Introduction to Hamiltonian systems

Transcription

1 Introduction to Hamiltonian systems Marlis Hochbruck Heinrich-Heine Universität Düsseldorf Oberwolfach Seminar, November 2008

2 Examples Mathematical biology: Lotka-Volterra model First numerical methods Mathematical pendulum Kepler problem Outer solar system Molecular dynamics First integrals Energy, linear invariants Quadratic and polynomial invariants Reversible differential equations Symmetric methods

3 Lotka-Volterra model I u(t) number of predators v(t) number of prey u = u(v 2) v = v(1 u) general autonomous system of odes ẏ = f (y) y point in phase space f (y) vector field (velocity in y) flow: ϕ t : y 0 y(t) if y(0) = y 0

4 Lotka-Volterra model II v v 5 v u u u u number of predators, v number of prey

5 Invariant of Lotka-Volterra model equations u = u(v 2), v = v(1 u) divide by each other and separation of variables with invariant 0 = 1 u u u v 2 v v = d I(u, v) dt I(u, v) = lnu u + 2 lnv v every solution lies on level curve of I level curves are closed thus all solutions are periodic

6 First numerical methods autonomous problem y = f (y) explicit Euler method: y n+1 = y n + hf (y n ) implicit Euler method: y n+1 = y n + hf (y n+1 ) implicit midpoint rule y n+1 = y n + hf ( ) yn + y n+1 2 discrete or numerical flow: Φ h : y n y n+1

7 Partitioned systems symplectic Euler partitioned system u = f (u, v), v = g(u, v) combine explicit and implicit Euler: symplectic Euler u n+1 = u n + hf (u n, v n+1 ) v n+1 = v n + hg(u n, v n+1 ) (SE1) or u n+1 = u n + hf (u n+1, v n ) v n+1 = v n + hg(u n+1, v n ) (SE2) SE1 becomes explicit if f (u, v) = f (u), g(u, v) = g(v) SE2 becomes explicit if f (u, v) = f (v), g(u, v) = g(u)

8 Lotka-Volterra model experiment explicit Euler implicit Euler symplectic Euler v v y 0 v 6 6 y 49 y y y 0 y y 0 y u 2 4 u 2 4 u

9 Hamiltonian problem Hamiltonian H(p, q) = H(p 1,...,p d, q 1,...,q d ) (total energy) q 1,...,q d positions p 1,...,p d momenta Hamiltonian equations of motion ṗ = H q, H q = q H = q = H p ( ) H T q energy conservation: H(p(t), q(t)) = const for all t

10 Mathematical pendulum mass m = 1, massless rod of length l = 1, gravitational acceleration g = 1 Hamiltonian H(p, q) = 1 2 p2 cos q cos q q l equations of motion ṗ = H q, q = H p m ṗ = sinq, q = p or q = sinq vector field 2π-periodic in q = phase space cylinder R S 1 flow ϕ t (p, q) is an area preserving mapping

11 Area preservation

12 Pendulum numerical experiment explicit Euler h = 0.2 symplectic Euler h = 0.3 Störmer-Verlet h = 0.6

13 Kepler problem two-body problem 1st body as center of coordinate system (p, q) coordinates of second body Hamiltonian H(p 1, p 2, q 1, q 2 ) = 1 2 (p2 1 + p 2 2) (q q 2 2) 1/2 equations of motion: first integrals q i = p i, ṗ i = H qi = q i (q q 2 2) 3/2 total energy H(p, q) angular momentum L(p 1, p 2, q 1, q 2 ) = q 1 p 2 q 2 p 1 (Kepler s second law)

14 Numerical example Kepler problem steps h = implicit midpoint 1 explicit Euler steps h = 0.05 symplectic Euler steps h = Störmer Verlet steps h = 0.05

15 Numerical example Kepler problem II.02 conservation of energy explicit Euler, h = symplectic Euler, h = global error of solution explicit Euler, h = symplectic Euler, h =

16 Qualitative long-time behavior Kepler problem method error in H error in L global error explicit Euler O(th) O(th) O(t 2 h) symplectic Euler O(h) 0 O(th) implicit midpoint O(h 2 ) 0 O(th 2 ) Störmer-Verlet O(h 2 ) 0 O(th 2 )

17 Outer solar system Hamiltonian H(p, q) = i=0 1 m i p T i p i g 5 i 1 i=1 j=0 m i m j q i q j astronomical units (1 A.U. = km) masses relative to mass of sun m 0 = (account for inner planets) g = gravitational constant initial positions and initial velocity from Sept. 5, 1994, 0h00

18 Outer solar system numerical example explicit Euler, h = 10 implicit Euler, h = 10 symplectic Euler, h = 100 Störmer Verlet, h = 200

19 Molecular dynamics Hamiltonian H(p, q) = 1 2 N i=1 1 m i p T i p i + N i 1 ) V ij ( q i q j i=2 j=1 V ij (r) potential function q i, p i positions and momenta of atoms m i atomic mass of ith atom in molecular dynamics: V ij Lennard-Jones potential.2 V ij (r) = 4ε ij ( (σij r ) 12 ( σij ) ) 6 r

20 Numerical experiment frozen argon crystal N = 7 argon atoms in a plane temperature T = 1 Nk B N m i q i 2 i=1

21 Numerical experiment argon crystal explicit Euler, h = 0.5[fs] symplectic Euler, h = 10[fs] total energy explicit Euler, h = 10[fs] symplectic Euler, h = 10[fs] temperature total energy temperature Verlet, h = 40[fs] Verlet, h = 80[fs] Verlet, h = 10[fs] Verlet, h = 20[fs]

22 First integrals Definition. A non-constant function I(y) is called a first integral of ẏ = f (y) if I (y)f (y) = 0 for all y. synonyms: invariant, conserved quantity, constant of motion

23 Examples of first integrals total energy H(p, q) in Hamiltonian systems total linear and angular momentum of N-body systems H(p, q) = 1 2 N i=1 1 m i p T i p i + N i 1 V ij (r ij ), i=2 j=1 r ij = q i q j equations of motion q i = 1 m i p i, ṗ i = N j=1 ν ij (q i q j ), ν ij = V ij(r ij )/r ij linear invariants I(y) = d T y, d constant, s.t. d T f (y) = 0

24 Quadratic and polynomial invariants consider Ẏ = A(Y )Y, A(Y ) skew symmetric for all Y where Y is a vector or a matrix Theorem. The quadratic function I(Y ) = Y T Y is invariant. In particular, orthogonality of Y 0 is conserved. Lemma. Let Y, A(Y ) R n,n. If tracea(y ) = 0 for all Y, then dety is an invariant. det Y represents volume of parallelepiped generated by columns of Y volume convervation for trace A(Y ) = 0

25 Reversible differential equations Definition. Let ρ be an invertible linear transformation in the phase space of ẏ = f (y). The differential equation and the vector field f (y) are called ρ-reversible if ρf (y) = f (ρy) for all y v y f (y) v y 0 ϕ t y 1 ρf (y) f (ρy) ρ u ρ ρ u ρy ρf (y) ρy 0 ϕ t ρy 1

26 Reversible vector fields examples partitioned system where u = f (u, v), v = g(u, v) f (u, v) = f (u, v), g(u, v) = g(u, v) is (ρ)-reversible for ρ(u, v) = (u, v) second order differential equations are (ρ)-reversible ü = g(u) u = v, v = g(u) Do numerical methods produce a reversible numerical flow when applied to a reversible differential equation?

27 Symmetric methods Definition. A numerical one-step method Φ h is symmetric or time reversible if Φ h Φ h = id. y 1 = Φ h (y 0 ) is symmetric if exchanging leaves the method unaltered y 0 y 1 and h h Examples: implicit midpoint rule, Störmer-Verlet method Theorem. If a numerical method applied to a ρ-reversible differential equations satisfies ρ Φ h = Φ h ρ then Φ h is ρ-reversible if and only if Φ h is a symmetric method.