Introduction to
MATHEMATICAL PHYSICS
———–
Lectures Notes 2021/2022
Degree in Physics, 2nd year, 2nd term
A. Ponno
Department of Mathematics “T. Levi-Civita”
University of Padua, Italy
Updated to:
April 22, 2022
Then Geometry, founded on the practice of mechanics, is nothing but that part of Univer- sal Mechanics dealing with and proving the art of careful measurements. And since manual arts mainly aim to move bodies, this caused people re- fer to Geometry the magnitude, and to Mechan- ics the motion. Thus Rational Mechanics is the science which deals with and proves in a care- ful way the movements resulting from whatever forces, and the forces requested to whatever mo- tions.
I. Newton, Preface to Philosophiae Naturalis Principia Mathematica, 1687.
Mathematics is a part of physics. Physics is an experimental science, a part of natural sci- ence. Mathematics is the part of physics where experiments are cheap. The Jacobi identity (which forces the altitudes of a triangle to meet in a point) is an experimental fact in the same way as the fact that the earth is round (that is, homeo- morphic to a ball). But it can be discovered with less expense.
V.I. Arnol’d, On teaching mathematics, Russ. Math.
Surv. 53, 229-236 (1998).
Contents
Preface 5
1 Newton mechanics 7
1.1 Basic concepts . . . 7
1.2 Newton principles . . . 9
1.3 Models of forces . . . 12
1.3.1 Interaction models . . . 12
1.3.2 Drag forces . . . 15
1.4 General properties of n particle systems . . . 16
1.4.1 Cardinal equations . . . 17
1.4.2 Conservative forces and energy . . . 19
1.5 Qualitative analysis of dynamics . . . 23
1.5.1 1D autonomous ODEs . . . 23
1.5.2 1D conservative Newtonian motions . . . 27
1.6 Quantization of the action and energy levels . . . 33
Appendices 34
A Rotations in Rd 35
Bibliography 37
3
Preface
All the fundamental laws (i.e. equations) of physics, e.g. classical mechanics, electromagnetism, electrodynamics, quantum mechanics, general relativity and quantum field theory, admit, at present, a mathematical formulation in terms of a stationary, or critical, action principle. The symbolic form of such a principle is
δS
δφ = 0 , (1)
where S[φ] is the so-called “action” functional of the specific theory, φ denotes the “field” of the theory, consisting of one or more dynamical variables, and δS/δφ denotes the derivative of S with respect to φ. The fundamental equations of physics are obtained detailing the form of S[φ] first, and then computing its derivative. The solutions ˆφ of equation (1) are called the critical points of S by analogy with the case of real functions of one or more real variables. From a mathematical point of view, equation (1) has the form of a system of ordinary differential equations (ODEs) [3, 21, 35], or partial differential equations (PDEs) [4, 20, 34]. If in the latter case one allows the field variable(s) φ to be operator valued, the theory extends to quantum field theory.
The deep justification of such a unified, mathematical approach to physical theories rests on classical mechanics, notably in the re-formulation of the Newtonian mechanics [29] due mainly to Euler, Lagrange and Hamilton. In particular, the theory of constrained mechanical systems developed by Lagrange was later shown to admit an action principle formulation by Hamilton, exploiting the methods of calculus of variations invented by Euler.
It must be emphasized that the fundamental equations of physics cannot be deduced: they are always invented and fixed on the basis of their ability to: provide an explanation of certain new experimental results that are incompatible with the existing theories; predict new phe- nomena that can be experimentally checked; include in some limit the old theory. In such a respect, the variational formulation (1) of physical theories (when possible) is often nothing but an equivalent, convenient reformulation of the known equations, which allows to deduce certain properties of the system in a simpler, deeper and more elegant form. By far the most important example of such an aspect is the N¨other theory connecting symmetries and conservation laws [30]. On the other hand, in looking for the correct equations of a new theory, the variational formulation may work as a guide, which is the case of relativistic electrodynamics and general relativity [26], for example.
The relevance of variational principles [37] and of calculus of variations [18] to theoretical physics is obvious. However, one has to pay attention to the fact that dissipative phenomena, but for certain simple cases, are not included in (1). Dissipation is a phenomenological aspect
5
of physics, thus usually excluded a priori in fundamental theories (this was the big conceptual jump made by Galileo and Newton with respect to Aristoteles). However, the list of physical phenomena and applications where dissipation plays a primary role is huge, ranging from the statics of structures, to the mechanics of bodies in any medium opposing resistance, passing through fluid mechanics [1, 27]. There is a fundamental mathematical reason for such an ex- clusion: the variational formulation (1) imposes certain symmetries on the equations of motion that are typically violated by dissipative forces. The interesting problem of determining the conditions under which certain classes of equations admit a variational formulation was first studied by Volterra [36].
The present notes deal with classical mechanics and its mathematical methods, which means making use of advanced tools of analysis, geometry and algebra. This is not a choice concerning rigor, or elegance: the focus is on interesting physical problems, and higher mathematics allows to get a deeper understanding of physics and to draw sharp conclusions avoiding, as much as possible, useless computations.
Chapter 1
Newton mechanics
Much of the material contained in this chapter should be known to the reader, perhaps at a more elementary level [23]. Reviewing Newtonian mechanics is necessary as a basis for its subsequent developments, namely the Lagrangian and the Hamiltonian formulation.
The main goals of Newton mechanics are:
• the mathematical foundations of the principles of mechanics partially formulated, on the basis of experiments and observations, by Galilei;
• the first theory of gravitation based on and able to explain the astronomical observations and, in particular, the laws of planetary motions empirically determined by Kepler;
• the formulation of relative kinematics with the explanation of the so-called apparent forces (the Newtonian version of general relativity);
• the laws of motions of a rigid bodies (Euler equations).
The presentation given below preserves the logic path followed by Newton, but is adapted to the present knowledge of physics and mathematics.
1.1 Basic concepts
Newtonian mechanics makes use of the following basic concepts.
• Mass: the mass of a given body is the quantity of matter contained in it, i.e. the sum of the masses of the atoms contained in the body. Each atom consists of a nucleus containing A = N +Z nucleons, where Z, the atomic number, is the number of protons, whereas N is the number of neutrons; A is called mass number. The masses of protons and neutrons are approximately equal in value. Such a common value is the atomic mass unit (or Dalton) mu ' 1.66 10−24g. A neutral atom contains Z electrons, whose mass me ' 0.9 10−27g is much smaller than mu. Thus, the mass of a single atom is approximately given by Amu, and the mass of a body isP
snsAsmu, where ns is the number of atoms of the species s with mass number As. The mass is the coupling constant of the gravitational interaction.
7
• Electric charge: this is another property of elementary particles constituting atoms. Elec- trons and protons have charge −e and +e, respectively, where e ' 4.8 10−10esu (CGS units are made use of), whereas neutrons have charge zero. The charge of a body is the algebraic sum of the charges of its protons and its electrons. Ordinary matter consists of neutral atoms, which implies that bodies are approximately neutral. The electric charge is the coupling constant of the electromagnetic interaction. Neutrality of macroscopic bodies implies that the gravitational interaction is the dominant one on macroscopic scales.
• Material point, or particle: a body whose physical dimensions are negligible with respect to the relevant distances involved in the specific problem. The same body can be consid- ered as point-like or extended in space depending on the kind of phenomena one wants to study. For example, the Earth can be regarded as a mass point when studying its motion around the Sun, whereas its form, extension and composition are relevant in studying tides, axis oscillations and so on.
• Reference frame: a system of coordinates with respect to which one can measure the position in space of a particle at any instant of time. For example a Cartesian system of mutually orthogonal axes whose origin is placed somewhere in space. This is what we do locally on the Earth. On the other hand, on a terrestrial scale, spherical polar coordinates such as latitude, longitude and altitude are the convenient ones. Important: the position of the same point in two different frames can be determined at the same instant of time:
time does not varies (i.e. is not transformed) in passing from a reference to another. In other words, one can measure the position of a particle in two different frames at the same time.
• Isolated system (of particles): a system consisting of one or more particles whose distance from all the other bodies in the universe is so large that their effect (whatever it means) can be neglected. We know that the gravity is a long range interaction and cannot be screened. Thus, this is a critical point in the Newtonian construction.
• Position (of a particle): the position of a particle in the space Rd (d = 1, 2, 3) at time t is described by the curve R 3 t 7→ x(t) ∈ Rd. The components of the vector position x(t), x1(t), . . . , xd(t), are determined with respect to a given reference frame, and each of them is supposed to be smooth enough in t.
• Velocity (of a particle): the local tangent vector to the curve t 7→ x(t), namely v(t) = dx(t)
dt = ˙x(t) := lim
h→0
x(t + h) − x(t)
h .
Since v(t) ∈ Rd, t 7→ v(t) is another curve in space.
• Acceleration (of a particle): the local tangent vector to the curve t 7→ v(t), namely a(t) = dv(t)
dt = ˙v(t) = d2x(t)
dt2 = ¨x(t) .
1.2. NEWTON PRINCIPLES 9
1.2 Newton principles
The principle of mechanics in the Newtonian construction, from a mathematical point of view, are hypotheses.
Principle 1 (Principle of inertia). There exists a privileged reference frame, called inertial frame, in which an isolated particle has zero acceleration: a(t) = 0.
By a double integration on [0, t] one gets x(t) = x(0) + v(0)t, the parametric equation of the line passing through x(0), the position of the particle at t = 0, and parallel to v(0), the velocity of the particle at t = 0. Such a motion is thus called rectilinear and uniform (with reference to the constant velocity: v(t) = v(0)). The existence of an inertial frame implies the existence of infinitely many inertial frames. Indeed, if K is the given inertial frame in which a(t) = 0, and if K0 is another reference frame translating with respect to K with constant velocity V , the position x0(t) of the particle in K0 is related to that in K by
x(t) = r + Rx0(t) + V (t − t0) , (1.1) where r is the vector connecting the origin O of K to the origin O0 of K0 at t = t0 (an arbitrary instant of time), R is a given rotation matrix (satisfying RRT = 1 and det R = 1) and V is the given translation velocity of K0 with respect to K. Then, by taking the first and the second derivative of (1.1) with respect to t, one gets v(t) = Rv0(t) + V , and a(t) = Ra(t), respectively.
Thus, a(t) = 0 implies a0(t) = 0. The first Newton principle implies the existence of infinitely many inertial frames: all frames translating with respect to a given inertial frame with constant velocity are inertial frames.
Principle 2 (Newton law). A non isolated particle of mass m moves in an inertial frame according to the law
ma(t) = f (x(t), v(t), t) . (1.2)
The vector valued function f : Rd× Rd× R → Rd is called the force acting on the particle.
The force f describes the effect of nearby bodies on the given particle. The motion of the particle is completely determined when f is a known function of its argument. In this case the Newton law of motion, equation (1.2), is a second order (vector) ODE, which is clear at sight if one rewrites it in the form
¨
x(t) = 1
mf (x(t), ˙x(t), t) . (1.3)
Exercise 1.1. Suppose that the initial conditions x(t0) := x0 and ˙x(t0) := v0 are assigned vectors in Rd. Assume f to be of class Ck, k ≥ 0, in all its arguments. Assume that a unique C1 solution x(t) of (1.3) corresponding to the initial condition (x0, v0) exists in a neighborhood I of t0. Show that then x(t) is Ck+2(I). Compute x(t) in a neighborhood of the initial time t0 with an error of the order (t − t0)k+2. Hint: Taylor expand x(t0+ h) with Lagrange remainder.
The second Newton principle defines the force: it is the object (a vector valued function) modeling the action of the external bodies on the given particle. The main problem at Newton’s time was to invent the form of the gravitational force.
Remark 1.1. The fact that the force f depends on position and velocity of the particle, and on time, and that all the quantity appearing in (1.3) are computed at the same time t is an assumption, which allows to solve the problem of predicting the motion and turns out to be in agreement with the experiments. There are situations where such an assumption does not work. An example of non Newtonian dynamics is found in electrodynamics: the force exerted on a charged particle by the electromagnetic field generated by the particle itself contains a term proportional to the derivative of the acceleration, namely the Abraham Lorentz radiation reaction force [22]
frad = 2q2
3c3˙a , (1.4)
where q is the charge of the particle and c ' 2.998 1010cm/s is the speed of light.
For an isolated system consisting of two particles in an inertial frame, by the first two principles one can write the system of Newton equations
m1a1 = f12(x1, x2, v1, v2, t)
m2a2 = f21(x1, x2, v1, v2, t) , (1.5) where mi, xi and vi denote mass, position and velocity of particle i, whereas fij denotes the force exerted by particle j on particle i, i, j = 1, 2. Here again, the dependence of fij on position, velocity and time allows to treat (1.5) as a system of second order, vector ODEs.
Principle 3 (Action and reaction). For an isolated system consisting of two particles in an inertial frame, which moves according to (1.5), the forces satisfy the following two conditions:
f12 = −f21, and f12k f21k x1− x2.
Notice that the conditions are two: 1) the forces are equal in modulus, have the same direction and opposite orientation, and 2) their common direction is that of the line passing through the two points. The consequence of the 3rd principle is that
f12 = ϕ12(x1, x2, v1, v2, t) x1− x2
|x1− x2| f21 = ϕ21(x1, x2, v1, v2, t) x2− x1
|x2− x1|
(1.6)
where ϕ12(x1, x2, v1, v2, t) = ϕ21(x1, x2, v1, v2, t), ensuring that f21 = −f12.
For an isolated system of n ≥ 2 particles in an inertial frame, one writes the Newton system in the form
m1a1 = f1(x1, . . . , xn, v1, . . . , vn, t) ...
mnan = fn(x1, . . . , xn, v1, . . . , vn, t)
, (1.7)
and a further hypothesis one the forces f1, . . . fn is needed if n ≥ 3.
1.2. NEWTON PRINCIPLES 11 Principle 4 (Superposition of forces). For an isolated system of n particles in an inertial frame, which moves according to (1.7), the force acting on particle i, i = 1, . . . , n satisfies the superposition principle
fi =
n
X
j=1 j6=i
fij , (1.8)
where each fij satisfies the 3rd principle: fij = −fji and fij k fjik xi− xj.
As a consequence of principles 3 and 4, the force fi acting on the ith particle of an isolated system in an inertial frame has the form
fi =
n
X
j=1 j6=i
fij =
n
X
j=1 j6=i
ϕij(x1, . . . , xn, v1, . . . , vn, t) xi− xj
|xi− xj| , (1.9)
where ϕij are scalar quantities satisfying ϕij = ϕji (symmetry is with respect to the exchange of the subscript indices only, not to the exchange of positions and velocities of particles i and j in the argument). A further constraint on the form of the forces is obtained by requiring the invariance of the Newton equations with respect to Galileian transformations (1.1).
In order to formulate the last Newton principle, it is convenient to rewrite the Newton equations (1.7), which is a non autonomous system of nd equations1 of second order, in the form of an equivalent autonomous system of 2nd + 1 equations, namely
˙xi = vi
˙vi = m1
i
Pn j=1 j6=i
ϕij(x1, . . . , xn, v1, . . . , vn, t)|xxi−xj
i−xj|
˙s = 1 ; s(0) = 0
i = 1, . . . , n , (1.10)
with the forces fi of the form (1.9). The dynamical variables are now the (vector) positions and velocities of the particles and the fictitious variable s(t), in terms of which the Galileian transformation (1.1) reads
xi(t) = r + Rx0i(t) + V s0(t) vi(t) = Rv0i(t) + V
s(t) = s0(t) + t0
i = 1, . . . , n . (1.11)
Such a transformation G : (x0, v0, s0) 7→ (x, v, s) obviously coincides with (1.1) when the condi- tions ˙s = 1 and s(0) = 0 are taken into account. G depends on 10 free parameters: r, V and t0 contributing for 3 + 3 + 1 = 7, the remaining 3 parameters being those determining the rotation matrix R; see Appendix A.
Principle 5 (Newton-Galilei special relativity). Given an isolated system of n particles in an inertial frame, its equations (1.10) must be invariant under any Galileian transformation (1.11).
1In referring to the number of equations of a given system of ODEs one counts the number of scalar equations.
The latter principle states that the dynamics of an isolated system appears identical in any inertial frame, or to any “inertial observer”. Notice that one requires the invariance of the equations, while the initial conditions change in an obvious way according to G.
Proposition 1.1. The relativity principle implies that the force scalars ϕij may depend only on the moduli of the difference of all (vector) positions and velocities of the particles, and must be independent of time: ϕij = ϕij(. . . , |xk− xl|, . . . , |vk− vl| . . . ), where the inideces k, l run over 1, . . . , n, with k 6= l.
Proof. Substituting (1.11) into (1.10) one gets
R ˙x0i(t) + V ˙s0(t) = Rv0i(t) + V R ˙vi0(t) = m1
i
Pn j=1 j6=i
ϕij(G(x0, v0, s0))Rx
0 i−x0j
|x0i−x0j|
˙s0 = 1 ; s0(0) = −t0
i = 1, . . . , n ,
where the transformed arguments of the ϕij have been pointed out symbolically.
An elementary simplification yields
˙x0i(t) = vi0(t)
˙vi0(t) = m1
i
Pn j=1 j6=i
ϕij(G(x0, v0, s0))x
0 i−x0j
|x0i−x0j|
˙s0 = 1 ; s0(0) = −t0
i = 1, . . . , n .
One now goes on as follows. First, setting R = 1, V = 0 and t0 = 0, G reduces to the translation by r of all the vector positions. Then ϕij(G(x0, v0, s0)) = ϕij(x0, v0, s0) if and only if ϕij depends only on the vector differences xk− xl. Second, setting r = 0, R = 1 and t0 = 0, G reduces to the translation by V of all the vector velocities. Then ϕij(G(x0, v0, s0)) = ϕij(x0, v0, s0) if and only if ϕij depends only on the vector differences vk− vl, Third, setting t0 = 0, and taking into account the previous results, G reduces to the rotation by R of all the vector differences xk− xl and vk− vl. Then ϕij(G(x0, v0, s0)) = ϕij(x0, v0, s0) if and only if ϕij depends only on the moduli of the such vector differences. Finally, setting r = 0, V = 0 and R = 1, G reduces to the translation by t0 of s0. The invariance of the ϕij implies that they do not depend on s0.
Corollary 1.1. For an isolated particle (n = 1) the relativity principle implies f ≡ 0.
Proof. It is left as an exercise. Hint: if u ∈ Rd is such that Ru = u for any rotation matrix R, then u ≡ 0 (why?).
1.3 Models of forces
1.3.1 Interaction models
All classical physics (and a large part of the quantum one) is built up by a further phenomeno- logical simplification: the force scalars ϕij are supposed to be independent of the velocities and
1.3. MODELS OF FORCES 13 to depend only on the distance |xi− xj| of particles i and j: ϕij = ϕij(|xi− xj|). This is due to the experimental fact that two out of the four fundamental interactions of nature, namely the gravitational and the electrostatic one, share such a property (within certain limits). Thus the internal forces are assumed, in general, of the so-called central form
fij = ϕij(|xi− xj|) xi− xj
|xi − xj| = ϕij(|xi− xj|)ˆxij , (1.12) where the unit vector ˆxij := |xxi−xj
i−xj| from j to i has been defined. The 3rd principle requires of course ϕij(r) = ϕji(r).
Exercise 1.2. Let Φij(r) be minus a primitive function of ϕij(r), i.e. Φij(r) = −R ϕij(r)dr, or Φ0ij(r) = −ϕij(r). Show that
fij = ϕ(|xi− xj|) xi− xj
|xi− xj| = − ∂
∂xiΦij(|xi− xj|) , (1.13) where ∂/∂xj = ∇xj denotes the gradient with respect to the components of the vector xj. Show that ϕij = ϕji implies that fji = −∂Φij/∂xj.
Example 1.1 (Elastic force). The simplest possible assumption is that the force scalar be a linear homogeneous function, i.e. ϕij(r) = −kijr, or ϕij(|xi − xj|) = −kij|xi − xj|, where kij > 0 is a given constant. Then
fij = −kij(xi− xj) , (1.14)
i.e. the so-called elastic force, or Hooke’s law. Notice that kij > 0 implies that such a force is always attractive. This force models the small motions of a given system around its equilibrium positions. Remark: the 3rd principle implies kij = kji.
Example 1.2 (Gravitational force). For the gravitational force, Newton deduced the form ϕij(r) = −Gmimj/r2, where mi and mj are the masses of particles i and j, respectively, whereas G = 6.67 10−8 cm3/(gs2) is the so-called gravitational constant, a fundamental con- stant of nature, like the speed of light, the Planck constant, and so on. Then
fij = −G mimj
|xi− xj|3(xi− xj) . (1.15) The Newton force satisfies the 3rd principle and is always attractive.
Example 1.3 (Electrostatic force). For the electrostatic force between charged particles, Coulomb deduced the form ϕij(r) = kqiqj/r2, where qi and qi are the electric charges of particles i and j, respectively, whereas the positive constant k depends on the system of units adopted: k = 1 in the CGS system (with charges measured in esu, the elementary charge being 4.8 10−10esu).
Thus
fij = k qiqj
|xi− xj|3(xi− xj) . (1.16) The Coulomb force satisfies the 3rd principle and is attractive for charges of opposite sign, and repulsive for charges of equal sign.
The form of the Newton force (1.15) can be “deduced” in different ways. A simple and direct one is the following. Newton was aware of the experiments of Boyle on falling bodies in vacuum tubes, from which it follows that the weight force on a particle is proportional to its mass:
m¨x = −mgˆe3, where ˆe3is the upward unit vector, and g = 980 cm/s2is the gravity acceleration, implies ¨x = −gˆe3, i.e. a falling law of the form x(t) = x(0) + v(0)t − 12gˆe3t2. The fundamental intuition of Newton is that the force law ruling falling bodies and, neglecting the resistance of air, the motion of a cannon ball, is an approximate form of the force law ruling the planetary motions. For the latter, assuming a central force law of the form m¨x = GMOmϕ(|x|)ˆx, where MO is the mass of a star (e.g. the Sun) placed approximately at rest in the origin. The proportionality to m is the one of the falling body problem, that to MO is due to the 3rd principle; at this level G is a dimensional proportionality constant. The other experimental information available to Newton was the third Kepler law: the ratio of the square of the revolution period T to the cube of the major semi-axis a of the (elliptic) planetary orbits is a constant, say 1/k, independent of the planet: T2/a3 = 1/k. Thus by rescaling the space and time variables in the Newton equation, measuring length in units of a and time in units of T , i.e. setting x = ax0, t = T t0, one has to get one and the same orbit, with unit period and unit semi-axis. The rescaling transforms the Newton law of the problem into
a T2
d2
dt02x0 = GMOϕ(a|x0|)ˆx0 .
Multiplying the latter by a2, and taking into account that a3/T2 = 1/k, one gets d2
dt02x0 = (kGMO) a2ϕ(a|x0|)ˆx0 . (1.17) Now, imposing that the right hand side of the latter equation does not depend on the particular semi-axis a, i.e. requiring that the equation itself transforms to d2x0/dt02 = (kGMO)ϕ(|x0|)ˆx0, yieldsa2ϕ(a|x0|) = ϕ(|x0|), or
ϕ(a|x0|) = a−2ϕ(|x0|) . (1.18)
The latter condition tells that ϕ(r) is a homogeneous function of degree −2, accoridng to the following definition.
Definition 1.1. A function f : Rd → R is said to be homogeneous of degree s ∈ R if f(λx) = λsf (x) for any λ > 0.
Theorem 1.1 (Euler’s theorem on homogeneous functions). If f : Rd → R is a homogeneous function of degree s, then x · ∇f (x) = sf (x). In particular, if d = 1, f (x) = c|x|s, where c is a constant.
Proof. Taking the derivative of f (λx) = λsf (x) with respect to λ, and setting λ = 1, one gets x · ∇f (x) = sf (x). If d = 1, the latter equation is an ordinary differential equation for f , namely f0 = sf /x. By separating variables, one can rewrite it as df /f = sdx/x, which integrated gives ln |f | = s ln |x| + c0, i.e. ln(|f |/|x|s) = c0. Taking the exponential and setting c = ±ec0 on gets the result.
1.3. MODELS OF FORCES 15 By Euler’s theorem, relation (1.17) gives ϕ(r) = cr−2(Notice that the argument of ϕ is positive).
Thus the Kepler 3rd law implies that the force is inversely proportional to the square of the distance of the particle from the center of attraction (the star, or planet). The sign of the constant is then fixed by requiring that the force is attractive, so that bodies close to the surface of the Earth, move downwards. The unification of the two phenomena (planetary motion and falling bodies) gives g = GMT/R2T, where MT and RT are the mass and the radius of the Earth, respectively.
Remark 1.2. The method just sketched is the first example of scale invariance of a physical law. The Newton gravitation law, with the known force law, is invariant under the re-scaling x = λx0, t = λ3/2t0 for any (dimensionless) λ > 0. Notice that |x|3/t2 = |x0|3/t02, which is another (more general) way of stating the 3rd Kepler law. In some texts scale-invariance is referred to as “mechanical similarity” [2, 25].
Exercise 1.3. Consider the d-dimensional (d = 1, 2, 3) harmonic motion, defined by the New- ton law m¨x = −kx, x ∈ Rd. Rescale space and time variables like x = λx0 and t = λαt0, and determine the value of the scaling exponent α such that the Newton law is invariant under the given scaling transformation. Discuss the physical meaning of the result.
1.3.2 Drag forces
Taking into account the resistance exerted by a medium (typically a fluid or a gas) on body motions is a difficult problem. For example, a body moving in an ideal (i.e. collisionless) gas is subject to a force opposing its motion which is due to the collisions with the particles of the gas hitting on it. For a sphere of radius r and mass much larger than that of the gas particles, which moves with an instantaneous velocity v much higher than the thermal velocity of the gas particles (which is proportional to the square root of the gas temperature), the average force opposing to the motion, or drag force, is found to be [9]
f = −2πρr2
3 |v|v , (1.19)
where ρ is the mass density of the gas (mass per unit volume). The law (1.19) is also due to Newton.
On the other hand, the drag force exerted by a fluid on a sphere moving in it with a small instantaneous velocity v, is given by the Stokes law
f = −(6πρνr)v (1.20)
where ν is the kinematic viscosity coefficient of the fluid [1, 27]. The Stokes law is valid if |v| ν/r. Both laws (1.19) and (1.20) admit corrections, to lower and higher velocities, respectively, whose derivations are not trivial. The final result can be resumed in a phenomenological formula for the drag force of the form
f = −k(|v|)v , (1.21)
where the drag coefficient k admits an expansion of the form k(|v|) = γ + k1|v| + k2|v|2 + · · · at small velocities, with coefficients depending on the form of the moving body, on the density of the medium and so on. Formula (1.21) holds for both fluids and gasses (though for fluids a less simple dependence of k on |v| holds). Taking into account the dependence of k on v in applications is difficult. By far the most widespread form of drag force in physics assumes a constant drag coefficient, i.e. is of the form f = −γv.
Exercise 1.4. Consider a particle of mass m falling in a gas subject to its weight force. Assume a drag coefficient of the form k = γ + k1|v|. Assume the vertical axis oriented downwards, so that in the falling motion v(t) > 0. The Newton equation of motion of the particle is m ˙v = −(γ + k1v)v + mg (why?). Solve the equation with the initial condition v(0) = 0.
Question: which is the limit velocity v∗ := limt→∞v(t)? Hint: determine v∗ before solving the Newton equation.
1.4 General properties of n particle systems
If a system consisting of n particles, in an inertial frame, is not isolated, it is assumed that the force fj on particle j can be decomposed into two components: an internal one, fj(i), due to the interaction of j with all the remaining n − 1 particles of the system, and an external component fj(e), due to the interaction with the “external world” (anything different from the other particles). Thus, the Newton equations of the system in this case read
mjaj = fj(i)+ fj(e) , j = 1, . . . , n , (1.22) with the internal forces fj(i) of the form implied by the principles 3 and 4. The case of isolated system is then recovered by setting fj(e)≡ 0 for all j = 1, . . . , n.
Suggestion 1.1. In the sequel, the scalar and vector products in Rd (d = 2, 3), denoted by · and ×, respectively, are extensively used. An excellent introduction to the subject, for physicists, is chapter 2 (Vectors) of [23]. Different symbols are used in the literature, especially in the mathematical one. The scalar product is often denoted by ( , ), h , i, or even no symbol.
The vector product is often denoted by a “wedge” symbol ∧ or a square bracket [ , ].
Use is made below of the following fundamental concept.
Definition 1.2. The derivative of a function I(ξ, η, s), I : Rnd× Rnd× R → R, along a solution of the Newton system mjaj = fj(x, v, t), j = 1, . . . , n is given by
d
dtI(x(t), v(t), t) =
n
X
j=1
∂I
∂xj · ˙xj(t) + ∂I
∂vj · ˙vj(t) +∂I
∂t
=
=
n
X
j=1
∂I
∂xj · vj(t) + ∂I
∂vj · 1
mjfj(x(t), v(t), t) + ∂I
∂t
.
(1.23)
The function I is said to be a first integral, or a constant of motion of the Newton system if dI/dt ≡ 0 along any solution of the Newton system.
1.4. GENERAL PROPERTIES OF N PARTICLE SYSTEMS 17 With some abuse of notation, the derivative of a scalar or vector function I along the solutions of the Newton equations is simply denoted by ˙I, its actual meaning being the one specified in (1.23). If ˙I = 0, then I(x(t), v(t), t) = I(x(0), v(0), 0), i.e. the constant value of I is determined by the initial conditions determining the specific solution of the Newtonian system.
1.4.1 Cardinal equations
Consider a system consisting of n particles, whose Newton equations, in an inertial frame, have the form (1.22). Let us define the following single-particle and collective quantities.
• M :=Pn
j=1mj, the total mass of the system.
• X(t) :=Pn j=1
mj
Mxj(t), the vector position of the center of mass, or barycenter C, of the system. Notice that C is a geometrical point in space, not coinciding, in general, with any particle of the system.
• F(i) :=Pn
j=1fj(i), the resultant internal force (vector sum of the internal forces).
• F(e) :=Pn
j=1fj(e), the resultant external force (vector sum of the external forces).
• `j := xj× mjvj, the angular momentum (or moment of momentum, the momentum being mjvj) of particle j.
• ` :=Pn
j=1`j, the total angular momentum.
• τj(i) := xj× fj(i), the moment of the internal force, or internal torque, of particle j.
• τj(e) := xj × fj(e), the moment of the external force, or external torque, of particle j.
• T(i) := Pn
j=1τj(i), resultant moment of internal forces, or resultant, or total internal torque.
• T(e) := Pn
j=1τj(e), resultant moment of external forces, or resultant, or total external torque.
Proposition 1.2 (Cardinal equations of dynamics). The following kinematical identities hold:
M ¨X = F(e) ; (1st cardinal eq. of dynamics) (1.24)
` = T˙ (e) . (2nd cardinal eq. of dynamics) (1.25) Proof. We make use of the Newton equations (1.22), with the internal forces of the form implied by the principles 3 and 4 (5 is not necessary).
M ¨X =
n
X
j=1
mjx¨j =
n
X
j=1
fj(i)+
n
X
j=1
fj(e)= F(i)+ F (e)
Now, one proves that F(i) ≡ 0. Indeed
F(i) =
n
X
j=1
fj(i) =
n
X
j=1 n
X
k=1k6=j
fjk =
n
X
j,k=1 j6=k
fjk = 1 2
n
X
j,k=1 j6=k
fjk+ 1 2
n
X
j,k=1 j6=k
fjk =
=1 2
n
X
j,k=1 j6=k
fjk+ 1 2
n
X
k,j=1 k6=j
fkj = 1 2
n
X
j,k=1 j6=k
(fjk + fkj) ≡ 0 ,
since, by the the 3rd principle, first assumption, fkj = −fjk. This proves equation (1.24). Taking the derivative of the total angular momentum, one gets
` =˙
n
X
j=1
˙xj × mjvj+
n
X
j=1
xj × mj˙vj =
n
X
j=1
τj(i)+
n
X
j=1
τj(e)= T(i)+ T(e) .
One now proves that T(i) ≡ 0. Indeed T(i) =
n
X
j=1
xj × fj(i) =
n
X
j,k=1 k6=j
xj× fjk = 1 2
n
X
j,k=1 k6=j
xj × fjk +1 2
n
X
j,k=1 k6=j
xk× fkj =
=1 2
n
X
j,k=1 k6=j
xj× fjk− 1 2
n
X
j,k=1 k6=j
xk× fjk = 1 2
n
X
j,k=1 k6=j
(xj− xk) × fjk ≡ 0 ,
since, by the third principle, second assumption, fjk k xj− xk. This proves equation (1.25).
Corollary 1.2. In an inertial frame, the center of mass of an isolated system moves of uniform rectilinear motion, and the total angular momentum is a constant of motion.
Proof. Equation (1.24) with F(e) ≡ 0 implies ¨X = 0, which by two successive integrations in [0, t] gives X(t) = X(0) + V (0)t. The values of the initial position X(0) and of the initial (and constant) velocity V (0) are determined by the initial positions and velocities of the particles: X(0) =P
j mj
Mxj(0); V (0) = P
j mj
Mvj(0).
Equation (1.25) with T(e) ≡ 0 implies ˙` = 0, i.e. `(t) = `(0). The constant value
`(0) is determined by the initial conditions: `(0) =P
jxj(0) × mjvj(0).
Exercise 1.5. Consider the case of a system of interacting particles subject to the weight force fj(e) = −mjgˆe3. Write the cardinal equations and solve them.
Exercise 1.6. Consider the previous case, for identical particles, and take into account the resistance of air, i.e. fj(e) = −γvj − mgˆe3 for each particle of mass m. Write the cardinal equations and solve them.
1.4. GENERAL PROPERTIES OF N PARTICLE SYSTEMS 19 Exercise 1.7. Consider the previous problem, and in particular the solution X(t) of the first cardinal equation. Suppose that: i) t/τ 1, where τ := m/γ; ii) |V (0)| |V∞|, where V∞= −gτ ˆe3. (or, as a simpler assumption, set V (0) = 0). Show that X(t), to second order in t/τ included, is the same as in the ideal problem, with no resistance of air.
Exercise 1.8. Suppose the external forces acting on the particles of the form fj(e) = ϕj(|xj|)ˆxj, where ˆxj := xj/|xj|. What can be concluded from the cardinal equations? Hint: compute τj(e) first.
Exercise 1.9. Consider the previous problem, with ϕj(r) = −GMOmj/r2, namely the case of a system of interacting particles subject to the gravitational attraction of a point of mass MO placed in the origin (modeling an extended body in the gravitational field of a far star, for example). Represent the positions of the particles as xj(t) = X(t) + x0j(t), and suppose
|x0j| |X| (what does it mean? Make a drawing.). Draw a conclusion on the motion of the center of mass, to leading order in the small ratio |x0j|/|X|. Write down the first correction to that.
1.4.2 Conservative forces and energy
Consider now a system of n particles which move according to the Newton equations
mjaj = fj , j = 1, . . . , n , (1.26) where no hypothesis is made now on the forces fj, but for their dependence on all the positions x1, . . . , xn ∈ Rd and velocities v1, . . . , vn ∈ Rd of all the particles of the system, and possibly on time t ∈ R. Let us introduce other two collective quantities of physical interest.
• K := 12Pn
j=1mj|vj|2, the kinetic energy of the system of particles with velocities v1, . . . , vn.
• P :=Pn
j=1fj· vj, the power provided by the system of forces fj (on the state of motion specified by the positions x1, . . . , xn and by the velocities v1, . . . , vn at time t).
Proposition 1.3. K = P .˙
Proof. Taking the derivative of the kinetic energy K along any solution of the Newton equations (1.26), one gets
dK dt =
n
X
j=1
mj 2
d
dt(vj· vj) =
n
X
j=1
mjaj· vj =
n
X
j=1
fj· vj = P .
Integrating in time on any interval [t1, t2] the identity ˙K = P one gets K(t2) − K(t1) =
Z t2
t1
n
X
j=1
vj · fj dt :=
Z
γ n
X
j=1
fj· dxj , (1.27) where γ : [t1, t2] 3 t 7→ (x1(t), . . . , xn(t)) ∈ Rnd is the actual curve of motion solving the Newton system. An equivalent formulation of the identity ˙K = P is obtained by defining
• Wγ :=R
γ
Pn
j=1fj· dxj, the work done by the system of forces fj along the motion curve, or path, γ.
One can then rewrite (1.27) in the compact form ∆K = Wγ, where ∆K := K(t2) − K(t1).
The kinematical identity ˙K = P , or its equivalent integral formulation, is completely useless unless some further assumptions on the forces are done. A fundamental property of forces is the following.
Definition 1.3. The positional forces fj(x1, . . . , xn, t), j = 1, . . . , n, are said to be potential if there exists a function U (x1, . . . , xn, t), U : Rnd+1 → R, called potential energy, such that
fj = −∂U
∂xj , j = 1, . . . , n . (1.28)
If the potential forces fj are independent of time (and U as well, as a consequence) they are said to be conservative.
Remark 1.3. ∂U/∂xj denotes the partial gradient of U with respect to the d components of the vector xj, so that (1.28) means, by components, fj1 = −∂U/∂xj1, . . . , fjd = −∂U/∂xjd. Proposition 1.4. Given a system of n particles subject to conservative forces of potential energy U , the total energy H := K + U is a constant of motion.
Proof. One has to show that ˙H ≡ 0 along the solutions of the Newton equations (1.26), where fj = −∂U/∂xj for any j = 1, . . . , n. Indeed
K = P = −˙
n
X
j=1
∂U
∂xj · ˙xj = −d
dtU (x1(t), . . . , xn(t)) , i.e. ˙H = ˙K + ˙U ≡ 0.
Remark 1.4. For time-dependent potential forces, U depends explicitly on time and the total energy H = K + U is not a constant of motion. Indeed: ˙H = ∂U/∂t (check it).
We now come back to the Newton model of forces for a system of n particles in an inertial frame: fj = fj(i)+ fj(e), where the internal forces are of the central type fjk = ϕjk(|xj− xk|)ˆxjk, where ϕjk(r) = ϕkj(r). Let Φjk be minus a primitive of ϕjk, and denote by
Φ(x1, . . . , xn) := 1 2
n
X
l,k=1 l6=k
Φlk(|xl− xk|) , (1.29)
the potential energy of the internal forces. Such a definition is justified by the following com- putation
−∂Φ
∂xj =
n
X
k=1 k6=j
− ∂
∂xjΦjk(|xj− xk|) =
n
X
k=1 k6=j
ϕjk(|xj − xk|)ˆxjk =
n
X
k=1 k6=j
fjk(i) = fj(i) (1.30)
which is easily obtained by taking into account (1.13).
1.4. GENERAL PROPERTIES OF N PARTICLE SYSTEMS 21 Proposition 1.5. K + ˙˙ Φ = P(e), where P(e) := Pn
j=1fj(e) · vj is the power provided by the external forces.
Proof. By taking into account the Newton equations mjaj = fj(i) + fj(e), together with (1.30), one has
K + ˙˙ Φ =
n
X
j=1
mjaj · vj +
n
X
j=1
∂Φ
∂xj · vj =
n
X
j=1
(mjaj − fj(i)) · vj =
n
X
j=1
fj(e)· vj .
The proof of the following two statements is obvious.
Corollary 1.3. The total energy H = K + Φ of an isolated system (in an inertial frame) is a constant of motion.
Corollary 1.4. If the external forces are conservative of potential energy U(e), the total energy K + Φ + U(e) is a constant of motion.
Example 1.4. For a system of particles subject to their weight force (gravity on the Earth), U(e)=Pn
l=1mlgxl3. Indeed
fj(e) = −∂U(e)
∂xj = −
∂U(e)
∂xj1
∂U(e)
∂xj2
∂U(e)
∂xj3
=
0 0
−mjg
= −mjgˆe3 .
Example 1.5. For a system whose particles are subject to external central forces, U(e) = Pn
l=1Φl(|xl|), where Φ0l(r) = −ϕl(r). Indeed fj(e)= −∂U(e)
∂xj = −Φ0j(|xj|)∂|xj|
∂xj = ϕj(|xj|)ˆxj .
Example 1.6. For system of interacting particles, whose potential energy of the internal forces is Φ, subject to the gravitational attraction of a mass MO placed in the origin, the total energy
H = K + Φ −
n
X
j=1
GMOmj
|xj| is a constant of motion.
Example 1.7. A confined gas is a system of n interacting particles constrained to move inside a specified vessel. One could model the confinement mechanism through a suitable wall potential Uw. In this case, the total energy of the system K+Φ+Uw is a constant of motion. However, it is much easier to assume that a particle of the system sees the vessel only when it hits the internal boundary, the collision being instantaneous and perfectly elastic. Supposing that particle j hits
the wall of the vessel at time tc, at a given impact point, one has vjk(tc+ 0) = vjk(tc− 0), and vj⊥(tc+ 0) = −vj⊥(tc− 0), where the subscript symbols k and ⊥ mean parallel and orthogonal to the wall at the impact point (the internal boundary is supposed to be smooth, thus admitting a tangent plane at any point). Instantaneous, perfectly elastic collisions preserve the kinetic energy K. As a consequence the total energy of the system H = K + Φ is a constant of motion (why?).
The total energy H = K + Φ of a Newtonian system of interacting particles can be a constant of motion even if the external forces are non conservative. An alternative possibility is that the power P(e) provided by the external forces vanishes.
Example 1.8. For a system of charged particles interacting through the Coulomb force (1.16) and subject to the external Lorentz force fj(e) = qcjvj × B(xj) caused by a stationary magnetic field B(x), the total energy
H =
n
X
j=1
mj|vj|2
2 + 1
2
n
X
j,k=1 j6=k
qjqk
|xj − xk|
is a constant of motion. Notice that the magnetic field is not assumed uniform in space, but constant in time (why?).
Remark 1.5. A given system, or field of positional forces f (x), where f : RN → RN, is conservative if there exists U (x), U : RN → R, such that f(x) = −∇U(x). This is equivalent to state that the linear differential form, or differential 1-form δW := f (x) · dx is integrable, or exact, i.e. it is an exact differential. Indeed, f (x) = −∇U (x) implies δW = −dU , and vice versa. The 1-form δW is interpreted as the work done by the field of force along the infinitesimal displacement dx, whereas its integral Wγ =R
γδW =Rs2
s1 f (ξ(s)) · ξ0(s) ds along a curve γ : [s1, s2] 3 s 7→ ξ(s) ∈ RN is the work done by the field of forces along the path γ. One is thus motivated to study the conditions characterizing exact differential 1-forms.
The conditions of exactness of a differential 1-form are summarized in the following propo- sitions, whose proof can be found in [31].
Proposition 1.6 (Independence of work of the path). Consider a field of forces f (x), contin- uous in an open connected set Ω ⊆ RN, and its associated differential 1-form δW = f (x) · dx.
The 1-form δW is exact, i.e. f (x) is conservative, if and only if the work Wγ = R
γδW is a function of the end-points of γ for any piece-wise differentiable curve γ (with image) in Ω.
Equivalently, δW is exact if and only if H
γδW = 0 for any piece-wise differentiable, simple closed curve in Ω.
Proposition 1.7 (Closure condition). Suppose the field of forces has continuous partial deriva- tives in an open, simply connected set Ω ⊆ RN. Then, f (x) is conservative if and only if its Jacobian is symmetric, namely ∂f /∂x = (∂f /∂x)T.
1.5. QUALITATIVE ANALYSIS OF DYNAMICS 23
1.5 Qualitative analysis of dynamics
Qualitative analysis of ODEs, deals with the determination of the general properties of the solutions of a given system, or of a given class of systems, for any initial condition, the explicit form of the solutions being not known (either because not interesting, or because impossible to compute, as usual). By general properties of the solutions one means, for example, stability of equilibria, asymptotic behavior for large times, existence of first integrals, and so on.
1.5.1 1D autonomous ODEs
The simplest ODE is of the form ˙x = f (x), where x ∈ R and f (x) is a smooth real function of a real variable. As is well known, such an equation can be solved by separating the variables, writing it as dx/f (x) = dt, for x such that f (x) 6= 0. Upon integration one gets F (x) = t + c, where F0(x) = 1/f (x), and by a local inversion one gets the general solution x(t) = F−1(t + c).
The arbitrary constant c is uniquely determined by the initial datum x(t0) = ξ: c = −t0 + F (ξ), and x(t) = F−1(t − t0 + F (ξ)). Particular solutions playing a fundamental role are the equilibrium, or stationary ones, defined as those constant functions x(t) = ¯x for any t, where ¯x is a zero of f (x), i.e. a solution of f (x) = 0. Check: ˙x(t) ≡ 0, and f (x(t)) = f (¯x) ≡ 0.
Such a method of solution works, in general, only in principle: both the explicit computation of a primitive function F (x) of 1/f (x), and its local inversion, are generally difficult when not impossible. More than this, were an explicit expression of x(t) available, it would be very difficult to extract interesting informations from it: one would have to study (with the standard tools of mathematical analysis) a complicated function.
On the other hand, one can understand everything about the behavior of the solutions of the equation ˙x = f (x) by interpreting x(t) as the position of a point moving on the real line.
The ODE is then interpreted as the law giving the velocity of the point when the position of the latter is known. One then concludes that ˙x(t) > 0, i.e. x(t) is a monotonically increasing function of time t, where f (x(t)) > 0. In other words, if the point occupies a position x(t) where f takes on a positive value, then the point is moving from left to right. The other way around, ˙x(t) < 0, i.e. x(t) is a monotonically decreasing function, where f (x(t) < 0, and the point occupying the position x(t) is moving from right to left. One can then draw the graph of f and point out the direction of motion of the point along the x axis. Such a picture is called the phase portrait, or diagram of the 1D ODE, or one dimensional dynamical system.
An example is reported in Figure 1.1 below.
Exercise 1.10. Plot the phase portrait of ˙x = f (x) for f (x) = −x, f (x) = −x3, f (x) = x(1 − x), f (x) = −x + x3, f (x) = sin(x).
Exercise 1.11. Plot the phase portrait of the equation for v of Exercise 1.4 on falling bodies in a gas.
The most important features of the phase diagram of ˙x = f (x) are the following.
• If ¯x is an isolated zero of f such that f0(x) < 0 for any x ∈ I \ {¯x}, then x(t) → ¯x as t → +∞ for any x(0) ∈ I \ {¯x}. The typical case of a simple zero, with f0(¯x) < 0, is
