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Historical and Modern Perspectives on Hamilton-Jacobi Equations

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Historical and Modern Perspectives on Hamilton-Jacobi Equations Historical and Modern Perspectives on Hamilton-Jacobi Equations Robin Ekman June 10, 2012 Abstract We present the Hamilton-Jacobi equation as originally derived by Hamilton in 1834 and 1835 and its modern interpretation as determining a canonical trans- formation. We show that the method of characteristics for partial differential equations, applied to the Hamilton-Jacobi equation, yields Hamilton’s canonical equations and the action as the solution. Canonical perturbation theory is ap- plied to the Sun-Earth-Jupiter system to calculate perturbations of the orbital elements. We also study tidal locking and find a damped harmonic oscillator equation for a moon that is almost locked. Finally, we present a modern (1980s), weaker notion of solutions to the Hamilton-Jacobi equation, viscosity solutions. We reproduce proofs that this concept is consistent with the classical concept and that visocsity solutions are unique, for locally Lipschitzian Hamiltonians. Contents I Historical Background for Hamilton-Jacobi Equations in Clas- sical Mechanics 4 I.1 Hamilton’s Characteristic and Principal Functions . 4 I.2 The Principal Function as a Canonical Transformation . 10 I.3 Hamilton’s Equations as Characteristics . 13 II Classical Perturbation Theory Applied to Planetary Orbits 15 II.1 CanonicalPerturbationTheory . 15 II.2 Separability and Action-Angle Variables . 17 II.3 Action-Angle Variables for the Kepler Problem . 18 II.4 Perturbation ............................. 24 II.5 Averaging............................... 25 II.6 PerturbationsofOrbitalElements . 26 II.7 TidalLocking............................. 29 IIIExistenceandUniquenessofSolutions 32 III.1Viscositysolutions .......................... 33 III.2 Classical Solutions are Viscosity Solutions . 34 III.3 Viscosity Solutions are Unique . 36 III.4OtherMethods ............................ 38 A ExpansionofthePerturbation oftheInclination 39 List of Figures II.1 Orbitalelements ........................... 22 II.2 Angles used for the Kepler problem. ON is the line of nodes; R is the position of the orbiting body. 22 II.3 Eccentric anomaly ψ and true anomaly ϕ. ............ 23 II.4 Average precession of the perihelion using values for the Sun, EarthandJupiter. .......................... 27 1 II.5 Trajectory for a planet perturbed by another planet . 27 II.6 An orbiting body of finite extent. (Not to scale.) . 30 III.1Asolutiontothewaveequation? . 32 2 [Lagrange showed] that the most varied consequences respecting the motions of systems of bodies may be derived from one radical for- mula; the beauty of the method so suiting the dignity of the results, as to make of his great work a kind of scientific poem. But the science of force, or of power acting by law in space and time, has undergone already another revolution, and has become already more dynamic ... and while the science is advancing thus in one direction by the improvement of physical views, it may advance in another direction also by the invention of mathematical methods. —William Rowan Hamilton, On a General Method in Dynamics, 1834. 3 Chapter I Historical Background for Hamilton-Jacobi Equations in Classical Mechanics In this chapter we will consider three perspectives on Hamilton-Jacobi equations in classical mechanics. The common denominator in these three presentations is showing that a system of ordinary differential equations (ODE) can be reduced to one partial differential equation (PDE), and vice versa. We shall discuss mechanical problems by which we mean a system of particles, interacting in some way described by a potential function. By a solution we mean a set of functions of time giving the motion of each particle, given the initial positions and velocities. First we will present Hamilton’s original derivation from his 1834 and 1835 papers On a General Method in Dynamics and A Second Essay on a General Method in Dynamics [9, 10] using the calculus of variations, extending the work of Lagrange. An important step here is Hamilton’s derivation of a system of first-order ODE, the canonical equations. Secondly, we present a later view where the Hamilton-Jacobi equation determines a coordinate transformation that also solves the mechanical problem at hand [7, 1]. This view assumes the canonical equations to have been found but uses no more methods from the calculus of variations. Lastly, we show that if we are given the Hamilton- Jacobi equation, the method of characteristics in the theory of PDE generates Hamilton’s canonical equations [6]. I.1 Hamilton’s Characteristic and Principal Func- tions Hamilton’s General Method In his 1834 paper [9] Hamilton considers a system of n particles (points, in his terminology) interacting by repulsion or attraction depending on their distances. 4 That is, from the equations of motion n m x00 δx = δU i i · i i=1 X 00 where mi denote the masses, xi is the acceleration in rectangular (Cartesian) 0 coordinates (likewise, xi will be the velocity), δxi an infinitesimal displacement and δU an infinitesimal change in the function U, which describes the interac- tions as U = mimj f(rij ). We have, unlike Hamilton, used vectorX notation and the dot product, and will continue to do so, as the modern reader should be familiar with this condensing notation. Hamilton called U the ‘force-function’, but we know it today as the potential and conventionally define it with the opposite sign; we would write mx00 = U. We shall follow Hamilton’s choice of sign. Next−∇ Hamilton considers 1 T = m x0 2, 2 i i 0 X xi being the velocity, which he calls half “the living force of the system”, in modern terms the kinetic energy. Then using the equations of motions we can write dT dT = dt = m x0 x00dt = m dx x00 = dU dt i i · i i i · i and we can conclude that X X T = U + H (I.1) where H is a constant of the motion. Hamilton calls this the “law of living force”; remembering that his definition of U differs from the modern by a sign we recognize it as the conservation of energy. If the initial conditions change, then H can change too and we have δT = δU + δH. Then integrating along the path taken by the system m dx δx0 = m dx0 δx + δHdt. (I.2) i i · i i i · i Z X Z X Z Hamilton presents his next result as following from “the principles of the calculus of variations”, skipping some steps. If we define, as Hamilton does, t 0 V = 2T dt = mixi dxi (I.3) 0 · Z Z X the variation is δV = m δx0 dx + m x0 δdx . i i · i i i · i Z X X The second term can be integrated by parts, putting another derivative on x: t 0 00 0 xi δV = miδxi dx mixi δxi dt + mixi δxi · − · · ai Z Z0 X X X 5 where we also used that the order of variation and differentiation can be inter- changed. Now the first integral can be replaced using (I.2) and since H is a constant of motion we can replace the integral with H with tδH. That is, t 0 00 0 xi δV = miδxi dxi + tδH mixi δxi dt + mixi δxi . · − · · ai Z Z0 X X X 00 0 But the two integrals cancel, since xi dt = dxi and we are left with δV = m x0 δx m a0 δa + tδH i i · i − i i · i X X where ai is the initial position. If V is considered a function of the initial and end points, we have the system of equations ∂V 0 = mixi (I.4) ∂xi ∂V 0 = miai (I.5) ∂ai − ∂V = t (I.6) ∂H which describes the system in terms of one single function, the characteristic function V . Now using this system of equations in the law of living force (I.1) Hamilton obtains two PDE, for V , the former which he calls final and the latter initial 1 1 ∂V 2 = U + H (I.7) 2 m ∂x i i X 2 1 1 ∂V = U0 + H (I.8) 2 mi ∂ai X where U0 is the inital force function. If the function V were known, the me- chanical problem would be solved, for we can verify that the system of equations for the characteristic function generates the same equations of motions that we started with. Then the problem of solving n ODE of second order has been transformed to solving one first order PDE. In the last section of his first paper, Hamilton introduces the function S defined by the relation V = tH + S. Then S, which Hamilton here calls the auxiliary function, using (I.1), (I.3) and t tH = 0 H dt since H is a constant of motion, satisfies R t S = (T + U)dt (I.9) Z0 ∂S and its partial derivative with respect to t is ∂t = H, its other partial deriva- tives satisfy the same equations as do the partial derivatives− of V . 6 Hamilton’s Second Essay In the Second Essay, Hamilton begins by showing that if the 3n Cartesian coordinates are “functions of 3n other and more general marks of position”, η1, η2,...,η3n, the form of the equations of motion is that given by Lagrange d ∂T ∂T ∂U 0 = (I.10) dt ∂ηi − ∂ηi ∂ηi 0 where T is now a function of the ηi and ηi. This is a set of n ODE of second order. Since for each x0 we have x0 = η0 ∂xi (Hamilton did not consider time- i i j ∂ηj dependent coordinate systems) and T is homogeneously quadratic when ex- 0 P 0 1 pressed in xi, it is homogeneously quadratic in ηj and a theorem of Euler gives 0 ∂T 2T = ηj 0 . (I.11) ∂ηj X Hamilton then notes that taking the variation of this, and subtracting the vari- ation of T in the form ∂T 0 ∂T δT = 0 δηi + δηi (I.12) ∂ηi ∂ηi X one obtains ∂T ∂T δT = η0δ δη .
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