Luciano Da
email: luciano@ifsc.usp.br

F Costa

Point Motion in Flat Spaces: An Ample Starting Point

In this work, we approach the interesting problem of representing and studying the position, velocity, acceleration and arc-length of trajectories of points defined in a two-dimensional geometrical space, which has been chosen for being simpler than three-dimensional spaces while retaining much of their structure, and richer than the one-dimensional line. In addition to providing an integrated revision of several aspects from linear algebra, differential calculus, and analytic geometry, the present work can also be understood as a starting point for studies of several ares in physics, especially particle dynamics, but also including electromagnetism, fluid dynamics and statistical physics. The presentation starts with mathematical concepts of spaces, bases and vectors, as well as the Dirac delta 'function' and its derivative, as a means for approaching motion discontinuity. The text then proceeds by presenting the concepts of position, motion (or trajectory) of points in terms of parametric curves, also addressing. The concept of arc-length of a trajectory is then presented, followed by a discussion of how point motion can interact with scalar and vector fields (including the concepts of gradient and line integrals). Brief presentations of the the concepts of relative position and speed, as well as moving frames conclude this work. Several examples and illustrations of the covered concepts have also been provided. "... the intrinsic affinity between mathematics and physics has intrigued humans from at least as early as Pythagora's time...

Introduction

One of the most basic concepts develed by humans while trying to model nature (e.g. [START_REF] Da | Modeling: The human approach to science[END_REF]) is that of a point, corresponding to a minute sphere whose radius can be as small as possible. It is probably not by chance that Euclid's Elements of Geometry (e.g. [START_REF] Cajori | A history of mathematics[END_REF][START_REF] Boyer | A history of mathematics[END_REF]) starts with the definition of a point, which is simply understood as 'having no part". Interestingly, Euclid's geometric understanding of a point was directly related to Democritus (c.460bC -c.370bC) definition of an atom as being indivisible. The fact that Euclid was active around 300bC suggests that his definition may have been inspired by Democritus's concept of atom.

Be that as it may, the fact that the point has long been motivated attention from the geometrical and physical perspective is of particular interest to the present work (see Fig. 1), as this concept represents one of the earliest Figure 1: Even though approaching point motion mostly from the mathematic-geometric perspective, the present work can be understood as being at the intersection between differential geometry (meaning also the areas of linear algebra, calculus, analytical geometry, and differential equations) and particle dynamics.

and most fundamental links between mathematics and physics, two areas which have since then been actively developed and merged into areas such as mathematical physics and applied mathematics.

The continuing importance of mathematical physics stems from the success of using mathematical concepts and constructs to model and better explain the physical world. Indeed, the intrinsic affinity between mathematics and physics has intrigued humans from at least as early as Pythagora's time (c.570bC -c.495bC), to the point that it has been suggested that mathematics could be the very language underlying the physical world. Though that possibility could be at least partially accounted for by the fact that mathematics was actually developed mostly to remain compatible with the real-world, its success an effectivity in describing and predicting physical phenomena remains as being particularly remarkable.

Despite the great interest motivated by better understanding the nature of the relationship between mathematics and physics, this captivating discussion shall not constitute the main focus of the present work, as it involves many areas including logic, ontology, foundations of mathematics, as well as philosophy. Our objective here is much simpler and specific, consisting of approaching the motion of a single massless point, more specifically a point, from a purely mathematical/geometrical perspective, i.e. without expressly considering mass (and therefore physical force, work, energy, etc.).

The separation between the mathematical and physical aspects of motion underlying the present work has the two following motivations. First, by being mostly independent of the physical related aspects, the concepts and methods presented here can provide an entry point not only to all areas of physics involving particle motion, but also many other related areas involving point motion, geometry of trajectories and scalar and vector fields. Examples of such areas include, but are not limited to: fluid dynamics, computer graphics, image processing/analysis, optimization, pattern recognition, and even artificial intelligence.

A more mathematically focused approach also allows the subjects of motion and fields to be approached from a broader perspective, since there are not specific physical constraints to be satisfied, especially Newton's laws of motion establishing how particles with mass interact with force fields (gravitational). Therefore, the present work provides an ampler perspective that can be subsequently contrasted with the physical world specificities. The adopted approach also helps realizing that some important physical concepts such as inertial reference frames and conservative fields originate directly from the mathematical point of view, without additional physical requirements.

In addition, though the initial section, addressing vector spaces, develops in R 3 , the remainder of this work develops exclusively on the R 2 space. This choice has been taken as a compromise between dealing on the ampler R 3 space (more equations would have been involved, also involving more challenging visualizations in R 3 ) and the much simpler R space of the line. At the same time, the concepts and results in R 2 are almost straightforwardly extensible to higher dimensional metric spaces. It is arguable that generalizing from R 2 to R 3 is simpler than from R into R 2 , not to mention R 3 .

Special attention is throughout given to the concepts of parametric curves (here understood as specifying the point motion), arc-length, as well as respective derivatives. Though we shall refer to the first and second derivative of a parametric curve also as instantaneous velocity and instantaneous acceleration respectively, these terms should not be necessarily related to their physical counterparts, reflecting the fact that we shall not be expressly considering mass. The adopted approach is therefore more closely related to the mathematical areas (e.g. [START_REF] Kreyszig | Advanced Engineering Mathematics[END_REF]) of linear algebra (e.g. [START_REF] Hoffman | Linear Algebra[END_REF]), calculus (e.g. [START_REF] Apostol | Calculus[END_REF][START_REF] Larson | Calculus. Cengage Learning[END_REF][START_REF] Stewart | Multivariable Calculus[END_REF][START_REF] Da | A mosaic of multivariate calculus[END_REF]), analytic geometry (e.g. [START_REF] Larson | Calculus with analytic geometry[END_REF]), differential Equations (e.g. [START_REF] Nagle | Fundamentals of Differential Equations[END_REF][START_REF] Da | Visualizing the content of differential equations[END_REF]), as well as differential geometry (e.g. [START_REF] Carmo | Differential geometry of curves and surfaces[END_REF]).

To complement our mathematical study of the properties of the motion of a single massless point, we will also consider how it can be influenced by scalar and vector fields, as well as how motion can be approached in relative terms and from distinct moving frames of reference.

Before we proceed with the presentation, it should be observed that it is expected that the reader his preliminary acquainted with basic mathematics (including some background on trigonometry, vectors, and matrices), up to rudiments of analytical geometry and differential calculus level. Some references are also suggested for complementary and subsequent studies.

Spaces, Bases, and Vectors

As studied in linear algebra (e.g. []), vector spaces are mathematical spaces whose elements, known as vectors, satisfy several conditions, including but not being limited to vector addition closure, associativity, commutativity, as well as existence of inverse. These properties refer to both the respective vectors in those spaces, but also to product with scalar values, which obey the distributive property, among others. Vector spaces are much broader than the traditional geometrical vector, also encompassing matrices and functions. The present work will focus on Cartesian spaces, especially R 2 and R 3 . These spaces have an additional interesting property that they incorporated the concept of inner product u, v between any two respective vectors u and v and respectively induced distance between vectors, therefore constituting geometrical spaces, or metric spaces.

The importance of vector spaces is that they have a respective structure, specified by the respective general properties as summarized above, that allows us to handle vectors in well-posed manners. For instance, this structure ensures that the sum of two vectors will always exist.

Given a vector space, it is possible to obtain one or more respective basis, namely a subset of vectors from which all other vectors can be obtained by respective linear combinations.

Let B be a basis of some 3D vector space, any vector u B in this space should be uniquely written as the following linear combination of its basis vectors:

u B = u x,B B 1 | B + u y,B B 2 | B + u z,B B 3 | B ( 1
)
where u x,B , u y,B , and u z,B , are the coordinates of the vector u B in basis B and B i |B are the components of that same basis. We can represent the vector in this basis more compactly as:

u B =   u x,B u y,B u z,B   = (u x,B , u y,B , u z,B ) (2)
Because there is an infinite number of possible basis, it is convenient to identify one of them that can be understood as being canonical (e.g. simplest, including that the basis elements have magnitude one and are mutually orthogonal). In the case of R 3 , its canonical basis consists of the three versors i, j, and k given as:

î =   1 0 0   ĵ =   0 1 0   k =   0 0 1   (3)
so that:

|| î|| = || ĵ|| = || k|| = 1 (4) î, ĵ = îT ĵ = 0 (5) î, k = îT k = 0 (6) ĵ, k = ĵT k = 0 (7)
where T stands for the transpose of a vector (or matrix). Observe that, though there is an infinite number of alternative basis composed by vectors that are mutually orthogonal and have unit magnitude (e.g. any rotation of the canonical basis above), they are not usually considered to be canonic. It is interesting to observe that there is no more precise definition of a canonical basis. Now, any vector u of R 3 can be uniquely expressed in terms of the following linear combination of the canonical basis elements:

u = x î + y ĵ + z k = = x   1 0 0   + y   0 1 0   + z   0 0 1   = = (x, y, z) (8)
Thus, we can represent the canonical basis in terms of the following matrix:

C C =   ↑ ↑ ↑ î ĵ k ↓ ↓ ↓   =   1 0 0 0 1 0 0 0 1   = I ( 9
)
where I is the identity matrix. However, it is also possible to uniquely express the vector u B respectively to the canonical basis, which can be done by translating the basis vectors B represented into the canonical basis, i.e.:

u C = u x,B B 1 | C + u y,B B 2 | C + u z,B B 3 | C = = u x,B   b 1,1 b 2,1 b 3,1   + u y,B   b 1,2 b 2,2 b 3,2   + u z,B   b 1,3 b 2,3 b 3,3   = =   u x,B b 1,1 + u y,B b 1,2 + u z,B b 1,3 u x,B b 2,1 + u y,B b 2,2 + u z,B b 2,3 u x,B b 3,1 + u y,B b 3,2 + u z,B b 3,3   = =   b 1,1 b 1,2 b 1,3 b 2,1 b 2,2 b 2,3 b 3,1 b 3,2 b 3,3     u x,B u y,B u z,B   = =⇒ u C = B C u B (10)
where:

B B =    ↑ ↑ ↑ B 1 | C B 2 | C B 3 | C ↓ ↓ ↓    =   b 1,1 b 1,2 b 1,3 b 2,1 b 2,2 b 2,3 b 3,1 b 3,2 b 3,3   (11)
Observe that the matrices representing basis are square. So, we have that:

u C = B C u B =⇒ (12) =⇒ u B = B -1 C u C (13)
where B C is assumed to be invertible (i.e. det(B C ) = |B C | = 0) in order to allow any vector expressed in the canonical basis to be uniquely translated into basis B (and vice-versa). As there is an infinite number of invertible real-valued matrices, it becomes plain that a same vector space can have an infinite number of respective bases.

Going back to our discussion of basis transformation, let us illustrate Equations 12 and 13 respectively to Figure 2, which assumes the vector space R 2 . Here, we have a vector u represented in terms of the canonical basis C as well as another basis B.

More specifically, we have that: The canonical coordinates of the vector u B can be obtained by Equation 13 as:

B C = 1 3 2 2 u B = 1.5 0.5
u C = B C u B = 3 4
It is interesting to observe that Equations 12 and 13 correspond to linear transformations of vectors u B and u C , respectively. Expressed more generally respectively to a vector x and a real-valued matrix A, a linear transformation of the elements of x can be written as:

y = A x (14)
A closely related more general type of transformation consists of affine transformations of a vector u by a matrix A and a displacement vector v:

y = A x + v (15)
Therefore, linear transforms can be understood as a special case of affine transformations. As affine transformations are known to preserve parallelism and straight lines, so are linear transforms.

Given that nothing specific about the canonical basis has been taken in consideration in the above developments and results, Equations 12 and 13 also hold for any other generic basis D other than the canonical, provided B D is invertible. Thus, we can generalize this result as:

u D = B D u B (16) u B = B -1 D u D (17)
Equations 12 and 13 (as well as Equations 16 and 17) can be immediately modified to translate two (or more) vectors from basis B to basis C in "parallel" fashion:

V B =   ↑ ↑ ↑ u B,1 u B,2 u B,3 ↓ ↓ ↓   (18) V C = B C V B ( 19
)
In particular, if V B is the triple of vectors of the basis B expressed respectively to itself, we have:

V B =    ↑ ↑ ↑ B 1 | B B 2 | B B 3 | B ↓ ↓ ↓    = B B B C = B C B B =⇒ (20) =⇒ B B = I = C C (21)
Thus, we have the interesting (and sometimes unnoticed) fact that a generic valid basis B becomes the canonical basis when its vectors are represented respectively to itself. At the same time, the vectors of B represented in terms of the canonical basis (i.e. the columns of B C ), in general, are not guaranteed to be orthogonal or to have unit magnitude. Now, let us consider the previous vector u translated into another basis D of the same vector space. We already know that:

u C = D C u D =⇒ (22) =⇒ u D = D -1 C u C (23)
By combining Equations 23 and 12, it follows that:

u D = D -1 C u c = D -1 C B C u B = = D -1 C B C u B = W u B ( 24
)
which provides a means for exchanging the coordinates of a vector from basis B to basis D, while matrix W -1 specifies the coordinate changes from frame D to B. These results can be summarized as:

u D = W u B = D -1 C B C u B u B = W -1 u D = B -1 C D C u D (25)
These equations remain valid when the canonical basis C is replaced by any valid other basis E.

An important additional aspect of vectors in a vector space concerns theirs respective position. Consider the situation shown in Figure 3. There are at least two possible respective understandings. One of them is that these vectors (except for that at the coordinate system origin) are not actually valid because their positions in the respective space cannot be specified by their coordinates, which only establish position of the arrowed extremity of a vector respectively to the coordinate systems origin (0, 0). In this sense, the only vector valid in Figure 3 would be that starting at (0, 0).

Another possible is that the vectors represent relative displacement, which allows them to be shifted in the respective space. However, this requires the definition an additional property of each vector, specifying its point of application, which ab be done by associating an additional vector expressing the coordinates (x p , y p , z p ) of the nonarrowed extremity of the vector.

There is an additional interesting issue regarding vector spaces, and it concerns specifying the position of the reference coordinate system or frame. Typically, this choice is performed respectively to some useful purpose. For instance, the coordinate system in Figure 3 is placed at a specific coordinate relatively to the respective so that is reasonably close to where it is discussed (i.e. the present paragraph). The figure remains at its absolute position until the page is moved (e.g. scrolled), while its position relative to the page is maintained.

In several situations, as the just mentioned example, the position of the coordinate system may vary with time, which constitutes yet another interesting issue that will be briefly approached in Section 11.

To conclude this section, we observe that the dot product (also inner product or scalar product) between any two vectors u and v in a vector space provided with this operation (i.e. a Hilbert space) can be expressed respec-tively to a same basis as:

u, v = v, u = x v x + y v y + z v z = = || u|| || v|| cos θ (26)
where θ is the smallest angle between the two vectors.

Observe that, in particular:

u, u = x 2 + y 2 + z 2 =⇒ || u|| = u, u (27)
where the norm || u|| of u allows us to define the distance between two points specified by respective vectors u and v as:

d( u, v) = || v -u|| (28)
3 The Dirac Delta "Function" and its Derivative

Strictly speaking, the Dirac delta "function" is not a function, but a distribution (e.g. [START_REF] Brigham | Fast Fourier Transform and its Applications[END_REF]). However, this function (for simplicity, the quotation marks will not be indicated henceforth) can be approximated in terms of the following limit of the normal function (statistical distribution, with unit area):

δ(t) = lim σ→0 1 σ √ 2π e -1 2 ( t σ ) 2 (29)
we then necessarily have that:

ˆ∞ -∞ δ(t)dt = 1 (30)
Figure 4(b) illustrates the function c δ(t -t 0 ). As will be informally discussed in the present section, the Dirac delta function provides a convenient way to represent discontinuities. We starta by integrating the more general Dirac delta function c δ(t -t 0 ) for t ∈ (-∞, T ), with a ∈ R:

g(t) = ˆt -∞ c δ(u -t 0 ) du = 0 t < t 0 c t ≥ t 0 (31)
The constant c in c δ(t) will be henceforth referred to as the amplitude or area of the Dirac delta function δ(t).

The result of the above integration is illustrated in Figure 4. Observe that, as could be expected, the Dirac delta has implied a discontinuity at t 0 with amplitude determined by the Dirac delta intensity (or area) c.

Interestingly, it is possible to express the first "derivative" of the Dirac delta function can be expressed as: with the understanding that:

Π(t) = d δ(t) dt = lim ∆t→0 δ(t) -δ(t + ∆t) ∆t (32)
ˆΠ(t -t 0 ) dt = δ(t -t 0 ) + c ( 33
)
where c is a constant implied by the indefinite integral.

The "function" Π(t) can be expressed in terms of the limit of the normal distribution as follows:

Π(t) = d δ(t) dt = lim ∆t→0 δ(t + ∆t) -δ(t) ∆t (34)
Therefore, Π(t) has null area. Figure 5 illustrates the function Π(t) as the limit of the difference of normal distributions as specified in Equation 32.

The Dirac delta provides a convenient resource for understanding discontinuities not of a function itself, but of Let us now consider a function that is continuous and has at least its two first derivatives also being continuous, such as is the case with h(t) = 0.5 t 2 + b. In case we want to incorporate a discontinuity with amplitude c at t 0 , we can simply write:

h(t) = ˆt 0 g(u) + c δ(u -t 0 ) du + b = h(t) = ˆt 0 u + c δ(u -t 0 ) du + b = = ˆt 0 ˆu 0 1 + c Π(v -t 0 ) dv du + b (35)
where

g(t) = d h(t) dt = t.
Observe that b has been considered externally to the integral representations.

Figure 7 illustrates the above integral representations of the discontinuity of h(t) in terms of an integral involving the Dirac delta function, or a double integral incorporating the Π(t) function, which is represented in the present work as a double arrow.

Position, Parametric Curves, and Motion

We start our approach considering a single, static infinitesimal point remaining in an arbitrary position in R 2 , as illustrated in Figure 8, respective to the Cartesian (orthogonal) coordinate system x × y. The natural manner to mathematically specify the position of the point is in terms of its coordinates (x, y), which can also be associated to a respective position (column) vector as follows:

u = x y = (x, y) (36)
Now, consider that the position of the point may change along time t. Mathematically, this can be expressed in terms of the following mapping Γ from a free variable t in real interval [a, b] into the space where the point is being contained, in our case R 2 :

Γ : t ∈ [a, b] ⊂ R -→ u(t) = x(t) y(t) ∈ R 2 (37)
with a, b ∈ R. For simplicity's sake, we shall henceforth consider that the position of a point in R 2 along the variable t as specified by the mapping Γ. Therefore, the sequence of points or vectors (x(t), y(t)) can be understood as the motion, displacement or trajectory of the point, while the parameter t corresponds to 'time'. Curves as Γ are often referred to as being parametric.

However, from the mathematical point of view the variable t has not physical interpretation other than being a parameter. The set of points (x(t), y(t)) defined by point through the mapping Γ and respective motion is henceforth associated to the motion of the point according to the time parameter t.

An important point to bear in mind is that t varies continuously in the real interval [a, b], going from a to b, except at both of its extremities a and b, where the mapping is discontinuous. Thus, the parameter t intrinsically imposes a sequence or order along the single point positions u(t) = (x(t), y(t)).

In principle, parameters can be transformed in any continuous manner, e.g. by introducing a respective relationship r = g(t), such as r = 2 t + 3. Then, it becomes possible to refer to thereversion of the parameter, namely considering the transformed parameter q = -r instead of r.

Let us consider a type of motion, defined as:

u(t) = x(t) y(t) = x 0 + v x t y 0 + v y t (38) with x 0 , y 0 ∈ R, v x , v y ∈ R + .
Figure 9 illustrates this trajectory for x 0 = 0, y 0 = 0, v x = 1 and v y = 1 considering t ∈ [0, 1]. The points in blue have been sampled at ∆t = 2 intervals along the trajectory.

For t = 0, we have that the point is at position (0, 0). As t increases, the point initiates a straight displacement along the orientation specified by vector v = (1, 1), up to the other extremity u(t = 1) = (1, 1). The type of motion expressed by Equation 38 is typically called linear.

Let us now consider the following modified version of the previous type of motion:

u(t) = x(t) y(t) = x 0 + a t 2 y 0 + b t 2 (39) with x 0 , y 0 ∈ R, v x , v y ∈ R + .
Figure 10 illustrates this type of motion considering x 0 = y 0 = 0, a = b = 1, and ∆ t = 0.04. Only the sampled points in blue are shown for the sake of enhanced visualization. Though the respectively defined set of points is identical to those obtained in the previous example, we do not necessarily have pointwise identity: except for t = 0 in the case of the present example, (x(t), y(t)) = (x(t), ỹ(t)) for each time instant t (in the particular case of the current example, pointwise identity is verified only at both extremities of the trajectory).

Also of interest in the trajectory in Figure 10 is the fact that the distance between the points along the trajectory, and therefore along time, successively grow, implying the points to be further apart. As it will be further discussed in Sections 5 and 6, this means that the velocity of the motion is not constant, implying in respective acceleration.

However, for any given t ∈ [0, 1], it is possible to establish a bijective correspondence between the two trajectories by solving the system:

x(t) = x(t) =⇒ x 0 + v x t = x 0 + a t 2 ⇒ v x = a t y(t) = ỹ(t) =⇒ y 0 + v x t = y 0 + b t 2 ⇒ v y = b t (40)
Another possible type of movement, known as circular, can be specified as: for t ∈ R, with x 0 , y 0 , f 0 ∈ R and ω = 2πf 0 , where f 0 = 2 is the frequency of the motion. This type of trajectory yields an orbit of radius ρ centered at the position (x 0 , y 0 ), as illustrated in Figure 11(a) for x 0 = y 0 = 0.1 and ρ = 0.7. The blue points have been sampled along the trajectory at intervals ∆t = 0.005.

u(t) = x(t) y(t) = ρ cos(ωt) + x 0 ρ sin(ωt) + y 0 (41) (a) (b)
In the case of the above type of uniform circular motion, it is possible to link the angular and the magnitude of the Cartesian speed v, which is known as scalar speed v s , which in the uniform case is also a constant, as:

v s = ω ρ (42)
The orientation of v always points toward the center of the defined circle. Yet another type of motion, closely related to the circular case, can be specified as:

u(t) =
x(t) y(t) = ρ t cos(ωt) + x 0 ρ t sin(ωt) + y 0 (43) with x c , y c , f 0 ∈ R and α ∈ R + . We shall refer to this type of motion, illustrated in Figure 11(b), as spiral. interestingly, the velocity along this motion varies not only in terms of its orientation, but its magnitude also progressively increases along time. Unlike the uniform circular motion, the velocity is not orthogonal to the acceleration in this type of spiral motion. Now, consider the following type of motion, assuming t ∈ [-1, 1]:

(x(t), y(t)) = (t, t) t < 0 (t, t + 1) t ≥ 0 (44)
Figure 12(a) depicts the trajectory defined by this type of motion.

The previous example related to a position discontinuity, but there are other possible types of discontinuities. For instance:

(x(t), y(t)) = (t, t) t < 0 (0.5, √ 2 -0.5 2 ) t ≥ 0 ( 45
)
is characterized by orientation discontinuity (see Fig. 12b).

Let us now consider the motion:

(x(t), y(t)) = (t, t) t < 0 (2t, 2t) t ≥ 0 (46)
which is characterized by a discontinuity at the first derivative of the position at t = 0. Yet another type of discontinuity is as follows:

(x(t), y(t)) = (t, t) t < 0 (t 2 + t, t 2 + t) t ≥ 0 ( 47
)
in which case the second derivative of the position is discontinuous at t = 0. This sequence of examples can extents to discontinuities at any successive derivative orders. These two types of motion are illustrated in Figures 12(b) and (c), respectively.

A completely discontinuous displacement can be specified as:

(x(t), y(t)) = (u(t), v(t)) (48)
where both u(t) and v(t) are random values uniformly distributed in the interval [0, 1], drawn at each time instant t. Though the respectively obtained trajectory is extremely dense and discontinuous, to a point that it cannot even be graphically discerned, it can be shown to cover the whole square (0 ≤ x ≤ 1, 0 ≤ y ≤ 1) ⊂ R 2 . One manner to start approaching this mapping is by considering that all [a, b], R and R 2 have Cantor cardinality equal to 2 ℵ0 = ℵ 1 , therefore having the "same number of points'.

First Derivative of Position

Given a motion specified parametrically, it is possible to consider its first derivative, in case it exists, which is also called instantaneous velocity or speed, which can be expressed in many ways:

v(t) = ˙ u(t) = dx(t) dt , dy(t) dt = = ( ẋ(t), ẏ(t)) = (x (t), y (t)) = (v x (t), v y (t)) = (49)
Observe that v(t) is a vector quantity. The following notation will be predominantly adopted in the present work:

v(t) = ˙ u(t) = v x (t) v y (t) = ux (t) uy (t) (50)
As an example, let us calculate the speeds of the examples of motion provided in Section 4. Let us start by the speeds corresponding to the previously considered continuous motions, respectively to Equations 38, 39, 41, and 43:

v(t) = (v x , v y ) (51) v(t) = (2t v x , 2t v y ) (52) v(t) = ρ ω (-sin(ωt), cos(ωt)) (53)
v(t) = ρ (cos(ωt) -ωt sin(ωt), sin(ωt) + ωt cos(ωt)) (54)

In both Equations 51 and 52, the obtained velocity is not a function of time, being therefore constant. As these motions are also linear, we have a uniform linear motion.

It is also interesting to consider the circular motion speed obtained in Equation 53 after transforming the Cartesian coordinates (x(t), y(t)) into respective polar coordinates. For simplicity's sake, we assume that x 0 = y 0 = 0, which leads to:

ρ = x(t) + y(t) = ρ θ = arctan y(t) x(t) = ωt (55)
from which we can obtain the respective angular velocity:

θ = ω (56)
which turns out to be constant. Therefore, the motion specified by Equation 41 can be called circular with constant angular velocity, or simply uniform circular motion, being illustrated in Figure 13(a). It can be observed that the speed vectors, shown in red in that figure, are always tangent to the vectors pointing toward the respective center of the circle defined by this specific type of motion, while their respective magnitude and angular speed remain both constant.

The speed of the spiral motion, given in Equation 43, are depicted in Figure 13(b). Interestingly, as the parameter t increases, both the magnitudes of the speed and acceleration also increase.

To conclude this section, we now provide the speed of the discontinuous motions specified by Equations 44, 45, 46, and 47: The velocity (red) and acceleration (green) vectors respective to spiral motion obtained by using Eq. 54, with (x0, y0) = (0.1, 0.1). In both cases, the magnitudes of the velocity and acceleration vectors are not shown to the original scale for improved visualization. It can be verified that the velocity is not orthogonal to the acceleration in this type of motion.

v(t) = (1, 1) t = 0 (0, δ(t)) t = 0 (57) v(t) = (1, 1) t < 0 (0.5, √ 2 -0.5 2 ) t ≥ 0 (58) v(t) = (1, 1) t < 0 (2, 2) t ≥ 0 (59) v(t) = (t, t) t < 0 (2t + 1, 2t + 1) t ≥ 0 (60) (a) (b)

Second Derivative of Position

The second derivative of the position of a motion u(t) can be expressed as:

a(t) = ˙ v(t) = ¨ u(t) = d 2 x(t) dt 2 , d 2 y(t) dt 2 = = (ẍ(t), ÿ(t)) = (x (t), y (t)) (61)
as well as:

a(t) = a x (t) a y (t) = vx (t) vy (t) = üx (t) üy (t) (62)
The accelerations obtained for the motions defined by Equations 38, 39, 41, and 43 are presented respectively as follows:

a(t) = (v x , v y ) (63) a(t) = (2t v x , 2t v y ) (64)
a(t) = ρ ω 2 (-cos(ωt), -sin(ωt)) ( 65)

a(t) = ρω (a x (t), a y (t)) ( 66
)
where: a x (t) = -2 sin(ωt) -ωt cos(ωt) a y (t) = 2 cos(ωt) + ωt sin(ωt)

In the case of the uniform circular motion, we can learn more about the relationship between the respective speed and acceleration vectors by performing their respective inner product: (v x (t), v y (t)), (a x (t), a y (t)) = = ( ẋ(t), ẏ(t)), (ẍ(t), ÿ(t)) = = ρ 2 ω 3 [cos(ωt) sin(ωt) -sin(ωt) cos(ωt)] = 0 (67) thus indicating that circular motion with constant angular speed is characterized by having the velocity vector being always orthogonal to the acceleration vector for any value of t, which can be observed in Figure 13.

The second derivatives of the motions in Equations 44, 45, 46, and 47 can be obtained as:

v(t) = (0, 0) t = 0 (0, Π(t)) t = 0 (68) v(t) =    (0, 0) t < 0 (-0.5δ(t), √ 2 -0.5 2 δ(t)) t = 0 (0, 0) t ≥ 0 (69) v(t) =    (0, 0) t < 0 (δ(t), δ(t)) t = 0 (0, 0) t > 0 (70) v(t) = (1, 1) t < 0 (2, 2) t ≥ 0 (71)
Though we shall constrain our more detailed study of point motion to its first and second derivative, higher derivatives can be obtained and analyzed in similar manners.

Arc-Length of Motion

Given a motion u(t) starting at t 0 , with respective speed v(t) = ˙ x(t), it is often interesting to known the total arclength of its trajectory from the initial time t 0 until a subsequent time t, which can be formally defined as:

s(t) = ˆt t0 ||d u(r)|| dr = ˆt t0 || ˙ u(r)|| dr (72)
where:

||d u(t)|| = || ˙ u(t)|| = [v x (t)] 2 + [v y (t)] 2 (73)
Conceptually, the arc-length between two distinct points of a trajectory can be understood as representing the trajectory as a thread, cutting it at those two points, extending it, and measuring the respective length.

As an example, let us calculate the arc-length of the trajectory defined by Equation 38:

x(t) = x 0 + v x t y(t) = y 0 + v y t (74)
We thus have that:

d x(t) dt = ẋ(t) = v x d y(t) dt = ẏ(t) = v y (75)
Assuming that t 0 = 0, we have:

s(t) = ˆt 0 ||d u(v)|| dt = ˆt 0 (v x ) 2 + (v y ) 2 dv = = ˆt 0 c dv = c t (76) with c = (v x ) 2 + (v y ) 2 .
Once we obtain the expression of the arc-length of a motion u(t) as a function of the free parameter t, i.e. s(t), we obtain t as a function of s, i.e.:

t = s -1 (t) = h(s) (77)
By replacing t by h(s) in the original parametric motion u(t), we get u(s), which therefore is parametrized by the arc-length s. This type of parametric curve will be necessarily characterized by:

|| ˙ u(s)|| = || v(s)|| = 1 (78)
In other words, the curve u(s) will have constant speed magnitude equal to 1, though the speed vector can undergo orientation changes.

Let us illustrate the above procedure respectively to the case in Equation 38. We already know that the arc-length of this motion is given by Equation 76, so that:

s(t) = c t =⇒ t = 1 c s = h(s) (79)
with:

c = (v x ) 2 + (v y ) 2 . ( 80
)
Substituting this expression into Equation 38, we obtain:

u(s) = (x(s), y(s)) = 1 c (v x s + x 0 , v y s + y 0 ) =⇒ =⇒ ˙ u(s) = ( ẋ(s), ẏ(s)) = 1 c (v x , v y )
From which it follows that:

|| ˙ u(s)|| = (v x ) 2 + (v y ) 2 (v x ) 2 + (v y ) 2 = 1
Now, let us proceed to the circular motion in Equation 41. Assuming that t 0 = 0, we have:

s(t) = ˆt 0 [-ρ ω sin(ωt)] 2 + [ρ ω cos(ωt)] 2 dt = = ρ ω ˆt 0 [sin(ωt)] 2 + [cos(ωt)] 2 dt = ρ ω t (81)
Recalling that ω = 2πf 0 , if we take t = 1/f 0 = T 0 , corresponding to a complete period of this motion, the arc-length results equal to s(T 0 ) = ρωT 0 = ρ2πf 0 T 0 = 2πρ, which indeed coincides with the perimeter of a circle with radius ρ.

Probing Fields

A scalar filed ψ(x, y) in R 2 can be understood as the mapping that associates a scalar value to each point (x, y) of R 2 , i.e.:

ψ(x, y) : (x, y) ∈ R 2 → ψ(x, y) = c ∈ R (82)
A simple example of scalar field is:

ψ(x, y) = a x y, a ∈ R (83)
A vector field is a mapping that associates a vector to each point (x, y) of R 2 , i.e.:

φ(x, y) : (x, y) ∈ R 2 → φ(x, y) = φ x (x, y) φ y (x, y) ∈ R 2 (84)
The gradient of a scalar field on R 2 is defined as:

∇ψ(x, y) = ∂ψ(x, y) ∂x i + ∂ψ(x, y) ∂y j ( 85
)
from which we conclude that the gradient defines a vector field on R 2 .

The gradient of a scalar field ψ(x, y) at any point (x, y) can be understood as pointing to the direction of maximum variation of that field at that point.

As a simple example of a scalar field, we have:

ψ(x, y) = a x y (86)
which induces the following respective gradient vector field:

∇ψ(x, y) = a y î + a x ĵ (87)
Given a scalar field ψ(x, y), and a generic small (differential) displacement vector d u at point (x, y):

d u = dx i + dy j (88)
the total derivative of ψ(x, y) implied by d u can be expressed as follows:

dψ(x, y)| d u = ψ(x, y), d u = ∇ψ(x, y) • d u = = ∂ψ(x, y) ∂x dx + ∂ψ(x, y) ∂y dy (89)
As an example, consider the small displacement vector placed at (x = 1, y = 2):

d u = 0.02 i -0.003 j (90)
and the scalar field defined as:

ψ(x, y) = 5 x y (91)
We have from Equation 89 that the total variation of the scalar field associated to the displacement d u is:

dψ(x = 1, y = 2)| d u = = 5(0.02)(2) + 5(0.003)(1) = 0.215 ≈ ≈ ψ(x + dx, y + dy) -ψ(x, y) = 0.2153
A particular important aspect regarding the possible relationship between trajectories and scalar and vector fields concerns the concept of line integrals.

Given a scalar field ψ(x, y) and a continuous trajectory u(t) sharing the same domain (e.g. R 2 ), the line integral of this scalar field from t i to t f can be written as:

ˆtf ti ψ(x(t), y(t)) || ˙ u(t)|| dt (92)
The resulting scalar value can be understood as the area beneath the scalar field sliced by the considered trajectory. It is sometimes interesting to thing of the moving point as a probe that is used to measure and analyze the vector field, in terms of its integration, along the respectively defined trajectory.

As an example, let us consider the line integral from t i = 0 to t f = 1 of the scalar field: ψ(x, y) = 2 x y considering the trajectory:

u(t) = (x(t), y(t)) = t -t + 2 so that: ˙ u(t) = ( ẋ(t), ẏ(t)) = 1 -1
The scalar field restricted by the considered trajectory can be written as:

ψ(x(t), y(t)) = 2 (t) (-t + 2) (93)
It follows from Equation 92 that:

ˆ1 0 ψ(x(t), y(t)) || ˙ u(t)|| dt = = ˆ1 0 2 t (-t + 2) (1) 2 + (-1) 2 dt = = 2 ˆ1 0 -t 2 + 2t √ 2 dt = = 2 √ 2 - t 3 3 + 2 t 2 2 1 0 = = 2 √ 2 - 1 3 + 1 = 4 3 √ 2 (94)
Given a vector field φ(x, y), it is possible to define its line integral, by a continuous trajectory u(t) from t i to t f as:

ˆtf ti φ(x(t), y(t)) • ˙ u(t) dt (95)
Therefore, the line integral of a scalar field along a given trajectory can be thought of as probing the 'net agreement' between the relative orientations of the gradient and the first derivative of the point motion along its respective trajectory.

As an example, let us calculate the line integral of the vector field corresponding to the gradient of the scalar filed used in the previous example and that same trajectory considering t i = 0 and t f = 1. We then have that:

φ(x, y) = ∇ψ(x, y) = 2 y 2 x (96)
The gradient field restricted by the considered trajectory can be expressed as:

φ(x(t), y(t)) = 2 (-t + 2) 2 t (97)
No, by using Equation 95, it follows that:

ˆ1 0 φ(x(t), y(t)) • ˙ u(t) , dt = = ˆ1 0 (-2t + 4, 2t) • (1, -1) dt = = 2 ˆ1 0 (-2t + 2) dt = = 2 -t 2 + 2t 1 0 = 2
Interestingly, it can be shown that the line integral between two points a = (x(t i ), y(t i )) and b = (x(t f ), y(t f )) of a vector field φ(x, y) corresponding to the gradient of a scalar field ψx, y does not depend on the specific trajectory between these two points, provided it is continuous. That is why this type of vector field is called conservative. We also have that:

ˆtf ti φ(x(t), y(t)) • ˙ u(t) , dt = = ψ(x(t f ), y(t f )) -ψ(x(t i ), y(t i )) = ψ( b) -ψ( a) (98)
Interestingly, it follows that even though the scalar field ψ(x, y) is not directly taken into account, it is still possible to calculate its relative change from its gradient scalar field by using the property above.

The line integral over a closed trajectory u(t) is called circulation of that field by the given trajectory. It follows from the previous property that:

˛ u(t) φ(x(t), y(t)) • ˙ u(t) dt = 0 (99)
Let us now switch R 3 , so that we can define the vector product between two vectors, as this will be used in the section about interaction between fields and trajectories. Thus, the vector product or cross product between two vectors u = (u x , u y , u z ) and v = (v x , v y , v z ) can be defined in terms of the following determinant:

u × v = î ĵ k u x u y u z v x v y v z = = u y v z î + u z v x ĵ + u x v y k- -u z v y î -u x v z ĵ -u y v x k = = (u y v z -u z v y ) î + (u z v x -u x v z ) ĵ + (u x v y -u y v x ) k (100)
We also have that the magnitude of the cross-product can be obtained as:

| u × v| = | u| | v| sin θ n ( 101
)
where θ is the smallest angle between the two vectors, and n is the unit magnitude vector that is normal to both vectors. Observe that, unlike the dot product, the cross product is not commutative.

The direction of the cross product between u and v can be obtained by the right-hand rule, in which we use our right hand, aligning vector u with the pointer figer, vector v with the middle finger, so that the resulting direction is given by the thumb. It is also possible to consider the corkscrew rule, in which vector u is rotated towards v, with the direction of the respective cross product being provided by the direction of a corkscrew rotating in that manner, as illustrated in Figure 14.

Field-Motion Interactions

So far in the present work, we have concentrated our attention on how motions/trajectories can be understood as providing means for better understanding and probing scalar and vector fields, which can be done in terms of respective line integrals. Now, we turn our attention to the particularly interesting question of how a given vector field φ(t) can be made to influence the motion of a point. Although there is an unlimited number of ways in which this can be done, here we first focus our interest on directly proportional relationships, through proportionality constants b, c ∈ R, c = 0, between the field vectors and the vectors obtained by time-differentiating the point location function u(t), i.e.:

b d n u(t) dt n = u [n] (t) = c φ(x, y, t) (102)
where the non-negative integer n indicates the order of the considered derivative at which the field/point motion relationship takes place.

Observe that the relationship specified by Equation 102 necessarily implies that the n-th derivative of the position vector x(t) will have the same orientation as that of the field or, in other words, will be necessarily parallel to that field. Therefore, only changes in the magnitude of the n-th derivative of x(t) can take place.

The establishment of a fully determined relationship between the point trajectory u and the field φ(t) therefore requires the definition of two parameters: (i) the proportionality constant c; and (ii) the order of the derivative n.

The specification of n implies some important consequences, including: (i) the derivatives from n upwards are determined by the vector field irrespectively to initial conditions; and (ii) the n derivatives n -1, n -2, 1, 0 of the point position are determined by the action of the the vector field over respective the initial conditions.

Given a specified differentiation of order n, we have that when φ(x, y) = 0 throughout, it will follow tha u [n-1] down to u [0] can be non-zero vectors. For instance, if u [n] (t) is a constant vector, u [n-1] (t) = ´x[n] dt will be linearly increasing, etc.

Assuming continuity of the derivatives, the choice of n determines the extent of the 'memory' of the motion along time t after the force field f (x) is removed or abruptly vanishes. Therefore, if a point is moving under influence of a field as discussed above respectively to a given value n, the point will henceforth keep its derivative u [n-1] constant as of at the instant when the field is completely removed, while the other lower order derivatives, as well as the point position, are progressive changed by the remaining constant derivative u [n-1] .

The above discussion is illustrated in Figure 15 For simplicity's sake, we henceforth take c = 1, t ∈ [0, 10], and that the initial point position in all cases is (x 0 , y 0 ) = (3, 2). We shall assume two possible situations: (i) the field remains during all considered time values, defining the trajectory shown in blue; and (ii) that the vector field φ(x, y) is completely removed at t r = 1.616162, shown in green. The share portion of the respective trajectories for t varying from 0 to t r are shown in red. The presented points have been equally sampled at time intervals δt = 0.1010101.

The situation depicted in Figure 15(a) concerns point motion with n = 1, in which case the tangent of the point velocity, at each time instant t, simply follows the imposed vector field until, when the latter is removed, the trajectory simply stops at its current position (last point along the trajectory shown in red). Now, we consider the case in which the field interaction takes place for n = 2, i.e. the action of the field implies directly proportional changes in the velocity of the point motion, which is illustrated in Figure 15 at the initial position. In case the field is kept along time, the point has its velocity (blue vectors) continuously changed, becoming progressively aligned with the field vectors. However, in the situation where the force is nullified at t r , the point continues its motion with constant vector speed (in green) equal to what it had when of the ceasing of the field .

Figure 15(c) depicts the case when n = 3. The initial velocity and acceleration vectors are shown at (v x (0), y y (0)) respectively in blue and dark red. The blue and green vectors along the two trajectories refer to the vector velocity, while the acceleration vectors are presented in cyan. When the field is kept constant, the point has its respective velocity and acceleration progressively aligned with the field vectors (blue trajectory). At the same time, the acceleration itself is increased along time. Interesting, and in marked contrast with the previous example, the point retains its acceleration when the field is removed at t r , which then acts so as to progressively change the velocity it had at that same time, while the velocity of the acceleration remains constant. Though not shown in the figure as a consequence of the considered spatial region, the green trajectory will thereon substantially diverge from the blue trajectory, while keeping its velocity vectors not necessarily aligned to the originally existing field.

Given a second vector field ψ(x, y, t) also influencing the point motion as specified in Equation 102 for a certain proportionality constant e and the same differentiation order n, in case these two effects can be linearly combined, we have:

b d n u(t) dt n = u [n] (t) = c φ(x, y, t) + e ψ(x, y, t) (104)
This equation therefore provides an example of the linear superimposition of effects.

Another interesting basic manner in which the motion of a point can be affected by a vector field corresponds to considering a vector field τ defined as a consequence of another vector field β acting on some of the derivatives m of the point position, such as in the following example, which necessarily assumes R 3 because of the cross product:

τ (x, y, z, t) = a d m u(t) dt m × β(x, y, z, t) (105)
The vector field τ (x, y, x, t) will therefore be orthogonal to both the considered derivative of the point position and the vector field β(x, y, x, t). The resulting vector field τ (x, y, z, t) can then influence the n-th derivative of the point motion as:

b d n u(t) dt n = a d m u(t) dt m × β(x, y, z, t) (106)
In case all the above vector fields φ(x, y, x, t), ψ(x, y, x, t) and τ (x, y, x, t) influence the point motion by acting on the same derivative n of its position by linear superimposition, the following single equation incorporating the interactions of the point motion with the three considered vector fields:

b d n u(t) dt n = c φ(x, y, z, t) + e ψ(x, y, z, t)+ + a d m u(t) dt m × β(x, y, z, t) (107)
10 Relative Position and Speed

Let two, possibly moving, points P and Q in R 2 be specified by respective canonical coordinates (x P (t), x Q (t)) and (x Q (t), y Q (t)). We can define the relative position of Q respectively to P as:

x Q,P (t) = x Q (t) -x P (t) y Q,P (t) = y Q (t) -y P (t) (108)
which allows us to determine the respective instantaneous relative velocity as corresponding to:

ẋQ,P (t) = ẋQ (t) -ẋP (t) ẏQ,P (t) = ẏQ (t) -ẏP (t) (109)
The instantaneous relative acceleration then corresponds to: ẍQ,P (t) = ẍQ (t) -ẍP (t) ÿQ,P (t) = ÿQ (t) -ÿP (t)

In case we translated these two points equally to another position (X, Y ) ∈ R 2 , we will have:

x P (t) = x P (t) + X y P (t) = y P (t) + Y (111) x Q (t) = x Q (t) + X y Q (t) = y Q (t) + Y (112)
It can be immediately verified that this translation will have no effect whatsoever on the relative position, velocity or acceleration, because the constant terms X and Y are respectively canceled by the subtraction involved in the definition of the relative position. Actually, any other function derived from the relative position will be preserved.

Moving Frames

Though we have so far studied the motion of a point respectively to a fixed Cartesian coordinate system, it is also interesting to consider how a same motion can be understood from different coordinate systems. One main motivation for this study is that there is no absolute position where to place our coordinate system in the real world. For simplicity's sake, we shall also refer to these coordinate systems as frames.

We start with the situation illustrated in Figure 16, where two Cartesian frames (x, y) and (u, v) are placed a distinct positions, namely (0, 0) and (1, 2).

More specifically, we have that the origin of the frame (u, v) is here specified in terms of the relative vector (X, Y ). which allows us to write:

x = X + u y = Y + v (113)
therefore relating the coordinates from both frames. We can now express the relationship between the position of the point P in the two considered frames as:

x P = X + u P y P = Y + v P (114)
It immediately follows that any motion observed in terms of the frame (x, y) will be identical to that tracked by using frame (u, v), except for a translation by the relative vector (X, Y ).

Let us now consider the situation in which frame (u, v) is moving with constant speed respectively to frame (x, y), as illustrated in Figure 17. In this case, we can write:

X(t) = (a t + b) Y (t) = (c t + d) (115)
In case point P is performing a generic motion, so that its coordinates become a function of t. In this case, it follows that:

x P (t) = (a t + b) + u P (t) y P (t) = (c t + d) + v P (t) (116)
The coordinates relative to the (u, v) frame become functions of time. As the motion of the frames is relative one another, we can rewrite the previous equations as:

We therefore also have that:

uP (t) = ẋP (t) -a vP (t) = ẏP (t) -c (118)
Thus, if ( ẋP , ẋP ) is constant, so will ( uP , vP ) be. This implies that a uniform linear motion in any of the frames will also be perceived as such by the other frame, though with possibly distinct constant speed implied by the relative motion between the two frames. Figure 18 illustrates this important property, showing a uniform motion as seen from frames (x, y) and (u, v), respectively. Now, let us consider that frame (u, v) is undergoing an accelerated motion relatively to frame (x, y), which can be expressed as: u P (t) = x p (t) -a t 2 + bt + c v P (t) = y p (t)d t 2 + et + f

In this case, if point P is moving with constant speed respectively to the frame (x, y), the following motion speed will be perceived from frame (u, Figure 19: A uniform linear motion in frame (x, y) can be perceived as an accelerated motion when observed from a relatively accelerated frame (u, v).

Therefore indicating that, unless a = d = 0, the uniform linear motion respectively to frame (x, y) will be understood as being accelerated (changing speed) when observed from frame (u, v). Figure 19 presents how a uniform linear motion in frame (x, y) will become when observed from frame (u, v).

Important implication of the above developments is that a relative uniform linear motion between two frames will transform linear motions into linear motions. In case one of the frames is accelerating relatively to the other, this will no longer be guaranteed, and it will be impossible for a observed placed in the accelerating frame to decide if the observed non-uniform motion is indeed nonuniform, of whether the observed variation of speed is a consequence of its frameworks not moving with uniform linear motion.

Concluding Remarks

Mathematics and physics are possibly the two most basic areas in the physical sciences. That there is an intrinsic, effective relationship underlying these two areas, es-pecially in the sense that the former provides a suitable main backbone for the latter, has not only paved the way to continuing discoveries and advances, but also intrigued scientists for a long time.

The present work has been conceived and developed from the perspective of approaching the interesting problem of point motion, as well as its properties including interaction with fields, from an almost exclusively general mathematical perspective. The choice of R 2 instead of R 3 has been motivated as a manner to reach a balance between the extreme simplicity of R and the substantially denser development considering R 3 . Thus, even though the present work is in a sense less general regarding the adopted space dimensionality, the extension to R 3 should be mostly direct, especially because the present approach focuses on differential aspects, and not integration, which would be intrinsically less directly extensible to R 3 .

The main reasons for the mathematics-based presentation is that it could provide useful background for subsequent studies in any area involving point motion, of which there are many, as well as for obtaining a treatment of motion that is more general than that in physics, as there is not need to satisfy constraints such as Newton's lass of motion. Therefore, in a sense the present work could be understood as approaching the issue of particle point motion, but without considering any related physical property such as mass or electric charge. Of additional interest is the fact that the derivation, along the current work, of important physics concepts such as inertial frames of reference and conservative fields while considering only mathematical aspects.

The enhanced generality of treating point motion as a preparation for approaching the physics of particle motion provides an additional interesting feature: it creates an ampler perspective that can then be made more specific when the physical constraints are brought in. For instance, the studied interaction between a moving point and scalar and vector fields considered a linear relationship taken at any possible motion derivative order. Thus, when Newtons's law of motion are considered, it becomes clear that they implement one choice among a much wider range of possibilities, emphasizing the importance of the discovery and application of these laws. The possibility of instantiating the broader concepts presented in the present work by incorporation of physical constraints is expected to be developed in a subsequent work.

Figure 2 :

 2 Figure 2: Example of basis transformation (or coordinates exchange). Observe that the plot coordinates refer to the canonical basis C, shown in blue. The green vectors illustrate the linear combination of the coordinates of vector u in basis B. It corresponds to the parallelogram law for vector addition after respective scalings of the basis elements specified by the coordinates of u B . A similar construction (not shown) also holds respectively to the canonical basis. The vector in in any valid coordinate system is shown in magenta. Please refer to the text for additional information.

Figure 3 :

 3 Figure 3: Though all the displayed vectors have the same magnitude and orientation, their respective varying positions cannot be determined from their coordinates.

Figure 4 :

 4 Figure 4: The integral function g(t) of the Dirac delta function with intensity (or area) c at position t 0 , namely g(t) = ´t 0 aδ(t -t 0 ) dt, consists of a discontinuous function characterized by being zero before t 0 , and becoming the constant function 1 afterwards. This property often provides a convenient means for expressing discontinuous functions.

Figure 5 :

 5 Figure 5: Illustration of the derivative of the Dirac delta, namely Π(t), can be understood as the limit of a difference of normal distributions. The values of ∆t are shown in the inset. Observe that any of the functions in this plot have area equal to zero.

Figure 6 :

 6 Figure 6: Illustration of how the first and second derivatives of a function f (t) with a discontinuity at t 0 can be understood in terms of the Dirac delta function. As with derivatives in general, the constant d cannot be directly recovered from f (t).

Figure 7 :

 7 Figure 7: The parabola h(t) = 0.5 t 2 + b with a discontinuity of amplitude c at t = t 0 can be expressed as the integral (with null initial condition) of the identity function u(t) = t plus one Dirac delta with amplitude c at t 0 , or as the double integral (also with null initial condition) of the constant function 1 plus the Pi function with amplitude c at that same position. Therefore, this figure also can also be understood as illustrating the first two derivatives of the function in (a).

Figure 8 :

 8 Figure 8: The position of a point in R 2 can be specified in terms of its Cartesian coordinates (x, y), which can be associated to a position vector u = [x y] T represented in the canonical basis. More specifically, in this case we have (x, y) = (0.5, 0.5).

Figure 9 :

 9 Figure 9: The motion or trajectory, shown in salmon, specified by Eq. 38 considering t ∈ [0, 1]. The blue points, sampled at ∆t = 2 intervals along the trajectory, have been included in order to provide more information about the motion which, in this case, is linear uniform, the constant speed magnitude being equal to v 2x + v 2 y ).

Figure 10 :

 10 Figure 10: Example of a uniformly accelerated linear motion, with the points being equally sampled along time. Observe that progressively larger displacements (increasing speed) implied by the acceleration.

Figure 11 :

 11 Figure 11: (a):The circular trajectory obtained respectively to Eq. 41 assuming x 0 = y 0 = 0.1 and ρ = 0.7. The blue points are equally spaced with ∆t = 0.005. Observe that this trajectory corresponds to a circle with radius ρ centered at the point (x 0 , y 0 ), shown as a red 'x'. As its angular speed ω = 2πf 0 is constant, this motion is often referred to as being circular uniform. (b): A spiral trajectory as described in Eq. eq:spiral, adopting x 0 = y 0 = 0.1, f 0 = 2, and ρ = 0.7. The blue points are sampled at equal intervals ∆t = 0.006.

Figure 12 :

 12 Figure 12: Four types of linear motions with discontinuities at t = 0. The case shown in (a) has a position discontinuity; the situation in (b) has speed orientation discontinuity; the motion in (c) is characterized speed magnitude discontinuity, while (b) concerns acceleration discontinuity.

Figure 13 :

 13 Figure 13: (a): The velocity (red) and acceleration (dark green) fields associated to a uniform circular motion with radius ρ = 0.7 and center (x0, y0) = (0.1, 0.1). (b):The velocity (red) and acceleration (green) vectors respective to spiral motion obtained by using Eq. 54, with (x0, y0) = (0.1, 0.1). In both cases, the magnitudes of the velocity and acceleration vectors are not shown to the original scale for improved visualization. It can be verified that the velocity is not orthogonal to the acceleration in this type of motion.

Figure 14 :

 14 Figure 14: The corkscrew rule for the cross product u × v between two vectors u and v.

  respectively to a point moving under the influence of the following time-constant field: φ(x, y) = φ x (x, y) = 0.1 y φ y (x, y) = 0.2 x (103)

Figure 15 :

 15 Figure 15: Different types of interaction between a point and a field. Motion of a point with (x 0 , y 0 ) =, (x 0 , y 0 ) = under the vector field φ(x, y) = considering n = 0 (a), n = 1 (b) and n = 2 (c).

Figure 16 :

 16 Figure 16: Two coordinate systems, or frames, with respective origins at different positions, are used to specify the position of the point p. Observe that we chose to express the position of the origin of the (u, x) frame in terms of the relative vector (X, Y ).

Figure 17 :

 17 Figure 17: Two distinct frames are considered for observing a point, but one is moving with constant speed relatively to the other.

u

  P (t) = x p (t) -(a t + b) v P (t) = y p (t) -(c t + d)

  v):uP (t) = ẋp (t) -(a t + b) vP (t) = ẏp (t) -(d t + e)(120)

Figure 18 :

 18 Figure18: A uniform linear motion in frame (x, y) will be neccessarily perceived as a uniform motion when observed from the frame (u, v) undergoing uniform linear motion relatively to the other frame.

Acknowledgments

Luciano da F. Costa thanks CNPq (grant no. 307085/2018-0) and FAPESP (grant 15/22308-2).

Observations

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Emilie Bout
email: emilie.bout@inria.fr

Loscri Valeria
email: valeria.loscri@inria.fr

Cybersecurity of the Low Power Wide Area Networks (LPWAN) Main English Title

Keywords:

Cyber attacks in wireless

Background

The growing popularity in recent years of the Internet of Things (IoT) and Machine-to-Machine (M2M) communications has fostered the development of major innovations in different sectors. However, the specifications and the constraints related to these new use cases have highlighted three major problems : the guarantee of low power consumption, a wide range, and a high data transmission rate. To respond to these constraints, a new class of wireless networks called Low Power Wide Area Network (LPWAN) has been created and regrouped several types of protocols such as : SigFox, NB-IoT or LoRaWAN. This new type of protocol is characterised with specific features (e.g., duty-cycle approaches to reduce the energy consumption), which can be seen by an attacker as new vulnerabilities.

Theory

Security in LPWAN has been a concern for several years due to their user resource limitations. Indeed, three main attack vectors can be cited : their energy, their management key, and their gateway. When energy is the attack vector, we speak of a subclass of attack called Energy Depletion Attacks. As their name suggests, this type of attack aims to completely drain the energy of a device with unexpected/illegal operation. To cause additional energy expenditure, the attacker has two main alternatives. The first is to provoke an energy cost overhead by forcing the victim to perform additional actions. In this category, we find all the denial of service (DOS) attacks in basic wireless protocols, such as jamming or replay attacks. Indeed, during a jamming attack, the attacker occupies a transmission channel to force the victim to delay its transmission and to remain active during this time. With a replay attack, the receiving node has to process an extra packet, and therefore it will consume more power than expected. In [START_REF] Mikhaylov | Energy Attack in LoRaWAN : Experimental Validation[END_REF], the authors discuss the energy consumption of a LoraWan network when it undergoes a jamming attack. They demonstrate that the attack increased the total power consumption of a single communication event from 36% to 576%. The second alternative for the attacker is to play with the different paradigms put in place to save energy in the LPWAN protocol. This type of category includes more targeted and elaborate attacks. Indeed, in recent years attackers have begun to integrate intelligent actions into their process to better respond to the specificities of victim protocols and make decisions autonomously. In [START_REF] Bout | HARPAGON : An energy management framework for attacks in IoT networks[END_REF], the authors develop a new attack based on an energy-saving mechanism, the duty-cycle. With a process of Markov chain theory, the authors demonstrate that thanks to the duty cycle mechanism, it is possible to deduce the optimal moment of the attack. They reveal that this type of attack considerably reduces network performance and causes numerous retransmissions which affect energy consumption. The second attack vector is the key management. Indeed, to ensure the integrity and confidentiality of data, the notion of session key has been integrated into the LPWAN protocols. For example, the LoraWan protocol includes two 128-bit pre-shared root keys stored in memory to derive the session key and ensures the integrity of payload data on the MAC layer. Thus, each device has a specific signature. However, like any new method of ensuring security, these can also become new targets or allow the inference of information. Indeed, most of the time these keys are directly stored in the memory of the device and are not re-initiated after their manufacture. Additionally, the encryption key and method no longer change once devices are deployed in most cases. Indeed, designing encryption algorithms that evolve over time and consume little energy is still an open research topic and few solutions have been integrated into reality. Therefore, if an attacker has the same type of device, he can easily deduce the encryption algorithm as well as the key. Moreover, an attacker having physical access to a node, can extract the key and decrypt the communication. The last attack vector on this type of network is the gateway. Indeed, the gateway component is an important part of the LPWAN network to ensure connectivity with end devices. However, this type of device contains valuable endpoint and management security information such as keys. An attacker can target the gateway to gain access to this information to later create other types of attacks such as spoofing attacks. Moreover, since the main purpose of this device is to provide connectivity between network nodes, this latter represents a single failure point for the network which could be used to disconnect hundreds of end-device from application. Finally, secret keys can also be deduced with a side channel attack. By analyzing the behavior of the embedded device such as the USIM cards on the gateway, the attacker is able to discover personal information. Indeed, in [START_REF] Liu | Small Tweaks Do Not Help : Differential Power Analysis of MILE-NAGE Implementations in 3G/4G USIM Cards[END_REF], the authors based on a Differential Power Analysis, deduce the secret key present in the gateway.

Future directions

Securing LPWAN networks remains an important issue and an open research topic. One of the first possible improvements would be the implementation of a system to reduce the energy consumption of more robust networks. A useful goal would be to create a system capable of detecting any type of attack underway and react appropriately. In this type of idea, several intrusion detection systems (IDS) have been developed in the literature [START_REF] Faisal Elrawy | Intrusion detection systems for IoT-based smart environments : a survey[END_REF]. However, problems such as the placement of the IDS solution in the network still remains to be resolved. Indeed, as we pointed out just above, if the IDS is installed directly in the gateway but it is corrupted, the latter will become ineffective. In order to respond to the security problem linked to the management key, several works are also in progress, as in [START_REF] Tsai | Low-Power AES Data Encryption Architecture for a LoRaWAN[END_REF]. In this work, the authors implement a novel low-power AES data encryption architecture to reduce power consumption and increase session key renewal security. More work has been done in this direction in recent years, however, as with IDS, there are still general points to be answered. For example, this new work improves the power consumption of the key generation and exchange process but significantly reduces the performance and data transmission rate of the devices. More generally, the improvement of security in recent years has gone through the creation of attacks. Many solutions to increase LPWAN networks have been integrated into the protocols. However, security has rarely been studied during their creation or integration. Recently, attacks targeting these systems have taken place, these are adversary attacks aimed at compromising security systems based on machine learning algorithms [START_REF] Deldjoo | A Survey on Adversarial Recommender Systems : From Attack/Defense Strategies to Generative Adversarial Networks[END_REF].