missouri and Northern Arkansas Railroad's reporting mark
 * Mna, a village in Georgia
 * Modified nodal analysis of electric circuits
 * Mouvement national algérien (Algerian National Movement), 1950s
 * Mouvement national de l'Azawad (National Movement for the Liberation of Azawad)
 * Museo Nacional de Antropología, the national museum of anthropology, Mexico
 * Muslim National Associations, a Muslim pro-Zionist organization
 * Myanmar National Airlines, the national flag carrier of Myanmar
 * Myanmar News Agency, the official news agency in Burma

title: MNA


text:
User:Cyan/sign

--                      /|  /|  |                          |                       ||__||  |       Please don't       | /  O O\__           feed           | /         \       the trolls        | /     \     \                        |                   /   _    \     \ --                  /    |\____\     \     ||                 /     | | | |\____/     ||                /       \|_|_|/   |    __||               /  /  \            |____| ||              /   |   | /|        |      --|              |   |   |//         |____  --|       * _    |  |_|_|_|          |     \-/    *-- _--\ _ \     //           |      /  _     \\ _ //   |        /    *  /   \_ /- | -     |       |      *      ___ c_c_c_C/ \C_c_c_c____________

title: User:Cyan/sign


text:
Talk:Neocortex

Untitled
This nugget belongs with Brain, with a redirect there.

 * I think that there's plenty of information about the neocortex that could be added here instead. For example, an overview of the different types of neocortical cells (pyramidal, chandelier, spiny stellate, aspiny stellate, etc.--  Some good overview papers came out a few years ago, and I'll post some links if anyone is interested.)  Also, a note about the evolution of the neocortex (is it sometimes called the "new mamallian brain" because some mammals don't have one?) MisterSheik 22:58, 8 August 2005 (UTC)

 * I think all mammals have neocortex. Here's an article that mentions "the neocortex of various species, representative of the major subdivisions of mammals".  Afaik it's called "new" just because its structure differs from that of the areas of cerebral cortex that are similar in all vertebrates.dsws 20:41, 14 December 2005 (UTC)
 * All mammals have neocortex, and some non-mammals do as well maybe. Neocortex is called "new" because it is phylogenetically newer than the rest of the CNS. Neocortex refers to any 6-layered cortex. Semiconscious 21:39, 15 December 2005 (UTC)
 * Cetacean intelligence claims that the dolphin only possesses 5 layers of the neocortex, but that layers I and VI are present among those. Is one of the six layers then missing in the dolphin? —The preceding unsigned comment was added by Wdfarmer (talk • contribs) 05:00, 29 April 2007 (UTC).

Neocortex still viable?
Is the notion of a neocortex still viable? In The Emotional Brain, Joseph LeDoux writes, "Anatomists like Harvey Karten and Glenn Northcutt were showing that so-called primitive creatures do in fact have areas that meet the structural and functional criteria of neocortex...As a result of these discoveries, it is no longer possible to say that some parts of the mammalian cortex were older than other parts. And once the distinction between old and new cortex breaks down, the whole concept of mammalian brain evolution is turned on its head." (100). --Mediaphd 02:00, 18 May 2006 (UTC)
 * I think the distinguishing characteristic of the neocortex is the six-layer sturcture and presence of neocortical columns, and while earlier mammals may have functional equivalents, they lack these specific structural details Tarcieri 02:20, 10 July 2006 (UTC)
 * Is neocortex so named because it is assumed to have developed later because it was not (thought) to be found in more 'primative' animals? Or can its devlopment actually be traced back? Perhaps neocortex is new, but both mammals and other animals have seperately developed structures which could be considered isocortex, and may be considered 'neocortex' for their respective classes. Consequently, the implication in the definition of mammalian neocortex as being evolutionarily newer than all non-mammalian cortex may be misleading, and the the comparison between mammalian and non-mammalian cortical 'ages' is misguided.Keepstherainoff (talk) 15:57, 18 July 2010 (UTC)
 * Isocortex and neocortex mean essentially the same thing. Neocortex is supposedly "new" in comparison to paleocortex (olfactory cortex) and archicortex (the hippocampus).  Paleocortex has a similar form across the full range of vertebrates.  Archicortex has changed its appearance dramatically in mammals but can be matched up with the "medial pallium" zone in nonmammals, as explained in our hippocampus article.  Neocortex, however, doesn't clearly match up with anything in non-mammals.  The deep layers of neocortex match up with a zone called the dorsal pallium, but the superficial layers seem to come out of nowhere.

 * What makes it difficult is that when we trace back the evolution of mammals, we face a huge "missing link" problem. Mammals are synapsids, which diverged from diapsids (birds, reptiles, dinosaurs) over 300 million years ago.  There are no existing synapsids other than mammals, but the earliest mammals come from 120-150 million years ago.  Furthermore, all mammals have a neocortex with essentially the same structure.  Thus, the neocortex took shape during a period from 300-150 million years ago about which we have no useful information -- fossils are useless for understanding this. Looie496 (talk) 18:26, 18 July 2010 (UTC)
 * The fish turned into birds 2600:6C48:757F:3F2B:75D1:AE06:3950:9B7D (talk) 12:30, 13 January 2024 (UTC)

80% of the brain's weight
I'm going to go ahead and flag this with Fact. According to the article:

"In humans [the neocortex] accounts for about 80% of the brain's weight."

However, according to this site:

http://faculty.washington.edu/chudler/facts.html

Average number of neurons in the brain = 100 billion Number of neocortical neurons (females) = 19.3 billion Number of neocortical neurons (males) = 22.8 billion

How is it that the neocortex could comprise approximately 20% of the brain's neurons but 80% of its weight?

I'll also go ahead and incorporate that information into the article. Tarcieri 02:20, 10 July 2006 (UTC)

Never write "go ahead and" again! Useless. Dumb. —Preceding unsigned comment added by 70.181.119.79 (talk) 23:33, 2 April 2010 (UTC)

RE: 80% of the brain's weight
Neurons are not the heaviest part of the brain... Glia weigh a lot too, and there is a great deal of myelination in the neocortex. but citation is still needed.

changed form percent to total number
I changed the part under anatomy from percentage of neurons in male and females to total number. Writing it as percentage assumes that the average man and the average woman have the same total number of neurons. The citation only talks about the average brain and does not specifiy whether men and women have the same total number of neurons. Correct me if I am wrong.

Images
Folks, it would be nice if someone could add an image here. I myself always wanted to see what really the neocortex is, but never found an image! —The preceding unsigned comment was added by 200.196.125.95 (talk) 02:52, 15 February 2007 (UTC).
 * I added a few images and hope to rouse myself up to expanding the article a bit.Colincbn (talk) 02:30, 12 November 2009 (UTC)
 * That would be great, this is a pretty important article. Looie496 (talk) 04:47, 12 November 2009 (UTC)

Image of the 6 layers ?
Can any of the specialists in making images come to help ? Thy, — Preceding unsigned comment added by SvenAERTS (talk • contribs) 11:19, 17 July 2027 (UTC)
 * You're probably better asking at Requested pictures - this help tag is more suited to requesting basic editing help than asking for specific contributions. stwalkerster (sock | talk) 10:19, 18 July 2023 (UTC)
 * Thy let me try ... SvenAERTS (talk) 14:11, 19 September 2023 (UTC)

male and female brain differences
could we remove this reference to different numbers of neocortical neurons in males and females? it is a silly fact (men are larger, and therefore have greater numbers of almost every cell type), and without context could come across as needlessly provocative (especially with the silly comment about "what effects this might have"). —Preceding unsigned comment added by PhineasG (talk • contribs) 06:25, 29 January 2008 (UTC)

This is what Feminism does to science. It is fact and not pointless that number of neurons is different for men and women. Difference being in brain it might have (most likely has) some effect on something.

Fact that men have more cells in general because they are larger is completely pointless and misleading. Mwn might have more muscle mass aswell ? what you seem to imply is that it does not make any difference ?

It is sad that we need to add "context" so women and some men would not be provoked by a simple scientific fact, when political correctness starts affecting how we write wikipedia articles then all is lost ? — Preceding unsigned comment added by 88.113.20.137 (talk) 09:57, 3 July 2014 (UTC)

 * I agree. If you give a brain to neurosurgeons, doesn't the original poster think they can't see the difference between a male or female brain and that the EEC's and frmi scans of a female versus male brain are different and respond differently?! 2A02:A03F:618D:6300:1142:E5B9:8C99:9BAB (talk) 11:10, 17 July 2023 (UTC)

Image redux
Thanks Traveler100 for adding an image, but, well, um, I don't like that image. Can I look around for a different one, showing the cell layering structure or something? Looie496 (talk) 19:10, 13 February 2011 (UTC)

Gyrification
Gyrification does not occur only because of spatial demands. This has been changed. Further illustration for doubters is the following: Macaques have partly and inhomogeneously gyrified cortex, which means there are other demands than spatial which determine the gyrification, because if space was the only demand, the cortex would be homogeneously gyrified. Furthermore, although I can't find the article, there exists a report on the consequence of removing part of the developing sheep cortex. This supposedly does not lead to altered gyrification, whereas selective severing of specific short distance white matter tracts does.

I also removed the part that stated that gyrification allowed for the development of working memory and speech and language. First of all, all animals have working memory. Second, there is absolutely no proof for this. — Preceding unsigned comment added by 86.89.79.42 (talk • contribs)

The last two sentences of the first paragraph under the Anatomy section state: "The mechanism by which the gyri form during embryogenesis is not entirely clear. However, it may be due to differences in cellular proliferation rates in different areas of the cortex early in embryonic development." I think the first sentence is fine, but the second sentence gives the impression that this is a singular hypothesis regarding gyri development. There are many such hypotheses, the most popular of which is probably Van Essen's (1997) theory of axonal tension. There is also the Richman (1975) theory of cortical buckling. The Ronan paper is a good one, but I don't see why we're giving this (rather new, by wikipedia standards) research special treatment in this section. If anything, this should state that there are several hypotheses. I recommend the removal of the last sentence. nben (talk) 20:53, 28 March 2014 (UTC)

Proposed merge with Proisocortex
Proisocortex is a stub and subtopic of neocortex where page would benefit from its inclusion Iztwoz (talk) 13:57, 5 May 2014 (UTC) ✅
 * Support, seems logical and stub is over 10 years old. --Spike Wilbury (talk) 16:11, 5 May 2014 (UTC)
 * Support. I agree. --Tryptofish (talk) 00:06, 10 May 2014 (UTC)

Etymology
I believe "cortex" in Latin can also mean "outer layer" or "shell". If I'm correct, calling neocortex "new rind" seems needlessly whimsical. At least, whimsical to anyone who imagines munching on a brain as whimsy. More importantly, it's uninformative unless it's known that the coiner had that etymology in mind. TalkPageWarrior (talk) 21:48, 1 March 2017 (UTC) TalkPageWarrior (talk) 21:48, 1 March 2017 (UTC)

Layer IV lacking in motor cortex is a misconception
See: 2014: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4201116/ 2014: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4290446/ 1997 (pdf): https://watermark.silverchair.com/070178.pdf?token=AQECAHi208BE49Ooan9kkhW_Ercy7Dm3ZL_9Cf3qfKAc485ysgAAAaUwggGhBgkqhkiG9w0BBwagggGSMIIBjgIBADCCAYcGCSqGSIb3DQEHATAeBglghkgBZQMEAS4wEQQM10TPK2t1aNgelOIuAgEQgIIBWIx7xdqlL5y2HYd4WZ4HyYXY9ddzfd2sMPVwVZn66vCYK39ByqaRF4no5pkkcIzp8KIQcd5QRoGEsbMIXuNT9EbtTDKmuwhnsvZu8Wp6Sat2bFXnIYJGzQJRrshMhrtLyDAjTs1xxOE1MguOrkQSN88h3l_1wN5hacYglYY6O_IjynGP0pltuFkA4aDZ5ysrHEoaloaNvEP8pd4axvZ_0QFTaX-3jMofsZSpGH52eFy8YXNrXQzGfLI-myRZXpZ_wgK50FXLFxSXA3LjorDO-HIG2aO2VUGd06XWGzCT-8zm2P00329rneKzkA1RqzKho16mtfATWt7L0Q2bkesHSS0WKywSPEIwP7KPbKOyDdy42CrknlRuoRPQyPgupBhgFX1DJPuy5lzuwB_QyLpuOyy-Azid4I-jp2RfYLn-XblV4Kbf9jm5KlDnXX2OBjb2OhDsDi5aF4Ep — Preceding unsigned comment added by 2600:1010:B00D:F5C:BD95:65E7:73E8:1B4C (talk) 23:53, 22 April 2018 (UTC)

Totally ridiculous
Wikipedia is very close to losing credibility with the poor quality of the images in articles across the project, rendering it unencyclopedic. Why let this article continue to slide (since 2011!) without a decent image such as this one, which actually shows where this structure is in the brain? 173.88.246.138 (talk) 04:31, 15 September 2020 (UTC)

title: Talk:Neocortex


text:
ΜClinux

μClinux is a variation of the Linux kernel, previously maintained as a fork, that targets microcontrollers without a memory management unit (MMU). It was integrated into the mainline kernel as of 2.5.46; the project continues to develop patches and tools for microcontrollers. The homepage lists Linux kernel releases for 2.0, 2.4 and 2.6 (all of which are end-of-life in mainline).

The letters "μC" are for "microcontroller": the name is pronounced "you-see-Linux", rather than pronouncing the letter mu as in Greek.

History
μClinux was originally created by D. Jeff Dionne and Kenneth Albanowski in 1998. Initially, they targeted the Motorola DragonBall family of embedded 68k processors (specifically the 68EZ328 series used in the 3Com PalmPilot) on a 2.0.33 Linux kernel. After releasing their initial work, a developer community quickly sprang up extending their work to newer kernels and other microprocessor architectures. In early 1999, support was added for the Motorola (now NXP) ColdFire family of embedded microprocessors. ARM processor support was added later.

Although originally targeting 2.0 series Linux kernels, it now has ports based on Linux 2.4 and Linux 2.6. The Linux 2.4 ports were forward ported from the 2.0.36 Linux kernel by Michael Leslie and Evan Stawnyczy during their work at Rt-Control. There were never any μClinux extensions applied to the 2.2 series kernels.

Since version 2.5.46 of the Linux kernel, the major parts of μClinux have been integrated with the mainline kernel for a number of processor architectures.

Greg Ungerer (who originally ported μClinux to the Motorola ColdFire family of processors) continued to maintain and actively push core μClinux support into the 2.6 series Linux kernels. In this regard, μClinux is essentially no longer a separate fork of Linux.

μClinux had support for many architectures, and forms the basis of many products, like network routers, security cameras, DVD or MP3 players, VoIP phone or gateways, scanners, and card readers.

Support for several of the original target architectures was dropped in 2018. The obsolete CPU architectures set to be removed in Linux 4.17 and subsequent releases included ADI Blackfin, Etrax CRIS, Fujitsu FR-V, Mitsubishi M32R, Matsushita/Panasonic MN10300, Imagination META (Metag), and Tilera TILE.

Hardware projects
The leanXCam was an open-source programmable smart camera used for industrial applications in the field of machine vision that ran under μClinux; the camera won an award at the 2008 VISION tradeshow. As of 2015, the project was discontinued.

Supported architectures
The current list includes:
 * Altera Nios/Nios II
 * Amber (open FPGA core)
 * ARM ARM7TDMI, ARM Cortex-M3/M4/M7, ARM Cortex-R
 * Lattice Mico32
 * NXP 680x0 (Motorola/Freescale 680x0)
 * Hyperstone E1/E2 (called hyLinux)
 * Intel i960
 * MIPS
 * NXP ColdFire (Motorola/Freescale ColdFire)
 * Xilinx MicroBlaze

No longer supported

 * NEC V850E - removed in Linux 2.6.27.
 * Hitachi/Renesas H8 ( h8300 ) - removed in Linux 5.19.

Before Linux 4.17 the following architectures were also supported:
 * ADI Blackfin ( blackfin )
 * Axis ETRAX ( cris )
 * Fujitsu FR-V ( frv )
 * Mitsubishi/Renesas M32R ( m32r )

title: ΜClinux


text:
Equality Party (Quebec)

The Equality Party (Parti Égalité) was a political party in Quebec, Canada, that promoted the use of English in Quebec on an equal basis with French. Four Equality Party members were elected to Quebec's National Assembly in 1989, as part of an anglophone reaction to changes made by the governing Liberals to Quebec's language law. The party had no success in subsequent elections, and stopped organizing after the 2003 Quebec election.

Foundation to 1989 election
The party was formed in 1989 as a reaction to then-Premier Robert Bourassa invoking the "Notwithstanding clause" of the Canadian constitution to override a Supreme Court ruling overturning parts of the Charter of the French Language (commonly known as "Bill 101"). The court ruling would have allowed languages other than French to appear on store signs and outdoor advertising; the government instead allowed other languages only on small signs inside shops. Another issue that fed Equality Party support was the Quebec Liberal Party's forcing anglophone Members of the Quebec National Assembly (MNAs) to condemn a report from the Official Languages Commissioner suggesting Quebec anglophones felt "humiliated" by laws such as Bill 101.

The Equality Party's platform called for equality of both languages (French and English) in Quebec, opposing Bill 101 which made French the sole official language of Quebec, imposed restrictions on the use of English on public signs, and required children to attend school in French unless one of their parents went to school in English in Canada. The Equality Party drew virtually all of its support from elements of Quebec's anglophone minority, and only ran candidates in electoral districts with very high anglophone populations.

The party first came to prominence in the 1989 general election, when it won four seats on Montreal Island in the National Assembly with 3.7% of the popular vote. Along with its then-sister party, the Unity Party (which ran candidates outside the Montreal Island), it won 4.7% of the provincial popular vote. The winning candidates were Gordon Atkinson, Neil Cameron, Richard Holden and party leader Robert Libman, who won popular votes ranging from 41 percent to 58 percent in their respective ridings.

In the National Assembly 1989-1994
The party did not receive official party status in Quebec's National Assembly, being eight members short of the required twelve necessary for recognition, nor had they received 20% of the popular vote which would have otherwise qualified them for official status. The Liberals and Parti Quebecois agreed that the Equality Party caucus would receive some of the privileges of an "official party", including having their members' seats in the National Assembly placed together and office space allocated close to each other, as well as some research funding; however, they were not allocated a guaranteed number of questions in the National Assembly's daily Question Period, meaning that their opportunity to speak was left to the Speaker's discretion. (This arrangement was repeated in later elections when Action démocratique du Québec won fewer seats in the National Assembly than required for official status).

The four members took an active role in National Assembly debates, most notably when Party leader Robert Libman made headlines by using his parliamentary privilege to reveal the details of confidential, money-losing contracts signed between Hydro-Québec and some of Quebec's aluminum producers.

The party voted against Bill 150, a law providing for a referendum on sovereignty for Quebec, (which was later cancelled in favour of a referendum on the Charlottetown Accord).

The Unity Party merged with the Equality Party on May 7, 1990, increasing the party's membership to what the party reported as 16,000. The party's published financial statements, however, reported that only $7795 in membership dues were collected in 1990, down 75% from 1989 and enough to account for only 1,559 dues-paying members.

From 1990 onward, the party suffered from frequent and public infighting, clashing over personalities and over issues such as whether to support the Meech Lake Accord and Charlottetown Accord. This peaked when Richard Holden, Equality MNA for the overwhelmingly Anglophone Westmount riding, left the party and eventually defected to the Parti Québécois, a party which had little support in Westmount due to its support of Bill 101 and Quebec independence. Three of the Equality Party's four elected members, including Libman, the party's leader, quit the party before the next election.

In addition to the infighting and defections, the Equality Party's raison d'être arguably vanished when the government amended the Charter of the French Language in 1993 to allow for more prominent English language text on commercial signs, so that the law no longer needed the use of the "notwithstanding clause" to withstand constitutional challenge. Furthermore, opinion polls prior to the 1994 election showed a close race between the two largest parties (the Liberals and the Parti Québécois), with the PQ explicitly promising an independence referendum if it was victorious. All of these factors encouraged Equality Party voters from 1989 to switch their support back to the Liberals in 1994.

Post-1994 activities
The Equality Party never repeated its electoral success of 1989. Cameron, the party's sole remaining MNA, was defeated in the 1994 general election. Two subsequent general elections in 1998 and 2003 did nothing to improve the party's fortunes.

The party continued to exist for another decade in a desultory fashion. Its remaining members held meetings, maintained a website and ran candidates for office until 2003. During these years, the party promoted political positions such as the reversal of the 2002 municipal mergers on the island of Montreal. Following the party's poor showing in the 2003 election, its leader, Keith Henderson, announced he would resign once a new leader was chosen, which never occurred. The party stopped holding meetings, updating its website or running candidates in elections, but continued to file annual reports with the Director General of Elections until it was removed from the list of registered political parties at the end of 2012.

In 2013, a group attempted to restart the party under the name of "Equality Party 2.0 - Parti Égalité 2.0", but abandoned efforts due to lack of support.

Leaders

 * Robert Libman (1989–1994)
 * Keith Henderson (1994-2012)

Members of the Quebec National Assembly

 * Gordon Atkinson (1989–1994)
 * Neil Cameron (1989–1994)
 * Robert Libman (1989–1994)
 * Richard Holden (1989–1992)

title: Equality Party (Quebec)


text:
List of fleets

The following is a list of fleets of navies from around the world.

Australian fleet

 * Fleet Command (Australia)

Canadian fleets

 * Commander, Canadian Fleet Atlantic (Maritime Forces Atlantic) (HQ Halifax, Nova Scotia)
 * Commander, Canadian Fleet Pacific (Maritime Forces Pacific) (HQ Esquimalt, British Columbia)

Historic

 * Chinese treasure fleet
 * Beiyang Fleet
 * Nanyang Fleet

Modern People's Liberation Army Navy fleets

 * East Sea Fleet
 * North Sea Fleet
 * South Sea Fleet

Oceanic Fleets

 * Naval Force of the Caribbean (HQ, Cartagena)
 * Naval Force of the Pacific (HQ, Bahía Málaga near Buenaventura)

Riverine Flotillas

 * Naval Force of the Orinoco (HQ, Puerto Carreño), Colombian part of the Orinoco and its tributaries.
 * Naval Force South (HQ, Puerto Leguízamo), Colombian part of the Amazon and its tributaries, mainly the Caquetá and Guainía

German fleets

 * High Seas Fleet – historic

Naval Commands

 * Western Naval Command
 * Eastern Naval Command
 * Southern Naval Command

Fleets

 * Western Fleet
 * Eastern Fleet

Indonesian fleets

 * 1st Fleet Command (HQ Jakarta)
 * 2nd Fleet Command (HQ Surabaya)
 * 3rd Fleet Command (HQ Sorong)

Iranian fleets

 * Northern Fleet (Iran)
 * Southern Fleet (Iran)

Pre World War I

 * Combined Fleet (聯合艦隊 Rengō Kantai?)
 * 1st Fleet (HQ Hashira-jima, Yamaguchi)
 * 2nd Fleet
 * 3rd Fleet (HQ Babeldaob, Palau)
 * 4th Fleet (HQ Truk, Micronesia)
 * Eastern Fleet (東部艦隊 Tōbu-Kantai?)
 * Guard Fleet (警備艦隊 Keibi Kantai?)
 * Medium Fleet (中艦隊 Chū-Kantai?)
 * Northern Expeditionary Fleet (北伐艦隊 Hokubatsu Kantai?)
 * Small Fleet (小艦隊 Shō-Kantai?)
 * Western Fleet (西部艦隊 Seibu-Kantai?)

World War I

 * Combined Fleet (聯合艦隊 Rengō Kantai?)
 * 1st Fleet
 * 2nd Fleet
 * 3rd Fleet

World War II

 * Combined Fleet, the main combatant component of the Imperial Japanese Navy during World War II.
 * 1st Fleet (HQ Hashira-jima, Yamaguchi)
 * 2nd Fleet
 * 3rd Fleet (HQ Babeldaob, Palau)
 * 4th Fleet (HQ Truk, Micronesia)
 * 5th Fleet
 * 6th Fleet (HQ Kwajalein, Marshall Islands)
 * Southern Expeditionary Fleet
 * 10th Area Fleet
 * Central Pacific Area Fleet
 * China Area Fleet, tasked with guarding the Chinese coastal area, had the same organizational level as the Combined Fleet
 * Northeast Area Fleet
 * Southeast Area Fleet

Modern Day

 * Self Defense Fleet, the main combatant component of the Japan Maritime Self-Defense Force.

North Korean fleets

 * East Coast Fleet headquartered at Toejo
 * West Coast Fleet headquartered at Nampo

Active fleets

 * Baltic Fleet
 * Black Sea Fleet
 * Caspian Flotilla
 * Russian Northern Fleet
 * Pacific Ocean Fleet (Russia)

Inactive fleets, Soviet and Russian

 * Amur Military Flotilla
 * Arctic Sea Flotilla
 * Soviet Red Banner Northern Fleet

South Korean fleets

 * Republic of Korea First Fleet (Headquarters Donghae) – East Sea
 * Republic of Korea Second Fleet (Headquarters Pyeongtaek) – Yellow Sea
 * Republic of Korea Third Fleet (Headquarters Mokpo) – South Sea
 * Submarine Force (HQ Jinhae, Jeju)
 * Component Flotilla FIVE (Headquarters Jinhae)
 * Mobile Flotilla SEVEN (Headquarters Jeju)
 * Mobile Squadron SEVEN ONE
 * Mobile Squadron SEVEN TWO
 * Fleet Combat Training Group EIGHT (Headquarters Jinhae)

Saudi Arabian fleets

 * Eastern Fleet (Saudi Arabia)
 * Western Fleet (Saudi Arabia)

Spanish

 * Spanish treasure fleet (HQ Seville)
 * Manila galleon (HQ Manila)
 * New Spain fleet (HQ Havana)
 * Royal Galley Fleet (HQ Cartagena and Naples)

Modern Day

 * La Flota (The Fleet), the main combatant component of the Spanish Navy (HQ Rota)
 * Canary Islands squadron (HQ Las Palmas)
 * Submarine flotilla (HQ Cartagena)

Swedish fleets

 * Coastal Fleet

Turkish fleets

 * Turkish Fleet Command
 * Northern Sea Area Command, İstanbul
 * Istanbul Strait Command, Anadolukavağı, İstanbul
 * Çanakkale Strait Command, Nara, Çanakkale
 * Black Sea Area Command, Karadeniz Ereğli, Zonguldak
 * Underwater Search and Rescue Group Command, Beykoz, İstanbul
 * Rescue Group Command
 * Underwater Defence Group Command
 * Naval Hydrography and Oceanography Division Command, Çubuklu, İstanbul
 * Bartın Naval Base Command, Bartın
 * Naval Museum Command, Beşiktaş, İstanbul
 * Istanbul Naval Shipyard Command, Pendik
 * Southern Sea Area Command, İzmir
 * Amphibious Task Group Command, Foça, İzmir
 * Amphibious Marine Brigade Command, Foça, İzmir
 * Amphibious Ships Command, Foça, İzmir
 * Aksaz Naval Base Command, Aksaz Naval Base, Marmaris
 * Mediterranean Area Command, Mersin
 * İskenderun Naval Base Command, İskenderun, Hatay
 * Aegean Sea Area Command, İzmir
 * Foça Naval Base Command, Foça, İzmir
 * Maintenance, Repair and Engineering Command, İzmir

Pre World War I

 * Atlantic Fleet – Atlantic Ocean, English Channel
 * Baltic Fleet, Baltic Sea
 * Channel Fleet – English Channel
 * Downs Station – Southern, North Sea
 * Irish Fleet Irish Sea
 * North Sea Fleet – North Sea
 * Reserve Fleet

World War I

 * Battle Cruiser Fleet – North Sea
 * First Fleet
 * Grand Fleet (HQ Scapa Flow, Orkney, Scotland) – North Sea
 * Mediterranean Fleet (HQ Malta) – Mediterranean Sea
 * Second Fleet
 * Third Fleet
 * Reserve Fleet

World War II

 * Home Fleet (HQ Scapa Flow, Orkney, Scotland) – North Sea
 * Mediterranean Fleet (HQ Alexandria, Egypt) – Mediterranean Sea
 * Eastern Fleet (HQ Trincomalee, Ceylon) – Indian Ocean
 * British Pacific Fleet (HQ Sydney, Australia) – South West Pacific
 * East Indies Fleet, (HQ Trincomalee, Ceylon) – Indian Ocean
 * Reserve Fleet

Post World War II

 * East Indies Fleet (HQ Trincomalee, Ceylon) – Indian Ocean
 * Far East Fleet (HQ Singapore) (to 1971)
 * Reserve Fleet
 * western Fleet (HQ Northwood, England) (1967–71)
 * Commander-in-Chief Fleet (1971–2002)

After 2002 a four-star commander-in-chief ceased to be appointed and the responsibilities of the post were transferred to the three-star Fleet Commander.

Naval commands

 * United States Fleet Forces Command (formerly Atlantic Fleet)
 * United States Naval Forces Central Command
 * United States Naval Forces Europe - Naval Forces Africa
 * United States Pacific Fleet
 * United States Pacific Fleet

Modern US Navy fleets
US Navy fleets are numbered odd in the Pacific or West, and even in the Atlantic or East:
 * United States Second Fleet (HQ Norfolk, Virginia) – North Atlantic Ocean, Arctic Ocean, & Homeland Defense.
 * United States Third Fleet (HQ San Diego, California) – East Pacific
 * United States Fourth Fleet (HQ Mayport, Florida) – South Atlantic
 * United States Fifth Fleet (HQ Manama, Bahrain) – Middle East
 * United States Sixth Fleet (HQ Naples, Italy) – Europe, including Mediterranean Sea & Black Sea.
 * United States Seventh Fleet (HQ Yokosuka, Japan) – West Pacific
 * United States Tenth Fleet (HQ Fort Meade, Maryland) – Reactivated as Fleet Cyber Command. Formerly anti submarine warfare coordinating organization.

Inactive and historic

 * United States First Fleet
 * United States Eighth Fleet
 * United States Ninth Fleet
 * United States Eleventh Fleet
 * United States Twelfth Fleet
 * United States Asiatic Fleet – historic
 * United States Navy Reserve Fleets
 * Great White Fleet – nickname for the United States Atlantic Fleet sent around the world by President Theodore Roosevelt in 1908
 * Great Green Fleet – nickname for the United States ships or current carrier strike group sent around the world to promote a new bio-energy sustainable fuel
 * East India Squadron
 * European Squadron
 * North Atlantic Fleet
 * Mediterranean Squadron
 * Scouting Fleet
 * South Atlantic Squadron

Informal fleet names

 * Spanish Armada – name used in England to describe the fleet assembled by Philip II of Spain for the invasion of England

title: List of fleets


text:
1998 Quebec general election

The 1998 Quebec general election was held on November 30, 1998, to elect members of the National Assembly of the Province of Quebec, Canada. The incumbent Parti Québécois, led by Premier Lucien Bouchard, won re-election, defeating the Quebec Liberal Party, led by Jean Charest. To date this is the last election where the Parti Québécois won a majority of seats in the Quebec Assembly, although not the last in which it formed a government.

After the narrow defeat of the PQ's proposal for political independence for Quebec in an economic union with the rest of Canada in the 1995 Quebec referendum, PQ leader Jacques Parizeau resigned. In January 1996, Bouchard left federal politics, where he was leader of the Bloc Québécois in the House of Commons of Canada, to lead the Parti Québécois and become premier.

Jean Charest had also left federal politics, where he had been leader of the Progressive Conservative Party of Canada. Charest was initially seen as a bad fit for the Quebec Liberal Party, and for provincial politics. He later overcame this perception.

In terms of the number of seats won by each of the two parties, the result was almost identical to the previous 1994 general election. However, this time the Liberals won a slightly larger percentage of the popular vote, but nonetheless far fewer seats.

Mario Dumont, leader of the Action démocratique du Québec, repeated his success in winning his own seat, as he had done in the 1994 general election. However, his party also repeated its failure to elect any other member apart from its leader.

Results
! colspan=2 rowspan=2 | Political party ! rowspan=2 | Party leader ! colspan=4 | MNAs ! colspan=4 | Votes ! Candidates !1994 !1998 !± !# ! ± !% ! ± (pp)

 * style="text-align:left;"|Lucien Bouchard
 * 124 || 77  || 76  || 1 || 1,744,240  || 7,202 || 42.87% || 1.88

 * style="text-align:left;"|Jean Charest
 * 125 || 47  || 48  || 1  || 1,771,858  || 34,160  || 43.55% || 0.85

 * style="text-align:left;"|Mario Dumont
 * 125 || 1  || 1  ||  || 480,636  || 227,915  || 11.81% || 5.36
 * style="text-align:left;"|Paul Rose
 * 97 || – || – || – || 24,097  || 9,172 || 0.59% || 0.26

 * style="text-align:left;"|Keith Henderson
 * 24 || – || – || – || 12,543  || 1,017  || 0.31% || 0.01

 * || 39 || – || – || – || 12,441  || 53,780 || 0.31% || 1.39

 * style="text-align:left;"|Marc-Boris St-Maurice
 * 24 || – || – || – || 9,944  || 9,944  || 0.24% ||New

 * style="text-align:left;"|Allen Faguy
 * 35 || – || – || – || 5,369  || 27,837 || 0.13% || 0.72

 * || 24 || – || – || – || 2,747  || 1,576  || 0.07% || 0.04

 * || 20 || – || – || – || 2,484  || 623  || 0.06% || 0.01

! colspan="3" style="text-align:left;" | Total ! " colspan="2"|125 ! " colspan="2"|4,068,472 ! " colspan="2"| 100.00%
 * || 20 || – || – || – || 2,113  || 1,051  || 0.05% || 0.02
 * 657||125
 * colspan="7" style="text-align:left;" | Rejected ballots
 * 46,991
 * 31,548
 * colspan="2"|
 * colspan="7" style="text-align:left;" | Voter turnout
 * 4,115,163
 * 123,136
 * 78.32
 * 3.26
 * colspan="7" style="text-align:left;" | Registered electors
 * 5,254,482
 * 361,017
 * colspan="2"|
 * }
 * 361,017
 * colspan="2"|
 * }

Synopsis of results

 * = open seat
 * = turnout is above provincial average
 * = winning candidate was in previous Legislature
 * = incumbent had switched allegiance
 * = previously incumbent in another riding
 * = not incumbent; was previously elected to the Legislature
 * = incumbency arose from byelection gain
 * = other incumbents renominated
 * = previously an MP in the House of Commons of Canada
 * = multiple candidates

title: 1998 Quebec general election