Generating functionsVariational integrators

Christian Lubich, Univ. Tubingen

Skeleton notes at Oberwolfach seminar, 27.11.2008

Generating function of a symplectic map

A mapping ϕ : (p, q) 7→ (P, Q) is symplectic if and only if thereexists locally a function S(p, q) such that

PT dQ − pT dq = dS .

In coordinates (q, Q) instead of (p, q): find that

P = ∇QS(q, Q), p = −∇qS(q, Q).

Mixed-variable generating function

near-identity symplectic maps generated by

P = p −∇qS1(P, q), Q = q +∇PS1(P, q).

• for symplectic Euler: S1(P, q, h) = hH(P, q)

• for symplectic Runge–Kutta method:

S1(P, q, h) = hs∑

i=1

biH(Pi , Qi )−h2s∑

i ,j=1

biaijHq(Pi , Qi )THp(Pj , Qj)

is globally defined (→ globally defined modified Hamiltonian)

• similar expression for symplectic partitioned Runge–Kuttamethods

Hamilton–Jacobi PDE

If S1(P, q, t) is a solution of the partial differential equation

∂S1

∂t(P, q, t) = H

(P, q +∇PS1(P, q, t)

), S1(P, q, 0) = 0,

then the mapping (p, q) 7→ (P(t), Q(t)

)defined by

P(t) = p −∇qS1(P(t), q, t), Q(t) = q +∇PS1(P(t), q, t),

is the exact flow of the Hamiltonian system for H.

Taylor expansion of S1 used by Feng Kang (1986) to constructhigh-order symplectic methods

Lagrange’s and Hamilton’s equations of motion

kinetic energy T = qTM(q)q, potential energy U(q),

Lagrangian L = T − U as function of (q, q)

Lagrange’s equationsd

dt

(∂L

∂q

)=

∂L

∂q,

conjugate momenta p = ∂L∂q (q, q) = M(q)q

Hamiltonian H = pT q − L(q, q) = T + U as function of (p, q)

Hamilton’s equationsp = −∇qH(p, q)

q = ∇pH(p, q)

Hamilton’s principle

Lagrange’s equations are the Euler–Lagrange equations forextremizing the action integral

S =

∫ t1

t0

L(q(t), q(t)) dt

among all curves q(t) that connect two given points q0 and q1

consider S as function of q0 and q1, for the solution q(t) of theE-L eqs., find

dS = pT1 dq1 − pT

0 dq0.

S(q0, q1) is the generating function for the symplectic flow map(p0, q0) 7→ (p1, q1):

p0 = −∇0S(q0, q1), p1 = ∇1S(q0, q1).

Discrete Hamilton’s principle: variational integrators

Extremize the approximated action integral (for fixed q0,qN)

N∑n=0

Sh(qn, qn+1) with Sh(qn, qn+1) ≈∫ tn+1

tn

L(q(t), q(t)) dt

The discrete Hamilton principle gives the symplectic method withgenerating function Sh:

pn = −∇0Sh(qn, qn+1), pn+1 = ∇1Sh(qn, qn+1).

symplectic methods = variational integrators

Examples: Stormer–Verlet and symplectic PRK

Stormer–Verlet: Sh(q0, q1) = h2 L

(q0,

q1−q0h

)+ h

2 L(q1,

q1−q0h

)symplectic partitioned Runge–Kutta method:

Sh(q0, q1) = hs∑

i=1

biL(Qi , Qi )

whereQi = q0 + h

s∑j=1

aij Qj

and the Qi are chosen to extremize the above sum under theconstraint

q1 = q0 + hs∑

i=1

bi Qi

Constrained systems: RATTLE as var. integrator

Extremizing the action integral over all curves q(t) withg(q(t)) = 0 that connect two given points q0 and q1, yieldsLagrange’s equations with L(q, q) replaced by L(q, q)− g(q)Tλ.

discrete analogue: replace Sh(qn, qn+1) by

Sh(qn, qn+1)− h

2g(qn)

Tλ+n −

h

2g(qn+1)

Tλ−n+1

obtain RATTLE for Sh of Stormer–Verlet:

pn = −∇0Sh(qn, qn+1) + h2 G (qn)

Tλ+n

pn+1 = ∇1Sh(qn, qn+1)− h2 G (qn+1)

Tλ−n+1

under constraints g(qn+1) = 0, G (qn+1)∇pH(pn+1, qn+1) = 0