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Hamiltonian Dynamics CDS 140B Hamiltonian Dynamics CDS 140b Joris Vankerschaver [email protected] CDS 01/22/08, 01/24/08 Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 1 / 36 Outline for this week 1. Introductory concepts; 2. Poisson brackets; 3. Integrability; 4. Perturbations of integrable systems. 4.1 The KAM theorem; 4.2 Melnikov's method. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 2 / 36 References Books I J. Marsden and T. Ratiu: Introduction to Mechanics and Symmetry. I V. Arnold: Mathematical Methods of Classical Mechanics. I M. Tabor: Chaos and Integrability in Nonlinear Dynamics. I P. Newton: The N-vortex problem. I F. Verhulst. Papers I J. D. Meiss: Visual Exploration of Dynamics: the Standard Map. See http://arxiv.org/abs/0801.0883 (has links to software used to make most of the plots in this lecture) I J. D. Meiss: Symplectic maps, variational principles, and transport. Rev. Mod. Phys. 64 (1992), no. 3, pp. 795{848. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 3 / 36 References Software I GniCodes: symplectic integration of 2nd order ODEs. Similar in use as Matlab's ode suite. See http://www.unige.ch/∼hairer/preprints/gnicodes.html I StdMap: Mac program to explore the dynamics of area preserving maps. See http://amath.colorado.edu/faculty/jdm/stdmap.html Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 4 / 36 Introduction Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 5 / 36 Transition to the Hamiltonian framework I Consider a mechanical system with n degrees of freedom and described by generalised coordinates (q1;:::; qn). 1 2 I Denote the kinetic energy 2 mv by T , and the potential energy by V (q). Define the Lagrangian L to be T − V . I Define the canonical momenta pi as @L p = : i @v i This defines a map from velocity space with coords (qi ; v i ) to phase i space with coords (q ; pi ), called the Legendre transformation. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 6 / 36 Transition to the Hamiltonian framework The associated Hamiltonian is given by i H(q; p) = pi v − L(q; v): To express the RHS as a function of q and p only, we need to be able to invert the Legendre transformation. By the implicit function theorem, this is the case if the matrix @2L (1) @v i @v j is invertible. Notes I Not every Hamiltonian is associated to a Lagrangian in this way. See next week's class on vortex dynamics! I Much of the above can be extended to the case where (1) is singular. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 7 / 36 Hamilton's equations I Variational interpretation: arise as extrema of the following action functional: Z i S(q(t); p(t); t) = pi (t)_q (t) − H(q(t); p(t); t)dt: I Equations of motion: i @H @H q_ = and p_ i = − i : @pi @q Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 8 / 36 Properties of Hamiltonian systems Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 9 / 36 Poisson brackets: definition Let f (q; p; t) be a time-dependent function on phase space. Its total derivate is df @f dqi @f dp @f _ i f ≡ = i + + dt @q dt @pi dt @t @f @H @f @H @f = i − i + @q @pi @pi @q @t @f = ff ; Hg + ; @t where we have defined the (canonical) Poisson bracket of two functions f and g on phase space as @f @g @f @g ff ; gg = i − i : @q @pi @pi @q Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 10 / 36 Poisson brackets: properties A Poisson bracket is an operation {·; ·} on functions satisfying the following properties: 1. ff ; gg = −{g; f g; 2. ff + g; hg = ff ; hg + fg; hg; 3. ff ; fg; hgg + fg; fh; f gg + fh; ff ; ggg = 0; 4. ffg; hg = f fg; hg + gff ; hg. Property 3 is called the Jacobi identity. 2n Properties 1, 2, and 3 make the ring of functions on R into a Lie algebra. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 11 / 36 Rewriting Hamilton's equations I For any function f on phase space, we have df = ff ; Hg: dt i I For f = q and f = pi , we recover Hamilton's equations: i i @H q_ = fq ; Hg = ; @pi and @H p_ = fp ; Hg = − : i i @qi Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 12 / 36 Conserved quantities Definition A function f is a conserved quantity if it Poisson commutes with the Hamiltonian: ff ; Hg = 0: Immediate consequences: I If the Hamiltonian is autonomous, then it is conserved, as fH; Hg = 0; I If f and g are conserved, then so is ff ; gg: fff ; gg; Hg = ffg; Hg; f g − fff ; Hg; gg = 0; using the Jacobi identity. Usually this doesn't give too much information. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 13 / 36 Not all Poisson brackets are canonical: Euler equations I Consider a rigid body with moments of inertia (I1; I2; I3) and angular velocity Ω = (Ω1; Ω2; Ω3). Define the angular momentum vector Π = (Π1; Π2; Π3) = (I1Ω1; I2Ω2; I3Ω3): I The equations of motion for the rigid body (Euler equations) are Π_ = Π × Ω: or, written out in components, I1Ω_ 1 = (I2 − I3)Ω2Ω3; I2Ω_ 2 = (I3 − I1)Ω3Ω1; I3Ω_ 3 = (I1 − I2)Ω1Ω2: I Clearly not canonical, since odd number of equations. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 14 / 36 Euler equations: Poisson form I Define the rigid body Poisson bracket on functions F (Π), G(Π) as fF ; Ggr:b:(Π) = −Π · (rF × rG): I The Euler equations are equivalent to F_ = fF ; Hgr:b:; where the Hamiltonian is given by 1 Π2 Π2 Π2 H = 1 + 2 + 3 : 2 I1 I2 I3 Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 15 / 36 Other examples of Poisson brackets 2n I Fruitful approach: start from canonical Poisson bracket on R or similar, consider a group action which leaves {·; ·} invariant, and define the bracket on the quotient. I Other examples: ideal fluids, MHD, the Toda lattice, . (Idea for project. ) Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 16 / 36 Characteristic property of Hamiltonian flows Theorem The flow of a Hamiltonian vector field preserves the Poisson structure: fF ; Gg ◦ Φt = fF ◦ Φt ; G ◦ Φt g: (thm. 10.5.1 in [MandS]) Take the derivative of u := fF ◦ Φt ; G ◦ Φt g − fF ; Gg ◦ Φt : du = ffF ◦ Φ ; Hg; G ◦ Φ g + fF ◦ Φ ; fG ◦ Φ ; Hgg − ffF ; Gg ◦ Φ ; Hg dt t t t t t du I Jacobi identity: dt = fu; Hg. I Solution: ut = u0 ◦ Φt , but u0 = 0. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 17 / 36 Liouville's theorem To each Hamiltonian H, associate the following vector field on phase space: @H ! @pi XH := @H − @qi Liouville's theorem The flow of XH preserves volume in phase space. I Consequence of the following fact: flow ofx _ = f (x) is volume preserving () divf (x) = 0: I @ @H @ @H divXH = i − i = 0: @q @pi @pi @q More is true: the flow of XH is symplectic. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 18 / 36 Consequences of Liouville's theorem 1. In a Hamiltonian system, there are no asymptotically stable equilibria or limit cycles in phase space. 2. Poincar´e'srecurrence theorem: Theorem Let Φ: D ! D be a volume preserving diffeomorphism from a bounded n region D ⊂ R to itself. Then for any neighborhood U in D, there is a point of U returning to U after sufficiently many iterations of Φ. The sets U; Φ(U); Φ2(U);::: cover D and have the same volume. So, for some k and l, with k > l, k l k−l Φ (U) \ Φ (U) 6= ? , Φ (U) \ U 6= ?: Take y 2 U \ Φk−l (U) and put x = Φ−n(y)(n = k − l), then x 2 U and Φn(x) 2 U. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 19 / 36 Aside: symplecticity 2N 2N I Define a map ! : R × R ! R as follows !(X ; Y ) = signed area of parallellogram spanned by X ; Y : 2N 2N I A map Φ : R ! R is symplectic if it preserves these areas. Remarks I symplecticity ) volume preservation, but not vice versa unless 2N = 2; I ! is a differential form. Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 20 / 36 Integrability and near-integrability Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 21 / 36 Integrability: definitions Definition A Hamiltonian system with n degrees of freedom and Hamiltonian H is called integrable if there exist n functionally independent integrals Fi , i = 1;:::; n, which are in involution: fFi ; Fj g = 0. I The odds of randomly picking an integrable Hamiltonian among the class of all analytic functions are zero. I Motivation: if system is integrable, one can find a canonical trafo to action-angle coordinates (p; q) 7! (I ; φ) and H(p; q) 7! H0(I ); i.e. H0 doesn't depend on φ. Trivially integrable eqns of motion: @H0 I_ = 0 and φ_ = := !(I ): @I Joris Vankerschaver (CDS) Hamiltonian Dynamics 01/22/08, 01/24/08 22 / 36 Integrability: examples I Simple example 1 1 H = (p2 + q2) + (p2 + !q2) 2 1 1 2 2 2 0 Integrable; action-angle form: H = I1 + I2!.
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