Discrete Hamiltonian Variational Integrators

0.34 MB·

by  Melvin Leok

DISCRETE HAMILTONIAN VARIATIONAL INTEGRATORS
MELVINLEOKANDJINGJINGZHANG
Abstract. Weconsider the continuous anddiscrete-timeHamilton’s variational principleonphasespace,
and characterize the exact discrete Hamiltonian which provides an exact correspondence between discrete
andcontinuous Hamiltonianmechanics. Thevariationalcharacterization oftheexact discreteHamiltonian
0
naturally leads to a class of generalized Galerkin Hamiltonian variational integrators, which include the
1
symplectic partitioned Runge–Kutta methods. We also characterize the group invariance properties of
0
discreteHamiltonianswhichleadtoadiscreteNoether’stheorem.
2

n
a
J
1. Introduction
9
1.1. Discrete Mechanics. Discrete-time analogues of Lagrangian and Hamiltonian mechanics, which are
]
A derivedfromdiscretevariationalprinciples,yieldaclassofgeometricnumericalintegrators[13]referredtoas
variationalintegrators[15, 21]. The discrete variationalapproachto constructingnumericalintegratorsis of
N
interest as they automatically yield methods that are symplectic, and by a backwarderror analysis, exhibit
.
h bounded energy errors for exponentially long times (see, for example, [12]). When the discrete Lagrangian
t or Hamiltonian is group-invariant,they will yield numerical methods that are momentum preserving.
a
m Discrete Hamiltonian mechanics can be derived from discrete Lagrangian mechanics by relaxing the dis-
crete second-order curve condition. The dual formulation of this constrained optimization problem yields
[
discrete Hamiltonian mechanics [15]. Alternatively, the second-order curve condition can be imposed using

1 Lagrange multipliers, and this corresponds to the discrete Hamilton–Pontryaginprinciple [16].
v
In contrast to the prior literature on discrete Hamiltonian mechanics, which typically start from the
8
Lagrangiansetting, we will focus on constructing Hamiltonian variational integratorsfrom the Hamiltonian
0
4 point of view, without recourse to the Lagrangian formulation. When the Hamiltonian is hyperregular, it
1 is possible to obtain the corresponding Lagrangianfunction, adopt the Galerkin construction of Lagrangian
1. variationalintegratorstoobtainadiscreteLagrangian,andthenperformadiscreteLegendretransformation
0 to obtain a discrete Hamiltonian. This is described in the following diagram:
0
1 H(q,p) FH //L(q,q˙)
: (cid:31)
v
(cid:31)
i
X (cid:31)
r (cid:15)(cid:15)(cid:31) (cid:15)(cid:15)
a H+(q ,p )oo L (q ,q )
d 0 1 FLd d 0 1
ThegoalofthispaperistodirectlyexpressthediscreteHamiltonianintermsofthecontinuousHamiltonian,
so that the diagram above commutes when the Hamiltonian is hyperregular. An added benefit is that such
an approach would remain valid even if the Hamiltonian is degenerate, as is the case for point vortices (see
[22], p.22), and no corresponding Lagrangianformulation exists.
The Galerkin construction for Lagrangianvariational integrators is attractive, since it provides a general
frameworkforconstructingalargeclassofsymplecticmethodsbasedonsuitablechoicesoffinite-dimensional
approximation spaces, and numerical quadrature formulas. Our approach allows one to apply the Galerkin
construction of variational integrators to Hamiltonian systems directly, and may potentially generalize to
variational integrators for multisymplectic Hamiltonian PDEs [4, 17, 18].
Discrete Lagrangian mechanics is expressed in terms of a discrete Lagrangian, which can be viewed as a
Type I generating function of a symplectic map, and discrete Hamiltonian mechanics is naturally expressed
in terms of discrete Hamiltonians [15], which are either Type II or III generating functions. The discrete
Date:January9,2010.
1
2 MELVINLEOKANDJINGJINGZHANG
Hamiltonianperspective allowsone to avoidsome ofthe technicaldifficulties associatedwith the singularity
associated with Type I generating functions at time t=0 (see [19], p.177).
Example 1. To illustrate the difficulties associated with degenerate Hamiltonians, consider
H(q,p)=qp,
with Legendre transformation given by FH : T Q → TQ, (q,p) 7→ (q,∂H/∂p) = (q,q). Clearly, in this
∗
situation,theLegendretransformationisnotinvertible. Furthermore,theassociatedLagrangianisidentically
zero, i.e., L(q,q˙)= pq˙−H(q,p)| = pq˙−qp| ≡0.
q˙=∂H/∂p q˙=q
The associated Hamilton’s equations is given by q˙ =∂H/∂p=q, p˙ =−∂H/∂q =−p, with exact solution
q(t) = q(0)exp(t), p(t) = p(0)exp(−t). This exact solution is, in general, incompatible with the (q ,q )
0 1
boundaryconditionsassociatedwithTypeIgeneratingfunctions,butitiscompatiblewiththe(q ,p )boundary
0 1
conditions associated with Type II generating functions.
In view of this example, our discussion of discrete Hamiltonian mechanics will be expressed directly in
terms of continuous Hamiltonians and Type II generating functions.
1.2. Main Results. We provide a characterization of the Type II generating function that generates the
exact flow of Hamilton’s equations, and derive the corresponding Type II Hamilton–Jacobi equation that it
satisfies. By considering a discrete Type II Hamilton’s variational principle in phase space, we derive the
discrete Hamilton’s equations in terms of a discrete Hamiltonian. We provide a variational characterization
of the exact discrete Hamiltonian that, when substituted into the discrete Hamilton’s equations, generates
samples of the exact continuous solution of Hamilton’s equations. Also, we introduce a discrete Type II
Hamilton–Jacobi equation.
From the variational characterization of the exact discrete Hamiltonian, we introduce a generalized
Galerkin approximation from both the Hamiltonian and Lagrangian sides, and show that they are equiv-
alent when the Hamiltonian is hyperregular. In addition, we provide a systematic means of implementing
these methods as symplectic-partitioned Runge–Kutta (SPRK) methods. We also establish the invariance
properties of the discrete Hamiltonian that yield a discrete Noether’s theorem. Galerkin discrete Hamilto-
nians derived from group-invariantinterpolatory functions satisfy these invariance properties, and therefore
preserve momentum.
1.3. Outline of the Paper. In Section 2, we present the Type II analogues of Hamilton’s phase space
variational principle and the Hamilton–Jacobi equation, and we consider the discrete-time analogues of
these in Section 3. In Section 4, we develop generalized Galerkin Hamiltonian and Lagrangian variational
integrators,and consider their implementation as symplectic-partitioned Runge–Kutta methods. In Section
5, we establisha discrete Noether’s theorem,andprovide a discrete Hamiltonianthat preservesmomentum.
2. Variational Formulation of Hamiltonian Mechanics
2.1. Hamilton’s Variational Principle for Hamiltonian and Lagrangian Mechanics. Considering
a n-dimensional configuration manifold Q with associated tangent space TQ and phase space T Q. We
∗
introduce generalized coordinates q = (q1,q2,...,qn) on Q and (q,p) = (q1,q2,...,qn,p ,p ,...,p ) on
1 2 n
T Q. Given a Hamiltonian H :T Q→R, Hamilton’s phase space variational principle states that
∗ ∗
T
δ [pq˙−H(q,p)]dt=0,
Z0
for fixed q(0) and q(T). This is equivalent to Hamilton’s canonical equations,
∂H ∂H
(1) q˙ = (q,p), p˙ =− (q,p).
∂p ∂q
If the Hamiltonian is hyperregular,there is a corresponding LagrangianL:TQ→R, given by
L(q,q˙)=ext pq˙−H(q,p)= pq˙−H(q,p)| ,
p q˙=∂H/∂p
whereext denotestheextremumoverp. Then,Hamilton’sphasespaceprincipleisequivalenttoHamilton’s
p
principle,
T
δ L(q,q˙)dt=0,
Z0
DISCRETE HAMILTONIAN VARIATIONAL INTEGRATORS 3
for fixed q(0) and q(T). The exact discrete Lagrangianis then given by,
h h
Lexact(q ,q )= ext L(q(t),q˙(t))dt= ext pq˙−H(q,p)dt,
d 0 1 q∈C2([0,h],Q) Z0 (q,p)∈C2([0,h],T∗Q)Z0
q(0)=q0,q(h)=q1 q(0)=q0,q(h)=q1
which correspond to Jacobi’s solution of the Hamilton–Jacobi equation. The usual characterization of the
exact discrete Lagrangian involves evaluating the action integral on a curve q that satisfies the boundary
conditions at the endpoints, and the Euler–Lagrange equations in the interior, however, as we will see, the
variational characterizationabove naturally leads to the construction of Galerkin variational integrators.
2.2. Type II Hamilton’s Variational Principle in Phase Space. The boundary conditions associated
with both Hamilton’s and Hamilton’s phase space variational principle are naturally related to Type I
generating functions, since they specify the positions at the initial and final times. We will introduce a
version of Hamilton’s phase space principle for fixed q(0), p(T) boundary conditions, that correspond to
a Type II generating function, which we refer to as the Type II Hamilton’s variational principle in phase
space. As would be expected, this will give a characterization of the exact discrete Hamiltonian. Taking
the Legendretransformationofthe JacobisolutionoftheHamilton–Jacobiequationleadsustoconsiderthe
following functional, S:C2([0,T],T Q)→R,
∗
T
(2) S(q(·),p(·))=p(T)q(T)− [pq˙−H(q(t),p(t))]dt.
Z0
Lemma 1. Consider the action functional S(q(·),p(·)) given by (2). The condition that S(q(·),p(·)) is
stationary with respect to boundary conditions δq(0) = 0, δp(T) = 0 is equivalent to (q(·),p(·)) satisfying
Hamilton’s canonical equations (1).
Proof. Direct computation of the variation of S over the path space C2([0,T],T Q) yields,
∗
δS=q(T)δp(T)+p(T)δq(T)
T ∂H ∂H
− q˙(t)δp(t)+p(t)δq˙(t)− (q(t),p(t))δq(t)− (q(t),p(t))δp(t) dt.
∂q ∂p
Z0 (cid:20) (cid:21)
By using integration by parts and the boundary conditions δq(0)=0,δp(T)=0, we obtain,
(3) δS=q(T)δp(T)+p(T)δq(T)−p(T)δq(T)+p(0)δq(0)
T ∂H ∂H
+ p˙(t)+ (q(t),p(t)) δq(t)− q˙(t)− (q(t),p(t)) δp(t) dt
∂q ∂p
Z0 (cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
T ∂H ∂H
= p˙(t)+ (q(t),p(t)) δq(t)− q˙(t)− (q(t),p(t)) δp(t) dt.
∂q ∂p
Z0 (cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
If(q,p)satisfiesHamilton’sequations(1),theintegrandvanishes,andδS=0.Conversely,ifweassumethat
δS=0 for any δq(0)=0,δp(T)=0, then from (3), we obtain
T ∂H ∂H
δS= p˙(t)+ (q(t),p(t)) δq(t)− q˙(t)− (q(t),p(t)) δp(t) dt=0,
∂q ∂p
Z0 (cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
and by the fundamental theorem of calculus of variations [2], we recover Hamilton’s equations,
∂H ∂H
q˙(t)= (q(t),p(t)), p˙(t)=− (q(t),p(t)).
∂p ∂q
(cid:3)
The above lemma states that the integral curve (q(·),p(·)) of Hamilton’s equations extremizes the action
functionalS(q(·),p(·))(2)forfixedboundaryconditionsq(0),p(T). WenowintroducethefunctionS(q ,p ),
0 T
which is given by the extremal value of the action functional S over the family of curves satisfying the
boundary conditions q(0)=q , p(T)=p ,
0 T
T
(4) S(q ,p )= ext S(q(·),p(·))= ext p q − [pq˙−H(q,p)]dt.
0 T T T
(q,p)∈C2([0,T],T∗Q) (q,p)∈C2([0,T],T∗Q) Z0
q(0)=q0,p(T)=pT q(0)=q0,p(T)=pT
4 MELVINLEOKANDJINGJINGZHANG
The next theorem describes how S(q ,p ) generates the flow of Hamilton’s equations.
0 T
Theorem 1. Given thefunctionS(q ,p )definedby (4), theexact time-T flow mapof Hamilton’s equations
0 T
(q ,p )7→(q ,p ) is implicitly given by the following relation,
0 0 T T
(5) q =D S(q ,p ), p =D S(q ,p ).
T 2 0 T 0 1 0 T
Inparticular, S(q ,p )isaTypeIIgeneratingfunctionthatgeneratestheexactflowofHamilton’sequations.
0 T
Proof. We directly compute
∂S ∂q T ∂p(t) ∂q˙(t) ∂q(t)∂H ∂p(t)∂H
T
(q ,p )= p − q˙(t)+ p(t)− (q,p)− (q,p) dt
0 T T
∂q ∂q ∂q ∂q ∂q ∂q ∂q ∂p
0 0 Z0 (cid:20) 0 0 0 0 (cid:21)
∂q ∂q ∂q T ∂p(t) ∂H ∂q(t) ∂H
T T 0
= p − p + p − q˙− (q,p) − p˙+ (q,p) dt
T T 0
∂q ∂q ∂q ∂q ∂p ∂q ∂q
0 0 0 Z0 (cid:20) 0 (cid:18) (cid:19) 0 (cid:18) (cid:19)(cid:21)
T ∂p(t) ∂H ∂q(t) ∂H
=p − q˙− (q,p) − p˙+ (q,p) dt,
0
∂q ∂p ∂q ∂q
Z0 (cid:20) 0 (cid:18) (cid:19) 0 (cid:18) (cid:19)(cid:21)
where we used integration by parts. By Lemma 1, the extremum of S is achieved when the curve (q,p)
satisfies Hamilton’s equations. Consequently, the integrand in the above equation vanishes, giving p =
0
∂ (q ,p ). Similarly, by using integration by parts, and restricting ourselves to curves (q,p) which satisfy
∂qS0 0 T
Hamilton’s equations, we obtain
∂S ∂p ∂q T ∂p(t) ∂q˙(t) ∂q(t)∂H ∂p(t)∂H
T T
(q ,p )= q + p − q˙(t)+ p(t)− (q,p)− (q,p) dt
0 T T T
∂p ∂p ∂p ∂p ∂p ∂p ∂q ∂p ∂p
T T T Z0 (cid:20) T T T T (cid:21)
∂q ∂q ∂q T ∂p(t) ∂H ∂q(t) ∂H
T T 0
=q + p − p + p − q˙− (q,p) − p˙+ (q,p) dt
T T T 0
∂p ∂p ∂p ∂p ∂p ∂p ∂q
T T T Z0 (cid:20) T (cid:18) (cid:19) T (cid:18) (cid:19)(cid:21)
=q .
T
(cid:3)
2.3. TypeIIHamilton–JacobiEquation. LetusexplicitlyconsiderS(q ,p )asafunctionofthetimeT,
0 T
which we denote by S (q ,p ). Theorem1 states that the Type II generatingfunction S (q ,p ) generates
T 0 T T 0 T
the exact time-T flow map of Hamilton’s equations, and consequently it has to be related by the Legendre
transformation to the Jacobi solution of the Hamilton–Jacobi equation, which is the Type I generating
function for the same flow map. Consequently, we expect that the function S (q ,p ) satisfies a Type II
T 0 T
analogue of the Hamilton–Jacobi equation, which we derive in the following proposition.
Proposition 1. Let
t
(6) S (q ,p,t)≡S (q ,p)= ext p(t)q(t)− [p(s)q˙(s)−H(q(s),p(s))]ds .
2 0 t 0
(q,p)∈C2([0,t],T∗Q)(cid:18) Z0 (cid:19)
q(0)=q0,p(t)=p
Then, the function S (q ,p,t) satisfies the Type II Hamilton–Jacobi equation,
2 0
∂S (q ,p,t) ∂S
2 0 2
(7) =H ,p .
∂t ∂p
(cid:18) (cid:19)
Proof. From the definition of S (q ,p,t), the curve that extremizes the functional connects the fixed initial
2 0
point (q ,p ) with the arbitrary final point (q,p) at time t. Computing the time derivative of S (q ,p,t)
0 0 2 0
yields,
dS
2
(8) =p˙(t)q(t)+p(t)q˙(t)−p(t)q˙(t)+H(q(t),p(t)).
dt
On the other hand,
dS ∂S ∂S
2 2 2
(9) =p˙(t) + .
dt ∂p ∂t
DISCRETE HAMILTONIAN VARIATIONAL INTEGRATORS 5
Equating (8) and (9), and applying (5) yields
∂S ∂S ∂S
2 2 2
(10) =p˙(t)q(t)+H(q(t),p(t))−p˙(t) =H(q(t),p(t))=H ,p .
∂t ∂p ∂p
(cid:18) (cid:19)
(cid:3)
The Type II Hamilton–Jacobi equation also appears on p.201 of [13] and in [8]. However, this equation
has generally been used in the construction of symplectic integrators based on Type II generating functions
byconsideringaseriesexpansionofS inpowersoft,substitutingtheseriesintheTypeIIHamilton–Jacobi
2
equation, and truncating. Then, a term-by-term comparison allows one to determine the coefficients in the
series expansion of S , from which one constructs a symplectic map that approximates the exact flow map
2
[7, 11, 24, 9].
However,approximatingJacobi’ssolutiononthe Lagrangianside,ortheexactdiscreterightHamiltonian
S(q ,p ) in (4), in terms of their variational characterization provides an elegant method for construct-
0 T
ing symplectic integrators. In particular, this naturally leads to the generalized Galerkin framework for
constructing discrete Lagrangiansand discrete Hamiltonians, which we will explore in the rest of the paper.
In Section 3, we will also present a discrete analogue of the Type II Hamilton–Jacobi equation, which
can be viewed as a composition theorem that expresses the discrete Hamiltonian for a given time interval
in terms of discrete Hamiltonians for the subintervals. This can be viewed as the Type II analogue of the
discrete Hamilton–Jacobi equation that was introduced in [10].
3. Discrete Variational Hamiltonian Mechanics
3.1. Discrete Type II Hamilton’s Variational Principle in Phase Space. The Lagrangianformula-
tion of discrete variationalmechanics is basedon a discretizationof Hamilton’s principle, and a comprehen-
sive review of this approach is given in [21]. The Hamiltonian analogue of discrete variational mechanics
was introduced in [15], wherein discrete Lagrangian mechanics was viewed as the primal formulation of a
constrained discrete optimization problem, where the constraints are given by the discrete analogue of the
second-order curve condition, and dual formulation of this yields discrete Hamiltonian variational mechan-
ics. AnanalogousapproachisbasedonthediscreteHamilton–Pontryaginvariationalprinciple[16],inwhich
the discrete Hamilton’s principle is augmented with a Lagrange multiplier term that enforces the discrete
second-order curve condition.
We beginby introducing a partitionofthe time interval[0,T]with the discrete times 0=t <t <...<
0 1
t = T, and a discrete curve in T Q, denoted by {(q ,p )}N , where q ≈ q(t ), and p ≈ p(t ). Our
N ∗ k k k=0 k k k k
discrete variational principle will be formulated in terms of a discrete Hamiltonian H+(q ,p ), which is
d k k+1
an approximation of the Type II generating function given in (4),
tk+1
(11) H+(q ,p )≈ ext p(t )q(t )− [pq˙−H(q,p)]dt.
d k k+1 (q,p)∈C2([tk,tk+1],T∗Q) k+1 k+1 Ztk
q(tk)=qk,p(tk+1)=pk+1
As we saw in Section 2, the curve in phase space with fixed boundary conditions (q ,p ) that extremizes
0 T
the functional (2),
T
S(q(·),p(·))=p(T)q(T)− [pq˙−H(q(t),p(t))]dt,
Z0
satisfies Hamilton’s canonical equations. Consequently, we can formulate discrete variational Hamiltonian
mechanics in terms of a discrete analogue of this functional, which is given by,
N−1 tk+1
(12) S ({(q ,p )}N )=p q − [pq˙−H(q(t),p(t))]dt
d k k k=0 N N
k=0 Ztk
X
N 1
−
=p q − p q −H+(q ,p ) .
N N k+1 k+1 d k k+1
k=0
X (cid:2) (cid:3)
Then, the Type II discrete Hamilton’s phase space variational principle states that δS ({(q ,p )}N ) = 0
d k k k=0
for discrete curves in T Q with fixed (q ,p ) boundary conditions.
∗ 0 N
6 MELVINLEOKANDJINGJINGZHANG
Lemma 2. The Type II discrete Hamilton’s phase space variational principle is equivalent to the discrete
right Hamilton’s equations
q =D H+(q ,p ), k =1,...,N −1,
k 2 d k 1 k
(13) −
p =D H+(q ,p ), k =1,...,N −1,
k 1 d k k+1
where H+(q ,p ) is defined in (11).
d k k+1
Proof. We compute the variation of S ,
d
N 1
−
δS =δ p q − (p q −H+(q ,p ))
d N N k+1 k+1 d k k+1
!
k=0
X
N 2 N 1
− −
=δ − p q + H+(q ,p )
k+1 k+1 d k k+1
!
k=0 k=0
X X
N 2 N 1
− −
=− (q δp +p δq )+ D H+(q ,p )δq +D H+(q ,p )δp
k+1 k+1 k+1 k+1 1 d k k+1 k 2 d k k+1 k+1
k=0 k=0
X X (cid:0) (cid:1)
N 1 N 1
− −
=− (q δp +p δq )+ D H+(q ,p )δq +D H+(q ,p )δq
k k k k 1 d k k+1 k 1 d 0 1 0
k=1 k=1
X X
N 1
−
+ D H+(q ,p )δp +D H+(q ,p )δp
2 d k 1 k k 2 d N 1 N N
− −
k=1
X
N 1 N 1
− −
=− q −D H+(q ,p ) δp − p −D H+(q ,p ) δq
k 2 d k 1 k k k 1 d k k+1 k
−
k=1 k=1
X (cid:0) (cid:1) X (cid:0) (cid:1)
+D H+(q ,p )δq +D H+(q ,p )δp .
1 d 0 1 0 2 d N 1 N N
−
where we reindexed the sum, which is the discrete analogue of integration by parts. Using the fact that
(q ,p ) are fixed, which implies δq =0, δp =0, the above equation reduces to
0 N 0 N
N 1 N 1
− −
(14) δS =− q −D H+(q ,p ) δp − p −D H+(q ,p ) δq .
d k 2 d k 1 k k k 1 d k k+1 k
−
k=1 k=1
X (cid:0) (cid:1) X (cid:0) (cid:1)
Clearly,ifthediscreterightHamilton’sequations,q =D H+(q ,p ),p =D H+(q ,p ),aresatisfied,
k 2 d k 1 k k 1 d k k+1
−
then the functional is stationary. Conversely, if the functional is stationary, a discrete analogue of the
fundamental theorem of the calculus of variations yields the discrete right Hamilton’s equations. (cid:3)
The above lemma states that the discrete-time solution trajectory of the discrete right Hamilton’s equa-
tions (13) extremizesthe discrete functional(12) forfixed q ,p . However,it does notindicate how the dis-
0 N
cretesolutionisrelatedtop , q . Note thatthe discretesolutiontrajectorythatrendersS (({(q ,p )}N )
0 N d k k k=0
stationary depends on the boundary conditions q , p . Consequently, we can introduce the function S
0 N d
which is given by the extremal value of the discrete functional S as a function of the boundary conditions
d
q(t ),p(t ), and is explicitly given by
0 N
(15) S (q(t ),p(t ))= ext S ({(q ,p )}N )
d 0 N (qk,pk)∈T∗Q d k k k=0
q0=q(t0),pN=p(tN)
N 1
−
= ext p q − (p q −H+(q ,p )).
q0=(qq(kt,0p)k,p)∈NT=∗pQ(tN) N N Xk=0 k+1 k+1 d k k+1
Then, using a similar approach to the proof of Theorem 1, we compute the derivatives of S (q ,p ) with
d 0 N
respect to q ,p . By reindexing the sum, which is the discrete analogue of integration by parts, we obtain
0 N
N 2 N 1
∂Sd(q ,p )= ∂ − − p q + − H+(q ,p )
∂q 0 N ∂q k+1 k+1 d k k+1
0 0 !
k=0 k=0
X X
DISCRETE HAMILTONIAN VARIATIONAL INTEGRATORS 7
N 1 N 1
=− − ∂pk q −D H+(q ,p ) − − ∂qk p −D H+(q ,p ) +D H+(q ,p ).
∂q0 k 2 d k−1 k ∂q0 k 1 d k k+1 1 d 0 1
k=1 k=1
X (cid:0) (cid:1) X (cid:0) (cid:1)
By Lemma 2, the extremum of S is obtained if the discrete curve satisfies the discrete right Hamilton’s
d
equations (13). Thus, by the definition of S (q ,p ), the above equation reduces to
d 0 N
(16) D S (q ,p )=D H+(q ,p ).
1 d 0 N 1 d 0 1
A similar argument yields
(17)
N 2 N 1
∂Sd(q ,p )= ∂ − − p q + − H+(q ,p )
∂p 0 N ∂p k+1 k+1 d k k+1
N N !
k=0 k=0
X X
N 1 N 1
=− − ∂pk q −D H+(q ,p ) − − ∂qk p −D H+(q ,p ) +D H+(q ,p )
∂pN k 2 d k−1 k ∂pN k 1 d k k+1 2 d N−1 N
k=1 k=1
X (cid:0) (cid:1) X (cid:0) (cid:1)
=D H+(q ,p ).
2 d N 1 N
−
Recall that the exact discrete Hamiltonian S(q ,p ) defined in (4) is a Type II generating function of the
0 T
symplecticmap,implicitlydefinedbytherelation(5),thatistheexactflowmapofthecontinuousHamilton’s
equations. To be consistent with this, we require S (q ,p ) satisfies the relation (5), which is to say
d 0 N
(18) q =D S (q ,p ), p =D S (q ,p ).
N 2 d 0 N 0 1 d 0 N
Comparing (16)–(17) and (18) we obtain
(19) q =D H+(q ,p ), p =D H+(q ,p ).
N 2 d N 1 N 0 1 d 0 1
−
Then, by combining (13) and (19), we obtain the complete set of discrete right Hamilton’s equations
(20a) q =D H+(q ,p ), k =0,1,...,N −1,
k+1 2 d k k+1
(20b) p =D H+(q ,p ), k =0,1,...,N −1.
k 1 d k k+1
It is easy to see that
0=ddH+(q ,p )=d D H+(q ,p )dq +D H+(q ,p )dp
d k k+1 1 d k k+1 k 2 d k k+1 k+1
=d(p dq +q dp )=dp ∧dq −dp ∧dq ,
k k k+1 k+1(cid:0) k k k+1 k+1 (cid:1)
for k =0,...,N −1. Then, successively applying above equation gives
dp ∧dq =dp ∧dq =···=dp ∧dq =dp ∧dq .
0 0 1 1 N 1 N 1 N N
− −
Thisimpliesthatthemapfromtheinitialstate(q ,p )tothefinalstate(q ,p )definedby(18)issympletic,
0 0 N N
since it is the composition of N symplectic maps (q ,p ) 7→(q ,p ),k =0,...,N −1, which are given
k k k+1 k+1
by (20a)–(20b). Alternatively, one can directly prove symplecticity of the map (q ,p )7→(q ,p ) by using
0 0 N N
(18) to compute 0 = d2S (q ,p ) = dp ∧dq −dp ∧dq . Given initial conditions q ,p , and under the
d 0 N 0 0 N N 0 0
∂2H+
regularityassumption d (q ,p ) 6=0,wecansolve(20b)toobtainp ,thensubstitutep into(20a)
∂qk∂pk+1 k k+1 1 1
to get q . By repeatedl(cid:12)y applying this pro(cid:12)cess, we obtain the discrete solution trajectory {(q ,p )}N .
1 (cid:12) (cid:12) k k k=1
(cid:12) (cid:12)
3.2. Discrete Type II Hamilton–Jacobi Equation. A discrete analogue of the Hamilton–Jacobi equa-
tionwasfirstintroducedin[10],andtheconnectionstodiscreteHamiltonianmechanics,anddiscreteoptimal
controltheorywereexploredin[23]. Inessence,thediscreteHamilton–Jacobiequationthereincanbeviewed
as a composition theorem that relates the discrete Hamiltonians that generate the maps over subintervals,
with the discrete Lagrangianthat generates the map over the entire time interval.
We will adopt the derivation of the discrete Hamilton–Jacobi equation in [23], which is based on intro-
ducing a discrete analogue of Jacobi’s solution, to the setting of Type II generating functions.
Theorem 2. Consider the discrete extremum function (15):
k 1
−
(21) Sk(p )=p q − p q −H+(q ,p ) ,
d k k k l+1 l+1 d l l+1
l=0
X(cid:2) (cid:3)
8 MELVINLEOKANDJINGJINGZHANG
which can be obtained from the discrete functional (12) by evaluating it along a solution of the right discrete
Hamilton’s equations (20). Each Sk(p ) is viewed as a function of the momentum p at the discrete end
d k k
time t . Then, these satisfy the discrete Type II Hamilton–Jacobi equation:
k
(22) Sk+1(p )−Sk(p )=H+(DSk(p ),p )−p ·DSk(p ).
d k+1 d k d d k k+1 k d k
Proof. From Eq. (21), we have
(23) Sk+1(p )−Sk(p )=H+(q ,p )−p ·q ,
d k+1 d k d k k+1 k k
where q is consideredto be a function of p and p , i.e., q =q (p ,p ). Taking the derivative of both
k k k+1 k k k k+1
sides with respect to p , we have
k
∂q
−DSk(p )=−q + k · D H+(q ,p )−p .
d k k ∂p 1 d k k+1 k
k
However, the term in the brackets vanish because the r(cid:2)ight discrete Hamilton(cid:3)’s equations (20) are assumed
to be satisfied. Thus we have
(24) q =DSk(p ).
k d k
Substituting this into (23) gives (22). (cid:3)
3.3. Summary of Discrete and Continuous Results. We have introduced the continuous and discrete
variationalformulationsofHamiltonianmechanicsinaparallelfashion,andthecorrespondencebetweenthe
two aresummarizedinFigure 1. Similarly,the correspondencebetweenthe continuous anddiscrete Type II
Hamilton–Jacobi equations are summarized in Table 1.
Figure 1. Continuous and discrete Type II Hamilton’s phase space variational principle.
In the continuous case, the variation of the action functional S over the space of curves
gives Hamilton’s equations, andthe derivatives of the extremum function S with respectto
the boundary points yield the exact flow map of Hamilton’s equation. In the discrete case,
the variation of the discrete action functional S over the space of discrete curves gives
d
the discrete rightHamilton’s equations and the derivatives of extremum functional S with
d
respect to the boundary points yield the symplectic map from the initial state to the final
state.
CotangentSpace Disc. CotangentSpace
(q,p) ∈ T∗Q (qk,pk+1) ∈ Q×Q∗
Hamiltonian Disc. RightHamiltonian
H(q,p) Hd+(qk,pk+1)
Disc. Action Disc. Extremum
ActionFunctionalS ExtremumFunctionS FunctionalSd FunctionSd
Disc. Right
Hamilton’sEqn. qT = D2S(q0,pT) Hamilton’sEqn. qN = D2Sd(q0,pN)
q˙ = ∂∂Hp,p˙ = −∂∂Hq p0 = D1S(q0,pT) pqkk ==DD21HHdd++((qqkk−,p1k,+pk1)) p0 = D1Sd(q0,pN)
Symplecticity Symplecticity
0 = ddS = dp0∧ 0 = ddSd = dp0∧
dq0 −dpT ∧dqT dq0 −dpN ∧dqN
DISCRETE HAMILTONIAN VARIATIONAL INTEGRATORS 9
Table 1. Correspondence between ingredients in the continuous and discrete Type II
Hamilton–Jacobi theories; N is the set of non-negative integers and R is the set of
0 0
≥
non-negative real numbers.
Continuous Discrete
(p,t)∈Q∗×R≥0 (pk,k)∈Q∗×N0
q˙=∂H/∂p, qk+1 =D2Hd+(qk,pk+1),
p˙=−∂H/∂q pk =D1Hd+(qk,pk+1)
t k−1
S2(p,t)≡p(t)q(t)− [p(s)q˙(s)−H(q(s),p(s))]ds Sdk(pk)≡pkqk− pl+1ql+1−Hd+(ql,pl+1)
Z
0 Xl=0(cid:2) (cid:3)
dS2 = ∂∂Sp2 dp+ ∂∂St2 dt Sdk+1(pk+1)−Sdk(pk)
qdp+H(q,p)dt Hd+(qk,pk+1)−pk·qk
∂S2 ∂S2 Sdk+1(pk+1)−Sdk(pk)
=H ,p
∂t (cid:18) ∂p (cid:19) =Hd+(DSdk(pk),pk+1)−pk·DSdk(pk)
4. Galerkin Hamiltonian Variational Integrators
4.1. Exact Discrete Hamiltonian. The exactdiscrete HamiltonianH+ (q ,p ) and the discrete right
d,exact 0 1
Hamilton’s equations (20) generate a discrete solution curve {q ,p }N that samples the exact solution
k k k=0
(q(·),p(·)) of the continuous Hamilton’s equations for the continuous Hamiltonian H(q,p), i.e., q = q(t )
k k
and p =p(t ).
k k
By comparing the definition (11) of a discrete right Hamiltonian function H+(q ,p ) on [0,h] and the
d 0 1
corresponding discrete Hamiltonian flow in (20) to the definition (4) of the extremal function on [0,T] and
corresponding symplectic map given by (5), and applying Theorem 1, it is clear that the discrete right
Hamiltonian function on [0,h], given by
h
(25) H+ (q ,p )= ext p q − [p(t)q˙(t)−H(q(t),p(t))]dt,
d,exact 0 1 (q,p)∈C2([0,T],T∗Q) 1 1 Z0
q(t0)=q0,p(t1)=p1
is an exact discrete right Hamiltonian function on [0,h].
4.2. Galerkin Discrete Hamiltonian. In general, the exact discrete Hamiltonian is not computable,
since it requires one to evaluate the functional S(q(·),p(·)) given in (2) on a solution curve of Hamilton’s
equations that satisfies the given boundary conditions (q ,p ). However, the variational characterization of
0 1
the exactdiscrete Hamiltoniannaturallyleadsto computable approximationsbasedonGalerkintechniques.
Inpractice,onereplacesthepathspaceC2([0,T],T Q),whichisaninfinite-dimensionalfunctionspace,with
∗
a finite-dimensional function space, and uses numerical quadrature to approximate the integral.
Let{ψ (τ)}s ,τ ∈[0,1],beasetofbasisfunctionsforas-dimensionalfunctionspaceCs. Wealsochoose
i i=1 d
1
a numerical quadrature formula with quadrature weights b , and quadrature points c , i.e., f(x)dx ≈
i i 0
s b f(c ). From these basis functions and the numerical quadrature formula, we will systematically
j=1 i i R
construct a generalized Galerkin Hamiltonian variational integrator in the following manner:
P
1. Use the basis functions ψ to approximate the velocity q˙ over the interval [0,h],
i
s
q˙ (τh)= Viψ (τ).
d i
i=1
X
10 MELVINLEOKANDJINGJINGZHANG
2. Integrate q˙ (ρ) over [0,τh], to obtain the approximation for the position q,
d
τh s s τ
q (τh)=q (0)+ Viψ (ρ)d(ρh)=q +h Vi ψ (ρ)dρ,
d d i 0 i
Z0 i=1 i=1 Z0
X X
where we applied the boundary condition q (0) = q . Applying the boundary condition q (h) = q
d 0 d 1
yields
s 1 s
q =q (h)=q +h Vi ψ (ρ)dρ≡q +h B Vi,
1 d 0 i 0 i
i=1 Z0 i=1
X X
1
where B = ψ (τ)dτ. Furthermore, we introduce the internal stages,
i 0 i
R s ci s
Qi ≡q (c h)=q +h Vj ψ (τ)dτ ≡q +h A Vj,
d i 0 j 0 ij
j=1 Z0 j=1
X X
where A = ciψ (τ)dτ.
ij 0 j
3. Let Pi = p(c h). Use the numerical quadrature formula (b ,c ) and the finite-dimensional function
Ri i i
space Cs to construct H+(q ,p ) as follows
d d 0 1
h
H+(q ,p )≈ ext p q − [p(t)q˙(t)−H(q(t),p(t))]dt
d 0 1 (q,p)∈C2([0,T],T∗Q) 1 1 Z0
q(t0)=q0,p(t1)=p1
(26)
s
H+(q ,p )= ext p q (h)−h b [p(c h)q˙ (c h)−H(q (c h),p(c h))]
d 0 1 qd∈Cds([0,h],Q),Pi∈Q∗( 1 d i=1 i i d i d i i )
X
s s s s
= ext p q +h B Vi −h b Pi Vjψ (c )−H q +h A Vj,Pi
Vi,Pi 1 0 i ! i j i  0 ij 
 Xi=1 Xi=1 Xj=1 Xj=1 
≡ ext K(q ,Vi,Pi,p ).   
 0 1 
Vi,Pi
ToobtainanexpressionforH+(q ,p ),wefirstcomputethestationarityconditionsforK(q ,Vi,Pi,p )
d 0 1 0 1
under the fixed boundary condition (q ,p ).
0 1
∂K(q ,Vi,Pi,p ) s ∂H
(27a) 0= 0 1 =hp B −h b Piψ (c )−hA (Qi,Pi) , j =1,...,s.
∂Vj 1 j i j i ij ∂q
i=1 (cid:18) (cid:19)
X
∂K(q ,Vi,Pi,p ) s ∂H
(27b) 0= 0 1 =hb ψ (c )Vi− (Qj,Pj) , j =1,...,s.
∂Pj j i j ∂p
!
i=1
X
4. By solving the 2s stationarity conditions (27), we can express the parameters Vi,Pi, in terms of
q ,p , i.e., Vi =Vi(q ,p ) and Pi =Pi(q ,p ). Then, the symplectic map (q ,p )7→(q ,p ) can be
0 1 0 1 0 1 0 0 1 1
expressed in terms of the internal stages
∂H+(q ,p ) ∂K(q ,Vi(q ,p ),Pi(q ,p ),p )
(28) p = d 0 1 = 0 0 1 0 1 1
0
∂q ∂q
0 0
∂K ∂K ∂Vi ∂K ∂Pi ∂K
= + + =
∂q ∂Vi ∂q ∂Pi ∂q ∂q
0 0 0 0
s
∂H
=p +h b (Qi,Pi),
1 i
∂q
i=1
X
Similarly, we obtain
∂H+(q ,p ) ∂K(q ,Vi(q ,p ),Pi(q ,p ),p )
(29) q = d 0 1 = 0 0 1 0 1 1
1
∂p ∂p
1 1
∂K ∂Vi ∂K ∂Pi ∂K ∂K
= + + =
∂Vi ∂p ∂Pi ∂p ∂p ∂p
1 1 1 1