1. MATH4248 Weeks 12-13 1 Topics: review of rigid body motion, Legendre transformations, derivation of Hamilton’s equations of motion, phase space and Liouville’s theorem and Poincare’s recurrence theorem, examples including free fall, harmonic oscillator, and pendulums - plane, spherical and rotational, survey of advanced topics including Poisson brackets, Hamilton-Jacobi equation, action-angle variables, integrable systems and chaos Objectives: To derive and understand Hamilton’s equations - in particular, their deep connections with symplectic geometry and topological dynamics

2. RIGID BODY MOTION-REVIEW 2 Definition The rotation group SO(3) consists of 3 x 3 matrices whose determinant equals 1 and whose inverses equal their transposes. Problem Show that SO(3) is a group under matrix multiplication using the following identities that hold for all 3 x 3 matrices A and B

3. RIGID BODY MOTION-REVIEW 3 Example Multiplication of a vector by the following matrix rotates the vector about the z-axis by angle Example Show that

4. RIGID BODY MOTION-REVIEW 4 Example A rotational motion about the z-axis can be described by choosing the to be a function of t It is convenient to rewrite this equation as where Problem Show that (identity matrix)

5. RIGID BODY MOTION-REVIEW 5 Now, we can dispense with the primes to obtain the following result: if a rigid body is rotating around the z-axis then there exists a function that satisfies Problem Show that the velocity of the particle satisfies body, the motion of that particle is described by the motion of its position vector by the following equation such that for any particle in where the matrix is skew-symmetric

6. RIGID BODY MOTION-REVIEW 6 where Problem Show that hence In this case we write Problem Analyse motion about the x and y axes

7. RIGID BODY MOTION-REVIEW 7 Problem Show that for general rotational motion of a rigid body about the origin, there exists a function O(t) with values O(t) in SO(3) such that O(0) = id (the identity matrix) and such that the motion of each particle in the rigid body is described by the equation Problem Differentiate this equation to obtain to show that S(t) is skew-symmetric then show that then differentiate the equation where there exists (angular velocity in space) such that hence

8. LEGENDRE TRANSFORMS 8 Definition Let f : R  R be convex and differentiable. The Legendre transform of f is the function g : R  R Example 1 Problem 1What is the Legendre transform of g ? Problem 2 Interpret f and p if x is replaced by

9. LEGENDRE TRANSFORMS 9 Theorem 1 Ifis convex then its is also a convex function and the Legendre transform of g equals f. Theorem 2 If Legendre transform, defined by is strictly convex, then determined fromby the equation is

10. LEGENDRE TRANSFORMS 10 Example 2 Let G be a positive definite symmetric matrix and construct the function Then by

11. HAMILTONIAN AS A LEGENDRE TRANSFORM 11 We will assume hereafter that the kinetic energy, and therefore the Lagrangian Definition The Hamiltonian is the Legendre transformation of with respect to is a strictly convex function offor all

12. HAMILTONIAN AND MOMENTUM 12 Is our new definition of the Hamiltonian, as a function of related to our old definition on page 16 of Weeks 8-9 vufoils as a function of ? where Answer Yes is uniquely chosen to satisfy New Old

13. HAMILTON’s EQUATIONS 13 with dH computed from the old definition of H Compare dH computed from the chain rule to obtain Hamilton’s equations

14. LIOUVILLE’S THEOREM 14 Theorem Assume that L, and therefore H, does not depend explicitly on t. Then Hamilton’s system of 2f equations defines a volume preserving flow on the 2f-dimensional phase space with coordinates Proof Letbe the flow defined by Hamilton’s equations so that is the solution of the equations with initial value Let D be any region in the phase space and define by It suffices to prove that

15. LIOUVILLE’S THEOREM 15 If h be the vector field Now then for small t, Therefore implies and D was arbitrary, therefore it suffices to prove that

16. LIOUVILLE’S THEOREM 16 and the result follows since Therefore

17. POINCARE’S RECURRENCE THEOREM 17 Proof We consider the images of U under iterates of g Theorem Let g be a volume preserving continuous one-to-one mapping that maps a bounded region D of Euclidean space onto itself : gD = D. Then in any nonempty open subset U of D there exists a point x which returns to U, i.e. for some If they were pairwise disjoint then D would have infinite volume, contradicting our bounded assumption hence for some

18. POINCARE’S RECURRENCE THEOREM 18 Corollary Poincare’s theorem shows that the flow is ergodic, that is the phase space can not be divided into two subsets that are invariant under g and have positive volume. The ergodic theorem implies that almost all points return will be recurrent Many mechanical systems have this property. A particle moving in a cup will (with probability one) return arbitrarily close to itself even though, if the cup is unsymmetric, the motion is unpredictably chaotic! These qualitative properties are studied in topological dynamics, a field motivated by the three body problem

19. PLANE PENDULUM 19 Lagrangian momentum Hamiltonian equations

20. PLANE PENDULUM 20 Phase space should be thought of as a cylinder Orbits are oscillations clockwise around (q,p)=(0,0) Orbits are fixed points with pendulum down Orbits are either separatrices (infinitely long swing), or unstable fixed points with pendulum up Orbits are rotations clockwise (top), or counterclockwise (bottom)

21. SPHERICAL PENDULUM 21 Lagrangian Hamiltonian Momenta

22. ROTATING PENDULUM 22 Lagrangian Hamiltonian Momenta

23. POISSON BRACKETS 23 Theorem A transformation is canonical (satisfies Hamilton’s equations for some Hamiltonian K(Q,P) iff the Poisson bracket or equivalently the following differential form is exact i.e. equals dF for some generating function F= F(q,Q)

24. CANONICAL TRANSFORMATIONS 24 Example (f=1) Let Q, P be ‘polar’ coordinates, so that Problem Show that this is a canonical transformation And compute its generating function Problem Show that the Hamiltonian for the harmonic oscillator becomes and therefore P, Q is an action variable, angle variable They give

25. HAMILTON-JACOBI THEORY INTEGRABLE SYSTEMS AND CHAOS 25 HJ theory provides the most powerful method to solve mechanical problems. Hamilton’s principle function is the minimal action is a generating function from (q,p) to It can be computed from Jacobi’s complete integral S by the Hamilton Jacobi eqn and the eqn A system is completely integrable if f-pairs of action- angle variables exist, else it tends to exhibit chaos