Generating function of a symplectic map Generating … · Generating function of a symplectic map...
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Generating functions Variational integrators Christian Lubich, Univ. T¨ ubingen Skeleton notes at Oberwolfach seminar, 27.11.2008 Generating function of a symplectic map A mapping ϕ :(p, q) → (P , Q) is symplectic if and only if there exists locally a function S (p, q) such that P T dQ - p T dq = dS . In coordinates (q, Q) instead of (p, q): find that P = ∇ Q S (q, Q), p = -∇ q S (q, Q). Mixed-variable generating function near-identity symplectic maps generated by P = p -∇ q S 1 (P , q), Q = q + ∇ P S 1 (P , q). • for symplectic Euler: S 1 (P , q, h)= hH(P , q) • for symplectic Runge–Kutta method: S 1 (P , q, h)= h s i =1 b i H(P i , Q i )-h 2 s i ,j =1 b i a ij H q (P i , Q i ) T H p (P j , Q j ) is globally defined (→ globally defined modified Hamiltonian) • similar expression for symplectic partitioned Runge–Kutta methods Hamilton–Jacobi PDE If S 1 (P , q, t ) is a solution of the partial differential equation ∂S 1 ∂t (P , q, t )= H P , q + ∇ P S 1 (P , q, t ) , S 1 (P , q, 0) = 0, then the mapping (p, q) → ( P (t ), Q(t ) ) defined by P (t )= p -∇ q S 1 (P (t ), q, t ), Q(t )= q + ∇ P S 1 (P (t ), q, t ), is the exact flow of the Hamiltonian system for H. Taylor expansion of S 1 used by Feng Kang (1986) to construct high-order symplectic methods Lagrange’s and Hamilton’s equations of motion kinetic energy T =˙ q T M(q)˙ q, potential energy U(q), Lagrangian L = T - U as function of (q, ˙ q) Lagrange’s equations d dt ∂L ∂ ˙ q = ∂L ∂q , conjugate momenta p = ∂L ∂ ˙ q (q, ˙ q)= M(q)˙ q Hamiltonian H = p T ˙ q - L(q, ˙ q)= T + U as function of (p, q) Hamilton’s equations ˙ p = -∇ q H(p, q) ˙ q = ∇ p H(p, q) Hamilton’s principle Lagrange’s equations are the Euler–Lagrange equations for extremizing the action integral S = t1 t0 L(q(t ), ˙ q(t )) dt among all curves q(t ) that connect two given points q 0 and q 1 consider S as function of q 0 and q 1 , for the solution q(t ) of the E-L eqs., find dS = p T 1 dq 1 - p T 0 dq 0 . S (q 0 , q 1 ) is the generating function for the symplectic flow map (p 0 , q 0 ) → (p 1 , q 1 ): p 0 = -∇ 0 S (q 0 , q 1 ), p 1 = ∇ 1 S (q 0 , q 1 ). Discrete Hamilton’s principle: variational integrators Extremize the approximated action integral (for fixed q 0 ,q N ) N n=0 S h (q n , q n+1 ) with S h (q n , q n+1 ) ≈ tn+1 tn L(q(t ), ˙ q(t )) dt The discrete Hamilton principle gives the symplectic method with generating function S h : p n = -∇ 0 S h (q n , q n+1 ), p n+1 = ∇ 1 S h (q n , q n+1 ). symplectic methods = variational integrators Examples: St¨ormer–Verlet and symplectic PRK St¨ ormer–Verlet: S h (q 0 , q 1 )= h 2 L q 0 , q1-q0 h + h 2 L q 1 , q1-q0 h symplectic partitioned Runge–Kutta method: S h (q 0 , q 1 )= h s i =1 b i L(Q i , ˙ Q i ) where Q i = q 0 + h s j =1 a ij ˙ Q j and the ˙ Q i are chosen to extremize the above sum under the constraint q 1 = q 0 + h s i =1 b i ˙ Q i
Transcript of Generating function of a symplectic map Generating … · Generating function of a symplectic map...
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
