Classical Mechanics. Prof. Dr. Alberto S. Cattaneo and Nima Moshayedi

Transcription

1 Classical Mechanics Prof. Dr. Alberto S. Cattaneo and Nima Moshayedi January 7, 2016

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3 Preface This script was written for the course called Classical Mechanics for mathematicians at the University of Zurich. The course was given by Professor Alberto S. Cattaneo in the spring semester I want to thank Professor Cattaneo for giving me his notes from the lecture and also for corrections and remarks on it. I also want to mention that this script should only be notes, which give all the definitions and so on, in a compact way and should not replace the lecture. Not every detail is written in this script, so one should also either use another book on Classical Mechanics and read the script together with the book, or use the script parallel to a lecture on Classical Mechanics. This course also gives an introduction on smooth manifolds and combines the mathematical methods of differentiable manifolds with those of Classical Mechanics. Nima Moshayedi, January 7,

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5 Contents 1 From Newton s Laws to Lagrange s equations Introduction Elements of Newtonian Mechanics Newton s Apple Energy Conservation Phase Space Newton s Vector Law Pendulum The Virial Theorem Use of Hamiltonian as a Differential equation Generic Structure of One-Degree-of-Freedom Systems Calculus of Variations Functionals and Variations Extremals Shortest Path Multiple Functions Symmetries and Conservation Laws The Action principle Coordinate-Invariance of the Action Principle Central Force Field Orbits Systems with Constraints and Lagrange Multipliers Differential forms Notations Definitions The wedge product The exterior derivative The Pullback The Lie derivative The contraction Properties Hamiltonian systems Introduction Legendre Transform Derivatives and Convexity Involution Total Differential of Legendre Transform Local Legendre Transformation Multivariable Case Canonical Equations

6 6 CONTENTS Hamiltonian Function Canonical Action Principle Previous Examples in Canonical Form The Poisson bracket Constants of motion The Poisson bracket in coordinate-free language Symplectic integrators Introduction The Euler method Hamiltonian systems The Noether Theorem Introduction Symmetries in Lagrangian mechanics Symmetries and the Lagrangian function Examples Generalized symmetries From the Lagrangian to the Hamiltonian formalism Symmetries in Hamiltonian mechanics Symplectic geometry The Kepler problem The Hamilton Jacobi equation Introduction The Hamilton Jacobi equation The action as a function of endpoints Solving the Cauchy problem for the Hamilton Jacobi equation Generating functions Introduction to Differentiable Manifolds Introduction Manifolds Maps Submanifolds Topological manifolds Differentiable manifolds The tangent space The tangent bundle Vector bundles Constructions on vector bundles Differential forms Applications to mechanics The Noether 1-form The Legendre mapping The Liouville 1-form Symplectic geometry Appendices

7 CONTENTS 7 Appendix A Topology and Derivations 75 A.1 Topology A.2 Derivations Appendix B Vector fields as derivations 79 Bibliography 80

8 8 CONTENTS

9 Chapter 1 From Newton s Laws to Lagrange s equations 1.1 Introduction Classical mechanics is a very peculiar branch of physics. It used to be considered the sum total of our theoretical knowledge of the physical universe (Laplace s daemon, the Newtonian clockwork), but now it is known as an idealization, a toy model if you will. Classical Mechanics still describes the world pretty well in the range of validity, which is for example that of our everyday experience. So it is still an indispensable part of any physicist s or engineer s education. It is so useful because the more accurate theories that we know of (general relativity and quantum mechanics) make corrections to classical mechanics generally only in extreme situations (black holes, neutron stars, atomic structure, superconductivity, and so forth). Given that GR and QM are much more harder theory to use and apply it is no wonder that scientists will revert to classical mechanics whenever possible. So, what is classical mechanics? 1.2 Elements of Newtonian Mechanics In the title classical means that there are no quantum effects. The simplest mechanical system is a mass point, which is a single point moving in space that has a finite mass m attached to it. You can think of the matter field belonging to a mass point as a delta-function in space: an infinitely concentrated, featureless lump of matter. The equation of motion for the mass point comes from physics and is expressed by Newton s law: force = mass accelaration, F = ma. Our first mass point system comes right at the beginning of mechanics, it is Newton s apple Newton s Apple Newton s apple is a mass point with mass m and vertical position z(t) at time t. Here z is a Cartesian coordinate pointing upwards. From physics it is known that the gravitational force on the apple is given by mg and points downwards. Here g is the gravity constant with typical value 9.81ms 1. Thus Newton s law for the apple is m z = mg z + g = 0, where the dot denotes a time derivative. We see that the mass of the apple does not affect its motion in the gravitational field. What is the solution to the equation given above? This is 9

10 10 CHAPTER 1. FROM NEWTON S LAWS TO LAGRANGE S EQUATIONS a second-order ODE in time, so the solution to the initial-value problem requires specifying two initial conditions. In our case these are given by the initial position z(0) and the initial velocity ż(0). Given these two numbers the solution to the above ODE is z(t) = z(0) + ż(0)t g 2 t2, which is the equation for a parabola that you may recognize. It is worth reflecting about what we have done so far. Newton s law tells us how to evolve a mechanical system in time. More specifically, let us define the state of our system as the collection of variables that completely specifies the conditions of our system at a moment in time. This is a key definition in mechanics. In the present case we have that state = {position, velocity} = {z, ż} because these were the initial conditions needed for our ODE. If our ODE is well posed for t [0, T ] with some T > 0 then there is a unique map such that state(0) state(t) for all t [0, T ]. Thus, in principle, the present state contains all the information needed to compute any future state; in other words the classical mechanical universe is deterministic and the future can in principle be predicted by solving a well-posed differential equation Energy Conservation To derive the energy conservation law from the solution of our ODE, we use a standard procedure: multiply the equation of motion by the velocity ż and manipulate. This yields ż z + gż = 0 d (ż2 ) dt 2 + gz = 0, which is a conservation law for the energy function H(z, ż) = ż2 2 + gz = const = E. This function is defined up to an integration constant. The meaning of it is that the energy function H(z, ż), which is called the Hamiltonian, is constant if z(t) satisfies Newton s law. Other ways of saying the same thing are: H is an invariant of the motion or H is a first integral of Newton s law. The value of H along a trajectory is denoted by E and is of course known from the initial state. In physics, the ż2 2 part of H is called the kinetic energy and the remainder is called the potential energy. The 1-dimensional potential energy is always given by a function V : R R of the given coordinate, noted V (z) if the coordinate is z, which is not always the same for different mechanical systems. Definition (Energy function (one dimensional)). Let z be the position of a point mass with mass m. The kinetic energy of the particle is given by T = 1 2 mż2 and the potential energy is a function V : R R, denoted V (z). Then the function H(z, ż) = T + U = 1 2 mż2 + V (z) is called the Hamiltonian energy function of the system.

11 1.2. ELEMENTS OF NEWTONIAN MECHANICS Phase Space The dynamics of our mechanical system is best visualized in phase space, which is the space spanned by two state coordinates z and ż. Phase space is attractive for the following reasons: any possible state of the system corresponds to a specific point in phase space, so phase space is also the space of all possible states; a solution traces out a trajectory in phase space, and these trajectories do not cross if Newton s law is well posed; If the energy is conserved, then the trajectories are contained in the contours (or level)of constant H. The last point means that plotting contours of constant H immediately produces the solution trajectories, albeit not their parametrization with respect to time t. This is very useful, because it allows us to learn something about the solution to another equation without solving the equation! That is what mathematics is all about. For Newton s apple we note that all trajectories are open and that there are no fixed points, i.e., no points at which ż = z = 0. By inspection, such a fixed point would correspond to a critical point of H, i.e., a point where H = Newton s Vector Law In general a mass point moves in three-dimensional space and its position is described by the cartesian coordinates x = (x, y, z). Newton s law is then the vector law mẍ = F(x, ẋ, t) subject to given x(0) and ẋ(0). Here the prescribed force vector F may in general depend on the state {x, ẋ} and on time t. However, we will consider only energy-conserving forces, i.e. forces that derive from a timeindependent scalar potential energy function V (x) via F = V. For instance, the gravitational potential associated with our ODE at the beginning was V = mgz. The vector character of the above equation means that Newton s law for a mass point corresponds to a system of three coupled ODE s in time and that the phase space is six-dimensional (two dimensions, position and velocity, for each spatial direction). Each coordinate adds a position-velocity pair to the definition of the system state. The number of coordinates is often called the degrees of freedom, and so the mass point has three degrees of freedom in general. In our apple example, the two degrees of freedom related to the horizontal directions were irrelevant because Newton s law reduced to ẍ = ÿ = 0 in them. Therefore, the system reduced to a single degree of freedom. another kind of reduction occurs through kinematical constraints, as the next example Pendulum A pendulum consists of a mass point with mass m attached to a rod with length l. The pendulum lies in the xz-plane and is fixed at the coordinate origin x = 0. There are two degrees of freedom for the position of the mass point, namely, x(t) and z(t), but they have to satisfy the constraint x 2 + z 2 = l, at all times. This can be used to eliminate one degree of freedom from consideration. Indeed, using the angle φ such that x = l sin φ and z = l cos φ we will derive the equation φ + g sin φ = 0, l

12 12 CHAPTER 1. FROM NEWTON S LAWS TO LAGRANGE S EQUATIONS where g is gravity as before. This nonlinear equation is harder to solve than the one before, but fortunately we can get most information about the solution from the Hamiltonian function. The state of the constrained system is described by the phase space coordinates {φ, φ} and energy conservation is derived exactly as before and yields the pendulum Hamiltonian H(φ, φ) = φ 2 2 g cos φ. l Clearly finding the potential energy amounts to setting the force in Newton s law equal to dv dφ and integrating. In systems with more than one degree of freedom this works only (locally) if the vector force F(x) satisfies the integrability condition F = 0. For small-amplitude oscillations φ 1 and therefore the potential energy term can be approximated by the first few terms of its Taylor series. Keeping only the first non constant term leads to the linear harmonic oscillator equations H = φ g l φ 2 2 and φ + g l φ = 0. with the simple general solution φ = A cos(ωt) + B sin(ωt), where the frequency Ω = g/l. In phase space the contours of this H are ellipses and therefore all orbits are bounded, which is consistent with the small-amplitude, low-energy approximation The Virial Theorem The pendulum allows demonstrating a second useful technique for extracting some knowledge about the solution from the governing equations without solving them. The procedure is similar to finding the energy conservation law, but with the difference that this time we multiply the ODE for the pendulum by φ instead of φ. Also, we then time-average the equation over the interval t [0, T ]. This first yields and then φ φ + g l φ sin φ = 0 d dt (φ φ) ( φ) 2 + g l φ sin φ = 0 1 T φ φ T T T 0 ( ( φ) 2 + g l φ sin φ ) dt = 0. The first term is evaluated at the endpoints of the time integral. Now, under the assumption that φ φ is bounded this first term goes to zero as T. If we denote the time-averaging as T by (...) then we obtain the virial theorem. ( φ) 2 = g φ sin φ. l In general, this shows a relationship that has to be true for all motions satisfying the assumption that φ φ is bounded. In particular, for small-amplitude oscillations the right hand side reduces to twice the averaged potential energy. This shows that for small oscillations there is equipartition of energy between its kinetic and potential forms. When does the virial apply? In general, the virial theorem is guaranteed to apply if the conservation law H = E can be used to derive an a priori bound on φ and φ. For the pendulum this occurs if H < E c, i.e. for the case of bounded orbits, where E c is the critical energy. It also applies to the case H = E c, because of the infinite travel time between saddle points that we noted before. However, it does not apply to the high-energy revolutions H > E c, which repeatedly spin over the top such that φ grows without bound.

13 1.2. ELEMENTS OF NEWTONIAN MECHANICS Use of Hamiltonian as a Differential equation The invariance of the function H(φ, φ) along a solution trajectory can be exploited to aid the integration of Newton s law. On a given trajectory the energy has value E and therefore the equation H(φ, φ) = E, can be solved as a first-order ODE. This means the second-order- ODE in Newton s law has been reduced to a first-order ODE by using the first integral H = E. In the pendulum case we obtain dφ dt = ± 2 (E + g ) l cos φ dφ 2 ( = ± gl E + cos φ) The sign is determined from the initial conditions. this reduces the solution procedure to a quadrature, which in this case can be performed using elliptic functions. For instance, this can be used to compute the period of finite-amplitude oscillations. This use of H as an ODE foreshadows the Hamilton-Jacobi PDE we will encounter later Generic Structure of One-Degree-of-Freedom Systems We can now summarize the structure of the equations for a generic coordinates q(t) satisfying Newton s law q + dv dq = 0 with potential energy function V (q). Note that we set the constant mass m = 1. The state of the system is determined by {q, q} and the Hamiltonian H(q, q) = T + V = q2 2 + V (q). is conserved if q is a solution of Newton s equation. Here T is a common shorthand for the kinetic energy. Newton s law expresses that the particle is accelerated towards decreasing values of the potential V (q). In general, this means acceleration towards a minimum of V, should one exist. In the case of Newton s apple there was no minimum and the push goes on forever. In the case of the pendulum minima of V occur at φ = 0 and its 2π-periodic repetitions. Upon reaching a minimum the particle overshoots due to its kinetic energy and then it climbs the potential on the other side. If this climbing motion is reversed before a maximum of V is reached then the particle turns back towards to minimum and an oscillation is observed. This is the low-energy case of the pendulum. If the motion goes over the next maximum of V then the particle aims for the neighboring minimum. This is the high-energy case of the pendulum going over the top. Turning for the particle corresponds to zero kinetic energy and energy conservation implies that V (q) = E at a turning point. Depending on the value of E this equation may or may not have a solution. For a given value of the energy H = E we can obtain the first-order equation q = ± 2(E V (q)) with quadrature dt. dq = ± 2(E V (q)) Finally, if q q is bounded along a trajectory, then we have the virial theorem q 2 2 = 1 2 q dv dq dt.

14 14 CHAPTER 1. FROM NEWTON S LAWS TO LAGRANGE S EQUATIONS for time-averaging over very long intervals on this trajectory. A useful special case arises if V is a power law V = Aq 2m with A > 0 and integer m > 0. Then q q remains bounded by energy conservation and the virial theorem yields q 2 2 = T = mv. For m = 1 this shows equipartition of energy, as in the linear harmonic oscillator but in any case it shows how the energy must, on average, be partitioned between its kinetic and potential forms. Like energy conservation, this is a powerful fact about the solution q(t) that can be derived without solving the governing equations. 1.3 Calculus of Variations Interesting things can be understood in at least two different ways. Mechanics can be understood from the point of view of Newton s law: solve a certain initial value problem for an ODE that takes from the present state at t = 0 to a future state at t = T. The true path of the coordinate q(t), say, is then a solution to this initial-value problem for the differential equation called Newton s law. Now, there is an alternative, complementary point of view that looks at the entire path q(t) for all t [0, T ] simultaneously and then gives a criterion for the true path as the solution to an optimization problem. To get to this point of view we need to recall the calculus of variations Functionals and Variations Consider a function y(x) defined on the interval x [a, b]. We will assume that y is always smooth enough to make possible all differential operations that we need to carry out 1. Typically, it will be sufficient that y is twice continuously differentiable. Define the integral J[y(x)] = b a F (y, y, x)dx in which the function F is assumed to be sufficiently smooth in all its three variable slots to allow partial derivatives up to second order to exist. We write derivatives of F with respect to the arguments in its slots as partial derivatives. For example F = x 2 y + (y ) 2 then F x = 2xy, F y = x2, F y = 2y, 2 F x y = 2 F y x = 2x and so on. The number J depends on the whole function y(x) and we say that J is a functional of y(x); this relationship is denoted by the square brackets. Thus a functional maps a function y(x) to a single number J. This is a familiar concept, for instance the usual integral norms are functionals. The calculus of variations is concerned with the change in J if the function y(x) is subject to a small variation, i.e., if y(x) is changed by a small amount. This means that y(x) is replaced by y(x) y(x) + δy(x), where the variation δy(x) is a smooth function that is small in the sense that δy 1 and δy 1. 1 Paraphrasing Einstein: y(x) should be as smooth as necessary for the problem at hand, but not any smoother. The general theory of calculus of variations accommodates functions (and generalized functions) that are less regular than we assume.

15 1.3. CALCULUS OF VARIATIONS 15 Here the variation of y equals the derivative of δy, i.e. δy = d dx δy. The change in J is the also small and can be computed from the Taylor expansion of F as J[y + δy] J[y] = = b a b a (F (y + δy, y + δy, x) F (y, y, x))dx ( δ F y (y, y, x) + δy F ) y (y, y, x) dx + o(δy, δy ). Using δy = d dxδy the integral can be rewritten using integration by parts b [ F δj = y d ( )] F dx y δydx + F y δy b. a a This expression is called the first variation of J around y(x) and it is usually denoted by δj. In general, it consists of an integral part and an endpoint part. The endpoint part vanishes if the admissible variations satisfy δy(a) = δy(b) = 0. For fixed y(x) the first variation δj is a linear functional in δy, and it plays the same role here as does the differential in ordinary calculus Extremals A function y(x) is an extremal of J if the first variation of J around y(x) vanishes for all δy that vanishes at the endpoints, i.e., δj = 0, δy s.t. δy(a) = δy(b) = 0. This is only possible if the integrand is zero everywhere. Otherwise, we could choose a variation that is zero at the endpoints but makes the integral nonzero, which leads to δj 0. Therefore an extremal must satisfy the celebrated Euler-Lagrange equation Definition (Euler-Lagrange equation (EL-equation)). The EL-equation for the extremal variational problem is given by EL : ( ) d F dx y = F y. Therefore the EL-equation is typically a second-order ODE for y(x). Remark The boundary conditions depend on the admissible functions y(x). For instance, if y(a) and y(b) are fixed then these are the boundary conditions. If y(b) is not fixed then the vanishing of δj for all δy implies that F y = 0 at x = b. This is called the natural boundary condition for the variational problem. An analogous statement applies at the other endpoint at x = a Shortest Path A simple example is to find the shortest path between two points (x A, y A ) and (x B, y B ) in twodimensional Euclidean space. Actually, we have not spoken about minima and maxima yet, and a good deal of work is needed to verify whether a given extremal corresponds to a minimum of the functional or not. Sometimes this is clear from the context, as is the case here. We assume that the curve can be parametrized by x, x B > x A and then the length functional can be written as

16 16 CHAPTER 1. FROM NEWTON S LAWS TO LAGRANGE S EQUATIONS J = B A ds = B A xb dx 2 + dy 2 = 1 + y 2 dx based on the function y(x) such that y(x A ) = y A and y(x B ) = y B. Thus F = 1 + y 2 and the EL equation is ( ) d F dx y = 0 F y = x A y 1 + y 2 = const. This implies that y is constant and therefore the extremal y(x) is a straight line through the two endpoints, as it should be. A variant of this problem allows the second endpoint to move freely in y at fixed x B ; i.e. y(x B ) is not fixed. This brings into play the natural boundary F condition at this point. i.e. y = 0 at x = x B. This implies that y = 0 there. Thus in this case the extremal is a horizontal straight line connecting the first point (x A, y A ) to the second point (x B, y B ). The y-location of the second point is itself part of the solution. Clearly, here we found the shortest path between the first point and the line x = x B Multiple Functions The EL-equation generalize easily to functionals that depend on multiple functions. Specifically, for a functional that depends on N functions y n (x) with integrand F (y 1, y 1,..., y N, y N, x), the EL-equations are ( ) d F n = 1,..., N : dx y n = F. y n Typically, this is a system of N coupled second-order ODEs for the functions y n (x). For instance, if we use a parametric representation of the path in the form (x(τ), y(τ)) with τ [0, 1] such that (x(0), y(0)) = (x A, y A ) and (x(1), y(1)) = (x B, y B ), then we naturally have a length functional that depends on N = 2 functions as 0 J[x(τ), y(τ)] = 1 0 F (ẋ, ẏ)dτ = 1 0 ẋ2 + ẏ 2 dτ. The dot denotes differentiation with respect to the parameter τ. The first variation of J is [ ( ) ( ) ] 1 d ẋ δj = dτ ẋ2 δx + d ẏ ẋδx + ẏδy 1 + ẏ 2 dτ ẋ2 δy dτ + + ẏ 2 ẋ2 + ẏ. 2 0 The variations δx and δy are independent and therefore the necessary conditions for an extremal are the two EL-equations F ẋ = ẋ ẋ2 + ẏ = const and F 2 ẏ = ẏ ẋ2 + ẏ = const. 2 These agree with the straight line condition. If the endpoints are variable then we now have an interesting new possibility in the case where the initial point (x A, y A ) is fixed whilst the final point (x B (s), y B (s)) can vary along a smooth curve parametrized by s. This imposes the condition (δx(1), δy(1)) = (x B (s), y B (s))ds on admissible endpoint variations, which implies that the critical path ends at a point (x B (s), y B (s)) such that ẋ(1)x B(s) + ẏ(1)y B(s) = 0.

17 1.3. CALCULUS OF VARIATIONS 17 Plugging everything together we get an equation for s and therefore for the endpoints (x B (s), y B (s)). The geometric interpretation of the last equation is simple: the shortest path must intersect the curve of possible endpoints with an angle of ninety degrees. This is a general property of the shortest path between a point and a smooth curve and it is consistent with the special case x B = const that we looked at before Symmetries and Conservation Laws We return to the generic N = 1 functional and define a conservation law to be a function G(y, y, x) that is constant along extremals of the functional. In other words, G is a first integral of the EL-equations that is dg dx = G x + G y y + G y y = 0 if y(x) satisfies the EL-equations. Conservation laws greatly simplify the problem of finding the extremal function y(x) in the first place, and they also allow us to understand something about the nature of the extremals without knowing them in detail. It turns out that many conservation laws can be linked to continuous symmetries of the functional J relative to transformation groups applied to x and y. The most general theorem to this effect is called Noether s theorem, but we will only use a few simple consequences of it. For instance, the distance functional is invariant under the continuous transformation group (x, y) (x, y) + (a, b) for arbitrary a, b R (this transformation includes the endpoints and the claimed invariance is then trivial). We say that J has a continuous symmetry with respect to translations in both x and y. This reflects the homogeneity of Euclidean space. By inspection, for a general functional based on the integrand F (y, y, x) this kind of translational symmetry is possible only if F does not depend explicitly on either x or y: i.e. we can have F (y ) only, and we saw that this led to the conservation law F y = const. In general, a translational symmetry in either x or y can be used to write down a generic conservation law. The form of the conservation law depend on whether the symmetry refers to a dependent or an independent variable. For example, a symmetry in the dependent variable y implies F (y, x) and this leads directly to a generic conservation law for G y = F y because dg y dx = d ( ) F dx y = F y = 0 by the EL-equations. Similarly, a symmetry in the independent variable x implies F (y, y ) and this also leads to a generic conservation law, but for the different quantity G x = y F y F. This follows from dg x dx ( ) F = y y + d F y dx y F x F y y F y y = F x = 0. In the distance example both these conservation laws reduce to y = const. Similarly conservation laws are obtained for arbitrary N 1. For instance, in the parametrized version of the distance problem we had N = 2 and the functional again had translational symmetries in x and y. However, in this case both x and y are dependent variables and therefore the associated conservation laws are simply the quantities F F ẋ and ẏ. There is an additional symmetry in the independent parametrization variable τ, but the associated conservation law for G τ = ẋ F ẋ + ẏ F ẏ F is irrelevant because G τ is identically zero for arbitrary functions x(τ) and y(τ). This is a consequence of the fact that the parametrization of the curve is irrelevant.

18 18 CHAPTER 1. FROM NEWTON S LAWS TO LAGRANGE S EQUATIONS 1.4 The Action principle We now return to Newton s apple and rephrase its fate as a variational problem. Compared to the generic theory, we replace x by t, y(x) by z(t) and F by a function L to be defined as follows. The action integral is S[z(t)] = T 0 L(z, ż)dt = T 0 (T V )dt = T 0 (ż2 ) 2 gz dt, where L = T V is the Lagrangian. Notice that L is not equal to H. It involves the dif f erence of kinetic and potential energy. The following statement is called the action principle: Newton s law is the EL-equation for an extremal of the action integral relative to all trajectories that have a fixed initial point z(0) and a fixed terminal point z(t ). The proof of this is immediate: the vanishing of the first variation implies the EL-equation d L dt ż = L z and L z = g, L ż = ż z = g. This is Newton s law. Because the endpoints are fixed there are no further terms to consider in δs. The peculiar thing is that the boundary conditions under which the trajectory of z(t) is an extremal of S are different from the initial conditions used to solve Newton s law. The former involve the position at both t = 0 and t = T whereas the latter involve the position and velocity simultaneously at time t = 0. The action functional S has a symmetry with respect to time because the Lagrangian L does not depend explicitly on t. This implies the conservation of ż L ż L = ż2 + gz = H(z, ż), 2 which is the familiar energy conservation law. Time symmetry implies energy conservation. Newton s apple immediately generalizes to a generic one-degree-of-freedom system with coordinate q(t). We have that the action S[q] = T 0 L(q, q)dt = T 0 (T V )dt = T 0 ( ) q 2 2 V (q) dt has vanishing first variation at the true path q(t) subject to fixed q(0) and q(t ). In other words, the true path q(t) is an extremal of S over the space of all admissible functions satisfying the fixed endpoint conditions. The EL-equation is d L dt q = L q q + V (q) = 0. The above remains true also for time-dependent potentials V (q, t) Coordinate-Invariance of the Action Principle There are a number of distinct properties of the action principle that make it attractive to use. First, the generic EL-equation are invariant under arbitrary coordinate transformations. This means that if we know that q(t) satisfies the generic EL-equation d L dt q = L q then any other coordinate Q(t) such that q = f(q) for some function f also satisfies the generic EL-equation based on the transformed Lagragian

19 1.4. THE ACTION PRINCIPLE 19 L(Q. Q, t) = L(f(Q), f (Q) Q, t). This transformed Lagrangian follows directly from the variable substitution and using q = f (Q) Q. The partial derivatives combine to yield ( d L ) dt Q = d dt L Q = L q f (Q) + L q f (Q) Q and L L = Q q f (Q) ( ) L f + L q q f Q = L q f + L q f Q = L Q. We have indeed obtained the generic EL-equation for Q. Of course, the final ODE for Q(t) will differ from that for q(t). The point is that the procedure that leads to the ODE is the generic EL-equation, which is always formed in the same way. A second property of the action principle is that, in conservative force fields, it generalizes easily to a particle moving in three dimensions. Then Newton s law is the vector law ẍ + V (x) = 0 for the particle location x = (x, y, z). However the action S remains the scalar with Lagrangian L given by L(x, ẋ) = 1 2 ẋ 2 V (x). The action principle then relates to independent variations of x(t), y(t) and z(t). corresponding EL-equations are the three coupled ODEs The d L dt ẋ = L x, d L dt ẏ = L y, d L dt ż = L z. This is already easy to solve in Cartesian coordinates but the key point is that these equations remain valid in arbitrary curvilinear coordinates (q 1, q 2, q 3 ). This is because of the coordinateinvariance of the EL-equations that was already noted. The same is not true for Newton s law, which needs to be reformulated in curvilinear coordinates according to the rules of vector calculus, usually a tedious task. An example will make this clear Central Force Field Orbits Consider a very heavy point mass sitting at the origin of a three-dimensional Cartesian coordinate system. A very light particle is orbiting it with position x(t). We neglect the motion of the heavy mass and want to compute the motion of the light mass, which we call our particle. This is a model for Earth orbiting the sun, for example. The heavy mass creates a gravitational field that depends only on distance r from the origin. It acts like a potential energy V (r) on the particle. Specifically, in Newton s theory of gravitation the potential is given by V (r) = G r, where G > 0 is a constant. This pulls the particle towards the origin all the time, with a central force that is proportional to 1. Now, if z(0) = ż(0) = 0 then one can easily show that r 2 the particle will never leave the xy-plane. We orient the coordinates such that this is the case

20 20 CHAPTER 1. FROM NEWTON S LAWS TO LAGRANGE S EQUATIONS and now have a system with two degrees of freedom for the particle, say polar coordinates with radius r(t) and angle θ(t) such that x = r cos θ and y = r sin θ. A naive application of Newton s laws would be to write r + V (r) = 0 and θ = 0, which captures the central force and the correct fact that there is no force in the azimuthal direction. However, the statement is clearly wrong as it leads to uniform angular motion θ = A + Bt and to finite-time collapse of the particle r = 0. Both are clearly wrong. Indeed, you would not be reading this if these equations were true, hence they must be false. The error was, of course, that Newton s law takes different forms in Cartesian and polar coordinates. Rather than using vector calculus, we correct our mistake by using the scalar action principle. Here all that is needed is to find the correct expression for the kinetic energy T in polar coordinates. This is easily done based on the Euclidean length element ds, which is given by ds 2 = dx 2 + dy 2 = dr 2 + r 2 dθ in Cartesian and polar coordinates. This elementary use of the metric is all we need to know about the nature of polar coordinates. The kinetic energy of the particle is T = 1 2 and therefore the Lagrangian is ( ) ds 2 = 1 dt 2 (ẋ2 + ẏ 2 ) = 1 2 (ṙ2 + r 2 θ2 ) L(r, ṙ, θ) = 1 2 (ṙ2 + r 2 θ2 ) V (r). We could now try to solve the two EL-equations d L dt ṙ = L r and d L L = dt θ θ. However, there is a shortcut and as mathematicians we know that taking shortcuts is always worth it, no matter how difficult it is to actually do it. In this case the shortcut comes from the symmetries of S. The present S is symmetric with respect to time t and angle θ, i.e., it is symmetric with respect to continuous translations in time and continuous rotations in space. By Noether s theorem there are two corresponding conserved quantities. The time symmetry gives the usual conserved energy H = ṙ L ṙ + θ L θ L = 1 2 (ṙ2 + r 2 θ2 ) + V (r) = E and the rotational symmetry gives a second conserved quantity via d dt L θ = 0 L θ = r2 θ = M. This quantity is called the angular momentum in physics and it is conserved 2 for all central potentials V (r). The presence of two conserved quantities allows us to integrate our system with two degrees of freedom by a sequence of quadratures. First, we substitute M for θ in the energy law. 2 The conservation of M is enough to explain Kepler s second law, the law of equal areas, which was extrapolated from astronomical observations before the laws of mechanics were known.

21 1.4. THE ACTION PRINCIPLE 21 H(r, ṙ, θ) = E to obtain H(r, ṙ, M/r 2 ) = E, which is a first order ODE for r(t) alone. Upon solving this ODE for r(t) we can in turn view the second conservation law θ = M/r 2 as a first-order ODE for θ(t). This procedure can be carried out and yields the full orbit r(t) and θ(t) in terms of quadratures. Here we will be satisfied with information on r(t) alone, i.e. 2ṙ2 1 + V (r) + M 2 2r 2 = E. This can be viewed as the energy equation for a generic one-degree-of-freedom system with an effective potential energy V eff = V (r) + M 2 2r 2. For nonzero M the behavior is fundamentally altered near r = 0, because now the effective potential accelerates r(t) away from the origin. This effect is what was missed by the naive approach, which is only correct in the case M = Systems with Constraints and Lagrange Multipliers A further property of the action principle is that it generalizes easily to systems with constraints. For instance, the pendulum is the result of considering a point mass with two degrees of freedom x(t) and z(t) exposed to gravity and subject to the constraint x 2 + z 2 = l. Without this constraint, using polar coordinates r, φ, we have the Lagrangian L = 1 2 (ṙ2 + r 2 φ2 ) + gr cos φ. With the constraint, we will assume that the action principle continues to apply in the following sense: the action S is extremal relative to all r(t) and φ(t) that satisfy the constraint as well as the usual endpoint conditions. This assumption has been corroborated by solving many model problems and finding no contradiction; we treat it as a basic statement of physics. In the present case the constraint r = l can be directly substituted in the Lagrangian above, which eliminates r and reduces the problem to a one-degree-of-freedom system with Lagrangian L = 1 2 l2 φ2 + gl cos φ. The variations in φ are unconstrained. However, it often happens that the constraint cannot be used to eliminate degrees of freedom. For instance, this could happen if the constraint involves time derivatives of coordinates. Then the EL-equations do not apply because they have been derived under assumption of unconstrained variations. Still, in such cases the constraint can be incorporated by the general method of Lagrange multipliers, as we shall see in this example. This is a mathematical method and not a physical law. It transforms a variational problem with constraints into a new unconstrained problem with more degrees of freedom. For this unconstrained system the EL-equations then apply in their standard form. Specifically, the method consists of using the homogeneous constrains r l = 0 to form the augmented Lagrangian ˆL = L λ(r l). Here λ(t) is a Lagrange multiplier function that must be added to the list of unknown functions r(t) and φ(t). Because the constraint acts at every moment in time there is not a single Lagrange multiplier but a full function. Indeed, by breaking down the action integral into a finite Riemann sum it is clear that there are as many constraints as members in that

22 22 CHAPTER 1. FROM NEWTON S LAWS TO LAGRANGE S EQUATIONS sum. They all have their own multiplier and in the continuous limit these multipliers become a function of t. Now, according to the theory of Lagrange multipliers the augmented action Ŝ based on ˆL is extremal relative to unconstrained variations of r, φ and λ precisely if the original action is extremal relative to the constrained variations in r and φ. Therefore, we consider the three EL-equations for the augmented action They are S[r, φ, λ] = T 0 [ ] 1 2 (ṙ2 + r 2 φ2 ) + gr cos φ λ(r l) dt. r : φ : λ : r = r φ 2 + g cos φ λ d dt (r2 φ) = gr sin φ r = l The last equation enforces the constraint. In the general case, these three equations need to be solved simultaneously. In the present simple case substituting from the third into the second equation again produces the EL-equation for φ alone. the first equation decouples and becomes λ = l φ 2 + g cos φ. This gives λ(t) once φ(t) has been found. It can be shown that λ is equal to the central force along the rod of the pendulum that is necessary to enforce the constraint. This turns out to be true more generally, i.e., the value of the Lagrange multiplier is proportional to the constraint forces. The method of Lagrange multipliers is an amazing construction. In your own time you might ponder a few peculiar questions such as whether Ŝ is numerically equal to S for the extremal path, what happens to λ when the constraint is rewritten as sin(r l) = 0 (or some such), what are the symmetries and associated conserved quantities of the augmented Lagrangian and what is the derivative of the extremalized action S with respect to l?

23 Chapter 2 Differential forms 2.1 Notations In these notes U will denote an open subset of R n, α a k-form, β an l-form, f a function and X a vector field on U. We will denote by (x 1,..., x n ) the coordinates on U and accordingly write α(x) = β(x) = X(x) = n i 1,...,i k =1 n j 1,...,j l =1 n α i1,...,i k (x) dx i 1 dx i k, β j1,...,j l (x) dx j 1 dx j l, X i (x) x i. For simplicity we will assume throughout that α, β, f and X are smooth, i.e., that all components α i1,...,i k, all components β j1,...,j l, all components X i and f are arbitrarily often continuously differentiable. Recall that functions and zero-forms are one and the same thing. Remark The symbols,..., x 1 x denote the basis of R n corresponding to our choice n of coordinates. The symbols dx 1,..., dx n denote the dual basis of (R n ) ; i.e., the canonical pairing of dx i with is 1 if i = j and 0 otherwise. The induced basis of k (R n ) is given by x j the symbols (dx i 1 dx i k) 1 i1 < <i k n. The wedge product of the symbols dx i is defined by the identity dx i dx j = dx j dx i. Using this identity one can rewrite all the terms in the above expansion of α into a linear combination of the basis elements (dx i 1 dx i k) 1 i1 < <i k n of k (R n ). Notice that the coefficients of each basis element is given by the complete antisymmetrization of the components with respect to their indices. This means that it is enough to consider completely antisymmetric components, but it is quite convenient (see, e.g., the formulae for the wedge product and for the exterior derivative below) to allow for more general (though redundant) components. Remark If k = 0, then α is a function; if k > n or k < 0, then α is 0. Remark The attentive reader might have noticed that we use consistently upper and lower indices (to denote components of vectors and forms, respectively). This helps in bookkeeping but is not essential. It also allows using Einstein s convention on repeated indices that a sum over an index is understood when the index is repeated, once as an upper index and once as a lower index. With this convention, which we will not use in this note, the last formula would, e.g., read: X(x) = X i (x) x i. 23

24 24 CHAPTER 2. DIFFERENTIAL FORMS 2.2 Definitions The wedge product The wedge product of α and β is the (k + l)-form α β(x) = n n i 1,...,i k =1 j 1,...,j l =1 Notice that if k + l > n, then α β is automatically zero The exterior derivative The differential or exterior derivative of α is the (k + 1)-form dα(x) = n j=1 i 1,...,i k =1 α i1,...,i k (x)β j1,...,j l (x) dx i 1 dx i k dx j 1 dx j l. n x j α i 1,...,i k (x) dx j dx i 1 dx i k. Notice that dx i denotes at the same time the i-th basis vector of (R n ) and the differential of the coordinate function x i. Also notice that if α is a top form, i.e., k = n, then automatically dα = The Pullback If V is an open subset of R m and φ a smooth map V U, the pullback of α is the k-form on V defined by φ α(y) := k dφ(y) α(φ(y)), y V, where dφ(y): R m R n denotes the differential of φ at y, dφ(y) its transpose and k dφ(y) the k-th exterior power of the latter. If (y 1,..., y m ) are coordinates on V, we have φ α(y) = m n j 1,...,j k =1 i 1,...,i k =1 α i1,...,i k (φ(y)) φi 1 y j 1 (y) φik y j k (y) dyj 1 dy j k. Observe that, if W is an open subset of R s and ψ a smooth map W V, we have The Lie derivative (φ ψ) = ψ φ. The Lie derivative with respect to X of α is the k-form φ X,t L X α = lim α α, t 0 t where φ X,t is the flow of X at time t. Explicitly we have L X α(x) = n i 1,...,i k =1 + X(α i1,...,i k )(x) dx i 1 dx i k + n i 1,...,i k =1 r,s=1 n ( 1) r 1 α i1,...,i k (x) Xir x s (x) dxs dx i 1 dx ir dx i k, where X(α i1,...,i k ) = n j=1 Xj α x j i1,...,i k denotes the directional derivative of the function α i1,...,i k in the direction of X and the caret denotes that the factor dx ir is omitted.

25 2.3. PROPERTIES The contraction The contraction of X with α is the (k 1)-form n n ι X α(x) = ( 1) r 1 α i1,...,i k (x)x ir (x) dx i 1 dx ir dx i k. i 1,...,i k =1 r=1 If α is a function, i.e., k = 0, then automatically ι X α = Properties The wedge product is bilinear over R n, whereas the differential, the pullback, the Lie derivative and the contraction are linear over R n. Moreover, we have the following properties: α β = ( 1) k+l β α, (2.1) d 2 α = 0, (2.2) d(α β) = dα β + ( 1) k α dβ, (2.3) φ f = f φ, (2.4) φ (α β) = φ α φ β, (2.5) dφ α = φ dα, (2.6) L X f = X(f), (2.7) L X (α β) = L X α β + α L X β, (2.8) L X dα = dl X α, (2.9) ι X (α β) = ι X α β + ( 1) k α ι X β, (2.10) L X α = ι X dα + dι X α (Cartan s formula). (2.11) Observe that the above properties characterize d, φ, L X and ι X completely: the formulae in the previous Section may be recovered from these properties. If Y is a second vector field, we also have ι X ι Y α = ι Y ι X α, (2.12) L X L Y α L Y L X α = L [X,Y ] α, (2.13) ι X L Y α L Y ι X α = ι [X,Y ] α, (2.14) where [X, Y ] is the Lie bracket of X and Y defined by [X, Y ](f) = X(Y (f)) Y (X(f)).

26 26 CHAPTER 2. DIFFERENTIAL FORMS

27 Chapter 3 Hamiltonian systems 3.1 Introduction The second formulation we will look at is the Hamilton formalism. In this system, in place of the Lagrangian we define a quantity called the Hamiltonian, to which the canonical equations of motion are applied. While the EL-equations describe the motion of a particle as a single second-order differential equation, the canonical equations describe the motion as a coupled system of two first-order differential equations. One of the many advantages of Hamiltonian mechanics is that it is similar in form to quantum mechanics, the theory that describes the motion of particles at subatomic distance scales. An understanding of Hamiltonian mechanics provides a good introduction to the mathematics of quantum mechanics. 3.2 Legendre Transform Consider a smooth real-valued function f(x) for x R that is strictly convex, i.e., f > 0 for all x. Then the Legendre transformation transforms the pair (x, f(x)) into a new pair (p, F (p)) by the definition Definition (Legendre Transform). The Legendre transform of a convex function f(x) is given by The necessary condition for a maximum is F (p) = max[px f(x)]. x p = f (x). This is to be viewed as an equation for x given p. The second x-derivative of the Legendre transform is f (x), which is negative by assumption. Therefore p = f (x) is necessary and sufficient for the unique maximum. Thus, an equivalent way to write F (p) is via the two equations This can be contracted to F (p) = px f(x) and p = f (x). F (p) = xf (x) f(x) provided one does not forget to solve p = f (x) for x in terms of p. This is invariably the tricky step. Note that the Legendre transformation is not a linear transformation despite 27

28 28 CHAPTER 3. HAMILTONIAN SYSTEMS the appearance of the contracted equation. this is because inverting p = f (x) is not a linear procedure in f(x). As an example, the Legendre transformation of f = xα α for α > 1 and x > 0 is computed to be F (p) = α 1 α xα and p = x α 1 > 0. Inverting the second equation and substituting in the first yields F (p) = pβ β where 1 α + 1 β = 1. We see that β > 1; i.e., F (p) for p > 0 is again convex. This is true in general as we shall see Derivatives and Convexity From the equations F (p) = px f(x) and p = f (x), the differential of F (p) can be written as df = pdx + xdp f (x)dx = xdp F (p) = x. This is a remarkable formula, which makes this transform useful for differential equations. The second derivative follows as F (p) = dx dp = 1 dp dx = 1 f (x) > 0, and therefore F (p) is strictly convex. We observe that the Legendre transform preserves convexity Involution This means that we can apply the Legendre transform to F (p) and the claim is that this returns f(x). This means that the Legendre transform is an involution; i.e., it is its own inverse. To show that this is true we rewrite the Legendre transform as f(x) = xf (x) F (p) = F (p)p F (p), where we have used the equation for the differential and p = f (x). expression is equal to However, the last max[xp F (p)] p for fixed x by the same argument as given before, which yields x = F (p) for the location of the maximum in p. Therefore f(x) = max[xp F (p)] and F (p) = max[px f(x)] p x both hold and this proves that the Legendre transform is an involution.