1001.1408v1

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arXiv:1001

.1408v1

[math.NA

]9Jan2010

DISCRETE HAMILTONIAN VARIATIONAL INTEGRATORS

MELVIN LEOK AND JINGJING ZHANG

Abstract. We consider the continuous and discrete-time Hamiltons variational principle on phase space,and characterize the exact discrete Hamiltonian which provides an exact correspondence between discrete

and continuous Hamiltonian mechanics. The variational characterization of the exact discrete Hamiltonian

naturally leads to a class of generalized Galerkin Hamiltonian variational integrators, which include thesymplectic partitioned RungeKutta methods. We also characterize the group invariance properties of

discrete Hamiltonians which lead to a discrete Noethers theorem.

1. Introduction

1.1. Discrete Mechanics. Discrete-time analogues of Lagrangian and Hamiltonian mechanics, which arederived from discrete variational principles, yield a class of geometric numerical integrators [13] referred to as

variational integrators [15, 21]. The discrete variational approach to constructing numerical integrators is ofinterest as they automatically yield methods that are symplectic, and by a backward error analysis, exhibitbounded energy errors for exponentially long times (see, for example, [12]). When the discrete Lagrangianor Hamiltonian is group-invariant, they will yield numerical methods that are momentum preserving.

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 yieldsdiscrete Hamiltonian mechanics [15]. Alternatively, the second-order curve condition can be imposed usingLagrange multipliers, and this corresponds to the discrete HamiltonPontryagin principle [16].

In contrast to the prior literature on discrete Hamiltonian mechanics, which typically start from theLagrangian setting, we will focus on constructing Hamiltonian variational integrators from the Hamiltonianpoint of view, without recourse to the Lagrangian formulation. When the Hamiltonian is hyperregular, itis possible to obtain the corresponding Lagrangian function, adopt the Galerkin construction of Lagrangianvariational integrators to obtain a discrete Lagrangian, and then perform a discrete Legendre transformationto obtain a discrete Hamiltonian. This is described in the following diagram:

H(q, p)FH //

L(q, q)

H+d (q0, p1) Ld(q0, q1)FLdoo

The goal of this paper is to directly express the discrete Hamiltonian in terms of the continuous Hamiltonian,so that the diagram above commutes when the Hamiltonian is hyperregular. An added benefit is that suchan approach would remain valid even if the Hamiltonian is degenerate, as is the case for point vortices (see[22], p. 22), and no corresponding Lagrangian formulation exists.

The Galerkin construction for Lagrangian variational integrators is attractive, since it provides a generalframework for constructing a large class of symplectic methods based on suitable choices of finite-dimensionalapproximation spaces, and numerical quadrature formulas. Our approach allows one to apply the Galerkinconstruction of variational integrators to Hamiltonian systems directly, and may potentially generalize tovariational integrators for multisymplectic Hamiltonian PDEs [4, 17, 18].

Discrete Lagrangian mechanics is expressed in terms of a discrete Lagrangian, which can be viewed as aType I generating function of a symplectic map, and discrete Hamiltonian mechanics is naturally expressedin terms of discrete Hamiltonians [15], which are either Type II or III generating functions. The discrete

Date : January 9, 2010.

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2 MELVIN LEOK AND JINGJING ZHANG

Hamiltonian perspective allows one to avoid some of the technical difficulties associated with the singularityassociated 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 byFH : TQ T Q, (q, p) (q,H/p) = (q, q). Clearly, in this

situation, the Legendre transformation is not invertible. Furthermore, the associated Lagrangian is identicallyzero, i.e., L(q, q) = pq H(q, p)|q=H/p = pq qp|q=q 0.

The associated Hamiltons equations is given by q = H/p = q, p = H/q = p, with exact solutionq(t) = q(0)exp(t), p(t) = p(0)exp(t). This exact solution is, in general, incompatible with the (q0, q1)boundary conditions associated with Type I generating functions, but it is compatible with the (q0, p1) boundaryconditions associated with Type II generating functions.

In view of this example, our discussion of discrete Hamiltonian mechanics will be expressed directly interms of continuous Hamiltonians and Type II generating functions.

1.2. Main Results. We provide a characterization of the Type II generating function that generates theexact flow of Hamiltons equations, and derive the corresponding Type II HamiltonJacobi equation that itsatisfies. By considering a discrete Type II Hamiltons variational principle in phase space, we derive thediscrete Hamiltons equations in terms of a discrete Hamiltonian. We provide a variational characterization

of the exact discrete Hamiltonian that, when substituted into the discrete Hamiltons equations, generatessamples of the exact continuous solution of Hamiltons equations. Also, we introduce a discrete Type IIHamiltonJacobi equation.

From the variational characterization of the exact discrete Hamiltonian, we introduce a generalizedGalerkin 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 implementingthese methods as symplectic-partitioned RungeKutta (SPRK) methods. We also establish the invarianceproperties of the discrete Hamiltonian that yield a discrete Noethers theorem. Galerkin discrete Hamilto-nians derived from group-invariant interpolatory functions satisfy these invariance properties, and thereforepreserve momentum.

1.3. Outline of the Paper. In Section 2, we present the Type II analogues of Hamiltons phase spacevariational principle and the HamiltonJacobi equation, and we consider the discrete-time analogues of

these in Section 3. In Section 4, we develop generalized Galerkin Hamiltonian and Lagrangian variationalintegrators, and consider their implementation as symplectic-partitioned RungeKutta methods. In Section5, we establish a discrete Noethers theorem, and provide a discrete Hamiltonian that preserves momentum.

2. Variational Formulation of Hamiltonian Mechanics

2.1. Hamiltons Variational Principle for Hamiltonian and Lagrangian Mechanics. Consideringa n-dimensional configuration manifold Q with associated tangent space T Q and phase space TQ. Weintroduce generalized coordinates q = (q1, q2, . . . , qn) on Q and (q, p) = (q1, q2, . . . , qn, p1, p2, . . . , pn) onTQ. Given a Hamiltonian H : TQ R, Hamiltons phase space variational principle states that

T0

[pq H(q, p)]dt = 0,

for fixed q(0) and q(T). This is equivalent to Hamiltons canonical equations,

q =H

p(q, p), p =

H

q(q, p).(1)

If the Hamiltonian is hyperregular, there is a corresponding Lagrangian L : T Q R, given by

L(q, q) = extp

pq H(q, p) = pq H(q, p)|q=H/p ,

where extp denotes the extremum over p. Then, Hamiltons phase space principle is equivalent to Hamiltonsprinciple,

T0

L(q, q)dt = 0,

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DISCRETE HAMILTONIAN VARIATIONAL INTEGRATORS 3

for fixed q(0) and q(T). The exact discrete Lagrangian is then given by,

Lexactd (q0, q1) = extqC2([0,h],Q)

q(0)=q0,q(h)=q1

h0

L(q(t), q(t))dt = ext(q,p)C2([0,h],TQ)

q(0)=q0,q(h)=q1

h0

pq H(q, p)dt,

which correspond to Jacobis solution of the HamiltonJacobi equation. The usual characterization of theexact discrete Lagrangian involves evaluating the action integral on a curve q that satisfies the boundaryconditions at the endpoints, and the EulerLagrange equations in the interior, however, as we will see, thevariational characterization above naturally leads to the construction of Galerkin variational integrators.

2.2. Type II Hamiltons Variational Principle in Phase Space. The boundary conditions associatedwith both Hamiltons and Hamiltons phase space variational principle are naturally related to Type Igenerating functions, since they specify the positions at the initial and final times. We will introduce aversion of Hamiltons phase space principle for fixed q(0), p(T) boundary conditions, that correspond toa Type II generating function, which we refer to as the Type II Hamiltons variational principle in phasespace. As would be expected, this will give a characterization of the exact discrete Hamiltonian. Takingthe Legendre transformation of the Jacobi solution of the HamiltonJacobi equation leads us to consider thefollowing functional, S : C2([0, T], TQ) R,

S(q(), p()) = p(T)q(T) T

0

[pq H(q(t), p(t))]dt.(2)

Lemma 1. Consider the action functional S(q(), p()) given by (2). The condition thatS(q(), p()) isstationary with respect to boundary conditions q(0) = 0, p(T) = 0 is equivalent to (q(), p()) satisfyingHamiltons canonical equations (1).

Proof. Direct computation of the variation ofS over the path space C2([0, T], TQ) yields,

S = q(T)p(T) +p(T)q(T)

T0

q(t)p(t) +p(t)q(t)

H

q(q(t), p(t)) q(t)

H

p(q(t), p(t)) p(t)

dt.

By using integration by parts and the boundary conditions q(0) = 0, p(T) = 0, we obtain,

S = q(T)p(T) +p(T)q(T) p(T)q(T) + p(0)q(0)(3)

+

T

0

p(t) +

Hq

(q(t), p(t))

q(t)

q(t) Hp

(q(t), p(t))

p(t)

dt

=

T0

p(t) +

H

q(q(t), p(t))

q(t)

q(t)

H

p(q(t), p(t))

p(t)

dt.

If (q, p) satisfies Hamiltons equations (1), the integrand vanishes, and S = 0. Conversely, if we assume thatS = 0 for any q(0) = 0, p(T) = 0, then from (3), we obtain

S =

T0

p(t) +

H

q(q(t), p(t))

q(t)

q(t)

H

p(q(t), p(t))

p(t)

dt = 0,

and by the fundamental theorem of calculus of variations [2], we recover Hamiltons equations,

q(t) =H

p(q(t), p(t)) , p(t) =

H

q(q(t), p(t)) .

The above lemma states that the integral curve (q(), p()) of Hamiltons equations extremizes the actionfunctionalS(q(), p()) (2) for fixed boundary conditions q(0), p(T). We now introduce the function S(q0, pT),which is given by the extremal value of the action functional S over the family of curves satisfying theboundary conditions q(0) = q0, p(T) = pT,

S(q0, pT) = ext(q,p)C2([0,T],TQ)q(0)=q0,p(T)=pT

S(q(), p()) = ext(q,p)C2([0,T],TQ)q(0)=q0,p(T)=pT

pTqT

T0

[pq H(q, p)] dt.(4)

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The next theorem describes how S(q0, pT) generates the flow of Hamiltons equations.

Theorem 1. Given the functionS(q0, pT) defined by (4), the exact time-T flow map of Hamiltons equations(q0, p0) (qT, pT) is implicitly given by the following relation,

qT = D2S(q0, pT), p0 = D1S(q0, pT).(5)

In particular, S(q0

, pT

) is a Type II generating function that generates the exact flow of Hamiltons equations.

Proof. We directly compute

S

q0(q0, pT) =

qTq0

pT

T0

p(t)

q0q(t) +

q(t)

q0p(t)

q(t)

q0

H

q(q, p)

p(t)

q0

H

p(q, p)

dt

=qTq0

pT qTq0

pT +q0q0

p0

T0

p(t)

q0

q

H

p(q, p)

q(t)

q0

p +

H

q(q, p)

dt

= p0

T0

p(t)

q0

q

H

p(q, p)

q(t)

q0

p +

H

q(q, p)

dt,

where we used integration by parts. By Lemma 1, the extremum ofS is achieved when the curve (q, p)satisfies Hamiltons equations. Consequently, the integrand in the above equation vanishes, giving p0 =

Sq0 (q0, pT). Similarly, by using integration by parts, and restricting ourselves to curves (q, p) which satisfyHamiltons equations, we obtain

S

pT(q0, pT) =

pTpT

qT +qTpT

pT

T0

p(t)

pTq(t) +

q(t)

pTp(t)

q(t)

pT

H

q(q, p)

p(t)

pT

H

p(q, p)

dt

= qT +qTpT

pT qTpT

pT +q0pT

p0

T0

p(t)

pT

q

H

p(q, p)

q(t)

pT

p +

H

q(q, p)

dt

= qT.

2.3. Type II HamiltonJacobi Equation. Let us explicitly consider S(q0, pT) as a function of the time T,which we denote by ST(q0, pT). Theorem 1 states that the Type II generating function ST(q0, pT) generates

the exact time-T flow map of Hamiltons equations, and consequently it has to be related by the Legendretransformation to the Jacobi solution of the HamiltonJacobi equation, which is the Type I generatingfunction for the same flow map. Consequently, we expect that the function ST(q0, pT) satisfies a Type IIanalogue of the HamiltonJacobi equation, which we derive in the following proposition.

Proposition 1. Let

S2(q0, p , t) St(q0, p) = ext(q,p)C2([0,t],TQ)

q(0)=q0,p(t)=p

p(t)q(t)

t0

[p(s)q(s) H(q(s), p(s))] ds

.(6)

Then, the function S2(q0, p , t) satisfies the Type II HamiltonJacobi equation,

S2(q0, p , t)

t= H

S2p

, p

.(7)

Proof. From the definition of S2(q0, p , t), the curve that extremizes the functional connects the fixed initialpoint (q0, p0) with the arbitrary final point (q, p) at time t. Computing the time derivative of S2(q0, p , t)yields,

dS2dt

= p(t)q(t) +p(t)q(t) p(t)q(t) + H(q(t), p(t)).(8)

On the other hand,

dS2dt

= p(t)S2p

+S2t

.(9)

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Equating (8) and (9), and applying (5) yields

S2t

= p(t)q(t) + H(q(t), p(t)) p(t)S2p

= H(q(t), p(t)) = H

S2p

, p

.(10)

The Type II HamiltonJacobi equation also appears on p. 201 of [13] and in [8]. However, this equationhas generally been used in the construction of symplectic integrators based on Type II generating functionsby considering a series expansion ofS2 in powers oft, substituting the series in the Type II HamiltonJacobiequation, and truncating. Then, a term-by-term comparison allows one to determine the coefficients in theseries expansion of S2, from which one constructs a symplectic map that approximates the exact flow map[7, 11, 24, 9].

However, approximating Jacobis solution on the Lagrangian side, or the exact discrete right HamiltonianS(q0, pT) in (4), in terms of their variational characterization provides an elegant method for construct-ing symplectic integrators. In particular, this naturally leads to the generalized Galerkin framework forconstructing discrete Lagrangians and 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 HamiltonJacobi equation, whichcan be viewed as a composition theorem that expresses the discrete Hamiltonian for a given time intervalin terms of discrete Hamiltonians for the subintervals. This can be viewed as the Type II analogue of the

discrete HamiltonJacobi equation that was introduced in [10].

3. Discrete Variational Hamiltonian Mechanics

3.1. Discrete Type II Hamiltons Variational Principle in Phase Space. The Lagrangian formula-tion of discrete variational mechanics is based on a discretization of Hamiltons principle, and a comprehen-sive review of this approach is given in [21]. The Hamiltonian analogue of discrete variational mechanicswas introduced in [15], wherein discrete Lagrangian mechanics was viewed as the primal formulation of aconstrained discrete optimization problem, where the constraints are given by the discrete analogue of thesecond-order curve condition, and dual formulation of this yields discrete Hamiltonian variational mechan-ics. An analogous approach is based on the discrete HamiltonPontryagin variational principle [16], in whichthe discrete Hamiltons principle is augmented with a Lagrange multiplier term that enforces the discretesecond-order curve condition.

We begin by introducing a partition of the time interval [0, T] with the discrete times 0 = t0 < t1 < . . .

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Lemma 2. The Type II discrete Hamiltons phase space variational principle is equivalent to the discreteright Hamiltons equations

(13)qk = D2H

+d (qk1, pk), k = 1, . . . , N 1,

pk = D1H+d (qk, pk+1), k = 1, . . . , N 1,

where H+d (qk, pk+1) is defined in (11).

Proof. We compute the variation ofSd,

Sd =

pNqN

N1k=0

(pk+1qk+1 H+d (qk, pk+1))

=

N2k=0

pk+1qk+1 +N1k=0

H+d (qk, pk+1)

= N2k=0

(qk+1pk+1 +pk+1qk+1) +N1k=0

D1H

+d (qk, pk+1)qk + D2H

+d (qk, pk+1)pk+1

= N1

k=1

(qkpk +pkqk) +N1

k=1

D1H+d (qk, pk+1)qk + D1H

+d (q0, p1)q0

+N1k=1

D2H+d (qk1, pk)pk + D2H

+d (qN1, pN)pN

= N1k=1

qk D2H

+d (qk1, pk)

pk

N1k=1

pk D1H

+d (qk, pk+1)

qk

+ D1H+d (q0, p1)q0 + D2H

+d (qN1, pN)pN.

where we reindexed the sum, which is the discrete analogue of integration by parts. Using the fact that(q0, pN) are fixed, which implies q0 = 0, pN = 0, the above equation reduces to

Sd = N1

k=1

qk D2H+d (qk1, pk) pk N1

k=1

pk D1H+d (qk, pk+1) qk.(14)Clearly, if the discrete right Hamiltons equations, qk = D2H

+d (qk1, pk), pk = D1H

+d (qk, pk+1), are satisfied,

then the functional is stationary. Conversely, if the functional is stationary, a discrete analogue of thefundamental theorem of the calculus of variations yields the discrete right Hamiltons equations.

The above lemma states that the discrete-time solution trajectory of the discrete right Hamiltons equa-tions (13) extremizes the discrete functional (12) for fixed q0, pN. However, it does not indicate how the dis-crete solution is related to p0, qN. Note that the discrete solution trajectory that renders Sd(({(qk, pk)}

Nk=0)

stationary depends on the boundary conditions q0, pN. Consequently, we can introduce the function Sdwhich is given by the extremal value of the discrete functional Sd as a function of the boundary conditionsq(t0), p(tN), and is explicitly given by

Sd(q(t0), p(tN)) = ext(qk,pk)

TQ

q0=q(t0),pN=p(tN )

Sd({(qk, pk)}Nk=0)(15)

= ext(qk,pk)TQ

q0=q(t0),pN=p(tN )

pNqN N1k=0

(pk+1qk+1 H+d (qk, pk+1)).

Then, using a similar approach to the proof of Theorem 1, we compute the derivatives of Sd(q0, pN) withrespect to q0, pN. By reindexing the sum, which is the discrete analogue of integration by parts, we obtain

Sdq0

(q0, pN) =

q0

N2k=0

pk+1qk+1 +N1k=0

H+d (qk, pk+1)

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= N1k=1

pkq0

qk D2H

+d (qk1, pk)

N1k=1

qkq0

pk D1H

+d (qk, pk+1)

+ D1H

+d (q0, p1).

By Lemma 2, the extremum ofSd is obtained if the discrete curve satisfies the discrete right Hamiltonsequations (13). Thus, by the definition of Sd(q0, pN), the above equation reduces to

D1Sd(q0, pN) = D1H+d (q0, p1).(16)

A similar argument yields

SdpN

(q0, pN) =

pN

N2k=0

pk+1qk+1 +

N1k=0

H+d (qk, pk+1)

(17)

= N1k=1

pkpN

qk D2H

+d (qk1, pk)

N1k=1

qkpN

pk D1H

+d (qk, pk+1)

+ D2H

+d (qN1, pN)

= D2H+d (qN1, pN).

Recall that the exact discrete Hamiltonian S(q0, pT) defined in (4) is a Type II generating function of thesymplectic map, implicitly defined by the relation (5), that is the exact flow map of the continuous Hamiltonsequations. To be consistent with this, we require Sd(q0, pN) satisfies the relation (5), which is to say

qN = D2Sd(q0, pN), p0 = D1Sd(q0, pN).(18)

Comparing (16)(17) and (18) we obtain

qN = D2H+d (qN1, pN), p0 = D1H

+d (q0, p1).(19)

Then, by combining (13) and (19), we obtain the complete set of discrete right Hamiltons equations

qk+1 = D2H+d (qk, pk+1), k = 0, 1, . . . , N 1,(20a)

pk = D1H+d (qk, pk+1), k = 0, 1, . . . , N 1.(20b)

It is easy to see that

0 = ddH+d (qk, pk+1) = d D1H

+d (qk, pk+1)dqk + D2H

+d (qk, pk+1)dpk+1

= d (pkdqk + qk+1dpk+1) = dpk dqk dpk+1 dqk+1,for k = 0, . . . , N 1. Then, successively applying above equation gives

dp0 dq0 = dp1 dq1 = = dpN1 dqN1 = dpN dqN.

This implies that the map from the initial state (q0, p0) to the final state (qN, pN) defined by (18) is sympletic,since it is the composition of N symplectic maps (qk, pk) (qk+1, pk+1), k = 0, . . . , N 1, which are givenby (20a)(20b). Alternatively, one can directly prove symplecticity of the map (q0, p0) (qN, pN) by using(18) to compute 0 = d2Sd(q0, pN) = dp0 dq0 dpN dqN. Given initial conditions q0, p0, and under the

regularity assumption 2H+dqkpk+1 (qk, pk+1)

= 0, we can solve (20b) to obtain p1, then substitute p1 into (20a)to get q1. By repeatedly applying this process, we obtain the discrete solution trajectory {(qk, pk)}Nk=1.

3.2. Discrete Type II HamiltonJacobi Equation. A discrete analogue of the HamiltonJacobi equa-tion was first introduced in [10], and the connections to discrete Hamiltonian mechanics, and discrete optimalcontrol theory were explored in [23]. In essence, the discrete HamiltonJacobi equation therein can be viewedas a composition theorem that relates the discrete Hamiltonians that generate the maps over subintervals,with the discrete Lagrangian that generates the map over the entire time interval.

We will adopt the derivation of the discrete HamiltonJacobi equation in [23], which is based on intro-ducing a discrete analogue of Jacobis solution, to the setting of Type II generating functions.

Theorem 2. Consider the discrete extremum function (15):

(21) Skd (pk) = pkqk k1l=0

pl+1ql+1 H

+d (ql, pl+1)

,

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which can be obtained from the discrete functional (12) by evaluating it along a solution of the right discreteHamiltons equations (20). Each Skd (pk) is viewed as a function of the momentum pk at the discrete endtime tk. Then, these satisfy the discrete Type II HamiltonJacobi equation:

(22) Sk+1d (pk+1) Skd (pk) = H

+d (DS

kd (pk), pk+1) pk DS

kd (pk).

Proof. From Eq. (21), we have

(23) Sk+1d (pk+1) Skd (pk) = H+d (qk, pk+1) pk qk,where qk is considered to be a function of pk and pk+1, i.e., qk = qk(pk, pk+1). Taking the derivative of bothsides with respect to pk, we have

DSkd (pk) = qk +qkpk

D1H+d (qk, pk+1) pk

.

However, the term in the brackets vanish because the right discrete Hamiltons equations ( 20) are assumedto be satisfied. Thus we have

(24) qk = DSkd (pk).

Substituting this into (23) gives (22).

3.3. Summary of Discrete and Continuous Results. We have introduced the continuous and discretevariational formulations of Hamiltonian mechanics in a parallel fashion, and the correspondence between the

two are summarized in Figure 1. Similarly, the correspondence between the continuous and discrete Type IIHamiltonJacobi equations are summarized in Table 1.

Figure 1. Continuous and discrete Type II Hamiltons phase space variational principle.In the continuous case, the variation of the action functional S over the space of curvesgives Hamiltons equations, and the derivatives of the extremum function S with respect tothe boundary points yield the exact flow map of Hamiltons equation. In the discrete case,the variation of the discrete action functional Sd over the space of discrete curves givesthe discrete right Hamiltons equations and the derivatives of extremum functional Sd withrespect to the boundary points yield the symplectic map from the initial state to the finalstate.

Cotangent Space

(q, p) TQDisc. Cotangent Space

(qk, pk+1) Q Q

Hamiltonian

H(q, p)

Disc. Right Hamiltonian

H+d

(qk, pk+1)

Action Functional SDisc. Action

Functional SdExtremum Function S Disc. ExtremumFunction Sd

Hamiltons Eqn.

q = Hp

, p = Hq

Disc. RightHamiltons Eqn.

qk = D2H+d

(qk1, pk)

pk = D1H+d

(qk, pk+1)

qT = D2S(q0, pT)p0 = D1S(q0, pT)

qN = D2Sd(q0, pN )

p0 = D1Sd(q0, pN )

Symplecticity0 = ddS = dp0 dq0 dpT dqT

Symplecticity0 = ddSd = dp0 dq0 dpN dqN

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Table 1. Correspondence between ingredients in the continuous and discrete Type IIHamiltonJacobi theories; N0 is the set of non-negative integers and R0 is the set ofnon-negative real numbers.

Continuous Discrete

(p,t) Q R0 (pk, k) QN0

q = H/p, qk+1 = D2H+d (qk, pk+1),

p = H/q pk = D1H+

d(qk, pk+1)

S2(p,t) p(t)q(t)

t

0

[p(s) q(s) H(q(s), p(s))] ds Skd (pk) pkqk k1l=0

pl+1ql+1 H

+d

(ql, pl+1)

dS2 =S2p

dp +S2t

dt Sk+1d (pk+1) Sk

d (pk)

q dp + H(q, p) dt H+d (qk, pk+1) pk qk

S2t = H

S2p , p

Sk+1d (pk+1) Skd (pk)= H+

d(DSkd (pk), pk+1) pk DS

k

d (pk)

4. Galerkin Hamiltonian Variational Integrators

4.1. Exact Discrete Hamiltonian. The exact discrete Hamiltonian H+d,exact(q0, p1) and the discrete right

Hamiltons equations (20) generate a discrete solution curve {qk, pk}Nk=0 that samples the exact solution(q(), p()) of the continuous Hamiltons equations for the continuous Hamiltonian H(q, p), i.e., qk = q(tk)and pk = p(tk).

By comparing the definition (11) of a discrete right Hamiltonian function H+d (q0, p1) on [0, h] and thecorresponding 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 rightHamiltonian function on [0, h], given by

H+d,exact(q0, p1) = ext(q,p)C2([0,T],TQ)q(t0)=q0,p(t1)=p1

p1q1

h0

[p(t)q(t) H(q(t), p(t))] dt,(25)

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 Hamiltonsequations that satisfies the given boundary conditions (q0, p1). However, the variational characterization ofthe exact discrete Hamiltonian naturally leads to computable approximations based on Galerkin techniques.In practice, one replaces the path space C2([0, T], TQ), which is an infinite-dimensional function space, with

a finite-dimensional function space, and uses numerical quadrature to approximate the integral.Let {i()}

si=1, [0, 1], be a set of basis functions for a s-dimensional function space C

sd. We also choose

a numerical quadrature formula with quadrature weights bi, and quadrature points ci, i.e.,1

0f(x)dx s

j=1 bif(ci). From these basis functions and the numerical quadrature formula, we will systematicallyconstruct a generalized Galerkin Hamiltonian variational integrator in the following manner:

1. Use the basis functions i to approximate the velocity q over the interval [0, h],

qd(h) =

si=1

Vii().

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2. Integrate qd() over [0, h], to obtain the approximation for the position q,

qd(h) = qd(0) +

h0

si=1

Vii()d(h) = q0 + h

si=1

Vi

0

i()d,

where we applied the boundary condition qd(0) = q0. Applying the boundary condition qd(h) = q1yields

q1 = qd(h) = q0 + hs

i=1

Vi

1

0

i()d q0 + hs

i=1

BiVi,

where Bi =1

0 i()d. Furthermore, we introduce the internal stages,

Qi qd(cih) = q0 + hs

j=1

Vjci

0

j()d q0 + hs

j=1

AijVj ,

where Aij =ci

0 j()d.

3. Let Pi = p(cih). Use the numerical quadrature formula (bi, ci) and the finite-dimensional functionspace Csd to construct H

+d (q0, p1) as follows

H+d (q0, p1) ext(q,p)

C2([0,T],TQ)

q(t0)=q0,p(t1)=p1

p1q1 h

0

[p(t)q(t) H(q(t), p(t))] dt

H+d (q0, p1) = extqdCsd([0,h],Q),PiQ

p1qd(h) h

si=1

bi [p(cih)qd(cih) H(qd(cih), p(cih))]

(26)

= extVi,Pi

p1

q0 + h

si=1

BiVi

h

si=1

bi

Pi s

j=1

Vjj(ci) H

q0 + h s

j=1

AijVj , Pi

extVi,Pi

K(q0, Vi, Pi, p1).

To obtain an expression for H+d (q0, p1), we first compute the stationarity conditions for K(q0, Vi, Pi, p1)

under the fixed boundary condition (q0, p1).

0 =K(q0, Vi, Pi, p1)

Vj= hp1Bj h

si=1

bi

Pij(ci) hAij

H

q(Qi, Pi)

, j = 1, . . . , s .(27a)

0 =K(q0, V

i, Pi, p1)

Pj= hbj

s

i=1

i(cj)Vi

H

p(Qj, Pj)

, j = 1, . . . , s .(27b)

4. By solving the 2s stationarity conditions (27), we can express the parameters Vi, Pi, in terms ofq0, p1, i.e., Vi = Vi(q0, p1) and Pi = Pi(q0, p1). Then, the symplectic map (q0, p0) (q1, p1) can beexpressed in terms of the internal stages

p0 =H+d (q0, p1)

q0=

K(q0, Vi(q0, p1), P

i(q0, p1), p1)

q0(28)

=

K

q0 +

K

Vi

Vi

q0 +

K

Pi

Pi

q0 =

K

q0

= p1 + h

si=1

biH

q(Qi, Pi),

Similarly, we obtain

q1 =H+d (q0, p1)

p1=

K(q0, Vi(q0, p1), P

i(q0, p1), p1)

p1(29)

=K

ViVi

p1+

K

PiPi

p1+

K

p1=

K

p1

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= q0 + hs

i=1

BiVi.

Without loss of generality, we assume that the quadrature weights bi = 0. Then, the stationarity condition(27b) reduces to

si=1 i(cj)V

i Hp (Qj , Pj) = 0. Moreover, by substituting (28) into the stationarity

condition (27a), we obtain, si=1 biP

ij(ci) p0Bj + hsi=1(biBj biAij)

Hq (Q

i, Pi) = 0.

In summary, the above procedure gives a systematic way to construct a generalized Galerkin Hamiltonianvariational integrator, which can be rewritten in the following compact form,

q1 = q0 + hs

i=1

BiVi,(30a)

p1 = p0 hs

i=1

biH

q(Qi, Pi),(30b)

Qi = q0 + h

sj=1

AijVj , i = 1, . . . , s ,(30c)

0 =s

i=1

biPij(ci) p0Bj + h

s

i=1

(biBj biAij)H

q(Qi, Pi), j = 1, . . . , s ,(30d)

0 =s

i=1

i(cj)Vi

H

p(Qj , Pj), j = 1, . . . , s ,(30e)

where (bi, ci) are the quadrature weights and quadrature points, and Bi =1

0 i()d, Aij =ci

0 j()d.This is the general form of a Galerkin Hamiltonian variational integrator. Issues of solvability, convergence,

and accuracy, depend on the specific Hamiltonian system, and the choice of finite-dimensional function spaceCsd and numerical quadrature formula (bi, ci). We will not perform an in depth analysis here, but we willillustrate how our proposed framework is related to the discrete Lagrangian based methods given in [ 21] andp.209 of [13].

4.3. Galerkin Variational Integrators from the Lagrangian Point of View. In this subsection,we investigate the generalized Galerkin variational integrators from Lagrangian point of view when the

Hamiltonian function is hyperregular. In this case, the exact discrete right Hamiltonian function is relatedby the Legendre transformation to exact discrete Lagrangian function, i.e.,

Lexactd (q0, q1) = extqC2([0,h],Q)

q(0)=q0,q(h)=q1

h0

L(q(t), q(t))dt(31)

= ext(q,p)C2([0,h],TQ)

q(0)=q0,q(h)=q1

h0

pq H(q, p)dt

= p1q1 H+d,exact(q0, p1)

q1=D2H

+d,exact(q0,p1)

.

We wish to see how Galerkin variational integrators that are derived from the Hamiltonian and Lagrangiansides are related. In order for the comparison to make sense, we will approximate the exact discrete La-grangian using the same basis functions and numerical quadrature formula as on the Hamiltonian side. Asbefore, let {i()}si=1, [0, 1], be a set of basis functions for a s-dimensional function space C

sd , and

choose a numerical quadrature formula with quadrature weights bi, and quadrature points ci. From thesebasis functions and the numerical quadrature formula, we will systematically construct a generalized GalerkinLagrangian variational integrator in the following manner:

1. Use the basis functions i to approximate the velocity q over the interval [0, h],

qd(h) =

si=1

Vii().

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2. Integrate qd() over [0, h], to obtain the approximation for the position q,

qd(h) = qd(0) +

h0

si=1

Vii()d(h) = q0 + hs

i=1

Vi

0

i()d,

where we applied the boundary condition qd(0) = q0. In the discrete Lagrangian framework, theboundary conditions are given by (q0, q1), so we will use a Lagrange multiplier to enforce the boundary

condition qd(h) = q1,

q1 = qd(h) = q0 + h

si=1

Vi1

0

i()d q0 + hs

i=1

BiVi,

where Bi =1

0 i()d. Furthermore, we introduce the internal stages,

(32) Qi qd(cih) = q0 + hs

j=1

Vjci

0

j()d q0 + hs

j=1

AijVj ,

where Aij =ci

0j()d, and their velocities,

Qi qd(cih) =s

j=1

j(ci)Vj .(33)

3. Use the numerical quadrature formula (bi, ci), and the finite-dimensional function space Csd to con-

struct Ld(q0, q1) as follows

Ld(q0, q1) extqC2([0,h],Q),

q(0)=q0

h0

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

+ (q1 q(h))

Ld(q0, q1) = extqdCsd([0,h],Q),

q(0)=q0

h

si=1

biL(qd(cih), qd(cih))

+ (q1 qd(h))(34)

= extVi,

h

s

i=1

biLq0 + hs

j=1

AijVj ,

s

j=1

Vjj(ci)+ q1 q0 hs

i=1

BiVi

extVi,

K(q0, Vi, , q1).

To obtain an expression for Ld(q0, q1), we first compute the stationarity conditions for K(q0, Vi, , q1)

under the fixed boundary condition (q0, q1).

0 =K(q0, Vi, , q1)

Vj= h

si=1

bi

L

q(Qi, Qi)hAij +

L

q(Qi, Qi)j(ci)

hBj , j = 1, . . . , s .(35a)

0 =K(q0, Vi, , q1)

= q1 q0 h

si=1

BiVi.(35b)

4. By solving the 2s stationarity equations (35), we can express the parameters Vi, , in terms ofq0, q1,

i.e., Vi

= Vi

(q0, q1), = (q0, q1). Then, the symplectic map (q0, p0) (q1, p1) can be expressed interms of the internal stages and the Lagrange multiplier

p0 = Ld(q0, q1)

q0=

K(q0, V

i(q0, p1), (q0, p1), q1)

q0

(36)

=

K

q0+

K

ViVi

q0+

K

q0

=

K

q0

= hs

i=1

biLq(Qi, Qi) + .

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Similarly, we obtain

p1 =Ld(q0, q1)

q1=

K(q0, Vi(q0, p1), (q0, p1), q1)

q1(37)

=K

q1+

K

ViVi

q1+ +

K

q1=

K

q1

= .By combining (36) and (37), we obtain p1 = p0 + h

si=1 biLq(Q

i, Qi). Substituting this into the station-

arity condition (35a), yieldss

i=1 biL/q(Qi, Qi)j(ci) p0Bj h

si=1(biBj biAij)L/q(Q

i, Qi) = 0.In summary, the above procedure gives a systematic way to construct a generalized Galerkin Lagrangian

variational integrator, which can be written in the following compact form,

q1 = q0 + hs

i=1

BiVi,(38a)

p1 = p0 + hs

i=1

biL

q(Qi, Qi),(38b)

Qi = q0 + hs

j=1

AijVj , i = 1, . . . , s(38c)

0 =

si=1

biL

q(Qi, Qi)j(ci) p0Bj h

si=1

(biBj biAij)L

q(Qi, Qi), j = 1, . . . , s(38d)

0 =

si=1

i(cj)Vi Qj , j = 1, . . . , s(38e)

where (bi, ci) are the quadrature weights and quadrature points, and Bi =1

0 i()d, Aij =ci

0 j()d.As expected, this is equivalent to the generalized Galerkin Hamiltonian variational integrator, as the

following proposition indicates.

Proposition 2. If the continuous Hamiltonian H(q, p) is hyperregular, and we construct a LagrangianL(q, q) by the Legendre transformation, then the generalized Galerkin Hamiltonian variational integrator

(30a)(30e) and the generalized Galerkin Lagrangian variational integrator (38a)(38e), associated with thesame choice of basis functions i and numerical quadrature formula (bi, ci), are equivalent.

Proof. Since we chose the same basis functions and numerical quadrature formula for both methods, theapproximations for q1 and Q

i are the same in both methods, as can be seen by comparing (30a) and (38a),(30c) and (38c). Since we assumed that the Lagrangian and Hamiltonian are related by L(q, q) = pqH(q, p),subject to the Legendre transformation q = H/p(q, p), we consider p to be a function of (q, q), and compute

L

q(q, q) = q

p

q

H

q(q, p)

H

p(q, p)

p

q=

H

q(q, p),

L

q(q, q) = q

p

q+p

H

p(q, p)

p

q= p.

Since these identities have to hold at the internal stages, we have that

H

p (Q

i

, P

i

) =

Q

i

,H

q(Qi, Pi) =

L

q(Qi, Qi),

Pi =L

q(Qi, Qi),

for i = 1, . . . , s. Clearly, substituting these identities into (30b), (30d), and (30e), yields (38b), (38d),and (38e). As such the two systems of equations, (30a)(30e) and (38a)(38e), are equivalent, once theLegendre transformation and the identities relating the continuous Lagrangian and Hamiltonian are takeninto account.

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4.4. Variational integrators and Symplectic Partitioned Runge-Kutta methods. In this subsec-tion, we consider a special class of Galerkin variational integrators, and demonstrate that they can beimplemented as symplectic partitioned RungeKutta methods.

Let Csd([0, 1], Q) be a s-dimensional function space, and consider a set of basis functions i() on [0, 1],and a set of control points ci, i = 1, . . . , s. We would like to construct a new set of basis functions i() thatspan the same function space, and satisfies i(cj) = ij , where ij is the Kronecker delta. This is possible

whenever the matrix

M =

1(c1) 1(c2) 1(cs)2(c1) 2(c2) 2(cs)

......

. . ....

s(c1) s(c2) s(cs)

(39)

is invertible. In particular, let () = [1(), . . . , s()]T, and construct a new set of basis functions () =[1(), . . . , s()]

T by () = M1(). It is easy to see that i(cj) = ij since

1(c1) 1(c2) 1(cs)2(c1) 2(c2) 2(cs)

......

. . ....

s(c1) s(c2) s(cs)

= M1M =

1 0 00 1 0...

.... . .

...

0 0 1

.(40)

We can construct a numerical quadrature formula that is exact on the span of the basis functions i() asfollows: Since i(cj) = ij , we can interpolate any function f() on [0, 1] at the control points ci by takingf() =

si=1 f(ci)i(). Then, we obtain the following quadrature formula,1

0

f()d

10

f()d =

10

si=1

f(ci)i()d =s

i=1

f(ci)

10

i()d

si=1

bif(ci),(41)

where bi =1

0i()d are the quadrature weights. By construction, the above quadrature formula is exact for

any function in the s-dimensional function space Csd([0, 1], Q). Now, if we apply this quadrature formula withquadrature points ci, we obtain a Galerkin variational integrator that can be implemented as a symplecticpartitioned RungeKutta (SPRK) method.

Theorem 3. Given any set of basis functions () = [1(), . . . , s()]T, that span Csd([0, 1], Q), considerthe quadrature formula given in (41). Then, the associated generalized Galerkin Hamiltonian variationalintegrator (30a)(30e), which is expressed in terms of the discrete right Hamiltonian function (26), canbe implemented by the following s-stage symplectic partitioned RungeKutta method applied to Hamiltonsequations (1):

q1 = q0 + hs

i=1

biH

p(Qi, Pi),(42a)

p1 = p0 hs

i=1

biH

q(Qi, Pi),(42b)

Qi = q0 + hs

j=1

aijH

p(Qj , Pj), i = 1, . . . , s ,(42c)

Pi = p0 hs

j=1

aijH

q(Qj , Pj), i = 1, . . . , s ,(42d)

where bi =1

0 i()d = 0, aij =ci

0 j()d, and aij =bibjbjaji

bi. The basis functions satisfy i(cj) = ij,

and are given by () = M1(), where () = [1(), . . . , s()]T, and M is defined in (39).

Proof. The new basis functions () are constructed from the original basis functions () by the relationship() = M1(). Thus, we have that () = M (), and in particular, i() =

sj=1 i(cj)j(). By

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substituting this into Bi =1

0 i()d, equation (30a) becomes

q1 = q0 + hs

i=1

BiVi

= q0 + hs

i=1

Vi 1

0

s

j=1

i(cj)j()d

= q0 + hs

j=1

si=1

i(cj)Vi

10

j()d

= q0 + h

sj=1

H

p(Qj, Pj)

10

j()d

q0 + hs

j=1

bjH

p(Qj , Pj),

where we used equation (30e) to go from the third equality to the fourth one. Similarly, by substitutingk() =

sj=1 k(cj)j() into Aik =

ci0

k()d and using equation (30e), equation (30c) becomes

Qi = q0 + hs

k=1

AikVk

= q0 + hs

k=1

Vkci

0

sj=1

k(cj)j()d

= q0 + hs

j=1

H

p(Qj , Pj)

ci0

j()d

q0 + hs

j=1

aijH

p(Qj , Pj),

where aij = ci0

j()d. Note that equation (30b) has the same form as (42b), so we only have to re-

cover equation (42d). Let E = [E1 , . . . , E s ]

T, where Ej s

i=1 biPij(ci) p0Bj + h

si=1(biBj

biAij)Hq (Q

i, Pi) = 0, which corresponds to (30d). By substituting Bi =1

0i()d =

10

sj=1 i(cj)j()d,

and bi =1

0 i()d into Ej , we obtain

0 = Ej =s

i=1

biPij(ci) p0Bj + h

si=1

(biBj biAij)H

q(Qi, Pi)

=s

i=1

biPij(ci) p0

10

j()d + hs

i=1

bi

10

j()d bi

ci0

j()d

H

q(Qi, Pi)

=s

i=1biP

ij(ci) p0 1

0

s

i=1j(ci)i()d

+ hs

i=1

bi

10

sk=1

j(ck)k()d bi

ci0

sk=1

j(ck)k()d

H

q(Qi, Pi)

=s

i=1

j(ci)

biP

i bip0 + hs

k=1

(bibk bkaki)H

q(Qk, Pk)

.

We swapped the role of the indices i and k in the second to last line to obtain the final equality. LetE = [E1 , . . . , E

s ]

T, where Ei biPi bip0 + h

sk=1(bibk bkaki)

Hq (Q

k, Pk). Then, the above equation

can be viewed as the j-th component of the system of equations M E E = 0, where M = [i(cj)] is

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invertible. Therefore, we have that E = 0, i.e., Ei = biPi bip0 + h

sk=1(bibk bkaki)

Hq

(Qk, Pk) = 0.

Since bi = 0, dividing by bi and recalling that aij =bibjbjaji

biyields (42d).

Comparison with Discrete Lagrangian SPRK Methods. Proposition 2 states that for hyperregularHamiltonians, if one chooses the same basis functions and quadrature formula, the generalized GalerkinHamiltonian variational integrator is equivalent to the generalized Galerkin Lagrangian variational integrator.

Therefore, the above theorem also applies in the Lagrangian setting. In particular, if one chooses theLagrange polynomials associated with the quadrature nodes ci as our choice of basis functions i(), thenthe coefficients of the SPRK method derived above agree with the method derived in [21] using discreteLagrangians. However, our approach remains valid in the case of degenerate Hamiltonians, for which it isimpossible to obtain a Lagrangian and apply the method in [21] to derive Hamiltonian variational integrators.

The derivation on p. 209 of the book [13], which is analogous to the result in [26], generalizes the approachin [21] by considering discrete Lagrangian SPRK methods without the restriction that the RungeKuttacoefficients are obtained from integrals of Lagrange polynomials. It is however unclear how one shouldchoose these coefficients. In contrast, our approach provides a systematic means of deriving the coefficientsby an appropriate choice of basis functions and quadrature formula. Our discrete Hamiltonian method isexpressed in terms of Type II generating functions and the continuous Hamiltonian as opposed to the discreteLagrangian approach based on Type I generating functions and the continuous Lagrangian.

Discrete Hamiltonian associated with Galerkin SPRK Method. For the symplectic partitionedRungeKutta method (42) described above, we can explicitly compute the corresponding Type II generatingfunction H+d (q0, p1) given in (26) as follows,

H+d (q0, p1) = p1qd(h) hs

i=1

bi

Piqd(cih) H(Qi, Pi)

(43)

= p1

q0 + h

si=1

biH

p(Qi, Pi)

h

si=1

bi

Pi

H

p(Qi, Pi) H(Qi, Pi)

= p1q0 + h

si=1

bi(p1 Pi)

H

p(Qi, Pi) + h

si=1

biH(Qi, Pi)

= p1q0 h2

si,j=1

biaij Hq

(Qi, Pi) Hp

(Qj , Pj) + h

si=1

biH(Qi, Pi).

This Type II generating function is consistent with the Type I generating function for SPRK methods thatwas given in Theorem 5.4 on p. 198 of [13].

Sufficient Condition for Consistency of the SPRK Method. If the constant function f(x) = 1 is in thefinite-dimensional function space Csd, then by interpolation, we have that 1 =

si=1 f(ci)i() =

si=1 i().

Thus,s

i=1 bi =1

0

si=1 i()d = 1. Partitioned RungeKutta order theory [5] states that the conditions

i=1 bi = 1 implies that the variational integrator (42) is at least first-order. Therefore, to obtain a consis-tent method, it is sufficient that the constant function is in the span of the basis functions we choose. Inparticular, if we let 1() = 1, we ensure that our method is at least first-order.

Construction of the SPRK Tableau. Let the symplectic partitioned RungeKutta method (42) be de-noted by the tableau.

c1 a11 a1s...

......

cs as1 assb1 bs

c1 a11 a1s...

......

cs as1 assb1 bs

Based on the above generalized Galerkin method, the coefficients in the partitioned RungeKutta tableaucan be constructed in the following systematic way:

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Generalized Galerkin Hamiltonian SPRK Method.

1. Choose a basis set i(), [0, 1], i = 1, . . . , s, with 1() = 1.2. Choose quadrature points ci, i = 1, . . . , s . Ensure that M = [i(cj)] is invertible.3. Let the column vector b = [b1, b2, . . . , bs]

T contain the coefficients in the SPRK tableau.There are two ways to obtain b:

i. Compute Bi =1

0 i()d and let B = [B1, B2, . . . , Bs]T. Then, b = M1B.ii. Compute a new basis set i() by using the relation () = M

1(), then computeb = [b1, b2, . . . , bs]

T by bi =1

0i()d.

4. Let the matrix A = [aij ] contain the coefficients of the SPRK tableau. As before, thereare two way to obtain A:

i. Compute coefficients A = [Aij ], where Aij =ci

0 j()d. Then, the matrix is

given by A = [aij ] = AMT.

ii. Compute A = [aij], where aij =ci

0 j()d, directly by using the new basis

functions () = M1().5. Compute the coefficients A = [aij ] by using aij =

bibjbjajibi

.

4.5. Examples. In this subsection, we will consider four examples to illustrate the above procedure for

constructing variational integrators.

Example 2. We consider one-stage methods. Choose the basis function1 = 1, then for any quadraturepoint c1, the matrix M = [1(c1)] = [1 ] is invertible.

i. If c1 = 0, then b1 = 1, a11 = 0, a11 = 1, which is the symplectic Euler method.ii. If c1 =

12 , then b1 = 1, a11 =

12 , a11 =

12 , which is the midpoint rule.

iii. If c1 = 1, then b1 = 1, a11 = 1, a11 = 0, which is the adjoint symplectic Euler method.

Example 3. Choose basis functions 1 = 1, 2 = cos(). If we choose quadrature points c1 = 0, c2 = 1,then we obtain

M =

1(c1) 1(c2)2(c1) 2(c2)

=

1 11 1

and M1 =

1

2

1 11 1

.

One can easily compute

B1 =

10

1()d = 1, B2 =

10

1()d = 0,

Therefore we get b = [b1, b2]T = M1B = [ 12 ,

12 ]

T, which is the trapezoidal rule. We also compute

A =

c10

1()dc1

02()dc2

0 1()dc2

0 2()d

=

0 01 0

.

Therefore, the matrix

A =

c10

1()dc1

02()d

c2

0 1()d

c2

0 2()d

= AMT =

0 012

12

,

By using the relationship aij =bibj

bjaji

bi , one obtains

A = [aij ] =

0 012

12

.

Thus, we obtain the StormerVerlet method. Interestingly, the StormerVerlet method is typically derived asa variational integrator by using linear interpolation, i.e., 1 = 1, 2 = , and the trapezoidal rule.

Example 4. If we choose basis functions 1 = 1, 2 = cos(), 3 = sin() and quadrature points c1 =0, c2 =

12 , c3 = 1, we obtain a new method which is second-order accurate, and the coefficients of the SPRK

method are given by

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5. Momentum Preservation and Invariance of the Discrete Right Hamiltonian Function

Momentum Maps. First, we recall the definition of a momentum map defined on TQ given in [1].

Definition 1. Let (P, ) be a connected symplectic manifold and : G P P be a symplectic actionof the Lie group G on P, i.e., for each g G, the map g : P P; x (g, x) is symplectic. We saythat a map J : P g, where g is a dual space of the Lie algebra g of G, is a momentum map for the

action if for every g, dJ() = iP, where J() : P R is defined by J()(x) = J(x) , and P isthe infinitesimal generator of the action corresponding to . In other words, J is a momentum map providedXJ() = P for all g.

For our purposes, we are interested in the case P = TQ, and = dqi dpi is the canonical symplectictwo-form on TQ. This gives a momentum map of the form J : TQ g, and we describe the constructiongiven in Theorem 4.2.10 of [1]. Notice that is exact, since = d = d(pidqi). Consider an actiong that leaves the Lagrange one-form invariant, i.e.,

g = for all g G. Then, the momentum map

J : TQ g is given by

J(x) = iP(x).(45)

We can show that this satisfies the definition of the momentum map given above by using the fact that gleaves the one-form invariant for all g G, and

Pis the infinitesimal generator of the action corresponding

to . This implies that the Lie derivative of along the vector field P vanishes, i.e., P = 0, for all g.By Cartans magic formula,

0 = P = iPd + diP,

therefore, diP = iPd = iP. As such, J()(x) = iP satisfies the defining property, dJ() = iP, of

a momentum map. Then, J(x) = J()(x) = iP(x). Moreover, by Theorem 4.2.10 of [1], this momentummap is Ad-equivariant.

Let : G Q Q be an action of the Lie group G on Q. We will give the coordinate expression for thecotangent lifted action T

Q. In coordinates, we denote g1 : Q Q by qi = ig1(Q), then its cotangent

lifted action TQ : G TQ TQ is given by

TQ(g, q, p) = Tg1(q, p) = ig(q), pj q

j

ig(q)

,(46)where Tg1 means the cotangent lift of the action g1 .

In the following proposition, we give the coordinate expression for the cotangent lifted action, and showthat it leaves the Lagrange one-form = pidq

i invariant.

Proposition 3. Given an action : G Q of a Lie group G on Q, the cotangent lifted action TQ :

G TQ TQ leaves the Lagrange one-form = pidqi invariant.

Proof. Given g G, let the cotangent lifted action of g on (q, p) be denoted by (Q, P) = TQ

g (q, p), the

components of which are given by Qi = ig(q) and Pi = pjqj

ig(q). Then, a direct computation yields

PidQi = Pid

ig(q) = pj

qj

ig

(q)

ig(q)

qjdqj = pjdq

j .(47)

This shows that TQ

g leaves the Lagrange one-form pidqi invariant.

Corresponding to the cotangent lift action TQ, for every g, the momentum map J : TQ g

defined in (45) has the following explicit expression in coordinates,

J(q) = iP(pidqi)(q) = p Q(q) = p

d

d

=0

exp()(qi),(48)

where q = (q, p) TQ.

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Discrete Noethers Theorem. In the discrete case, consider the one-step discrete flow map FH+d

:

(q0, p0) (q1, p1) defined by the discrete right Hamiltons equations,

p0 = D1H+d (q0, p1), q1 = D2H

+d (q0, p1).(49)

We will show that if the generalized discrete Lagrangian, Rd(q0, q1, p1) = p1q1 H+d (q0, p1), is invariant under

the cotangent lifted action, then we have discrete momentum preservation, which is the discrete analogue of

Noethers theorem for discrete Hamiltonian mechanics.

Theorem 4. Let TQ be the cotangent lift action of the action on the configuration manifold Q. If

the generalized discrete Lagrangian Rd(q0, q1, p1) = p1q1 H+d (q0, p1) is invariant under the cotangent lifted

action TQ, then the discrete flow map of the discrete right Hamiltons equations preserves the momentum

map, i.e., FH+d

J = J.

Proof. In coordinates, let (q0, p0) :=

TQexp()(q0, p0) and (q

1, p

1) :=

TQexp()(q1, p1). From the invariance of

p1q1 H+d (q0, p1), we have that

0 =d

d

=0

{p1q1 H

+d (q

0, p

1)}(50)

= p1d

d=0

q1 + q1

d

d=0

p1 D1H

+d (q0, p1)

d

d=0

q0 D2H

+d (q0, p1)

d

d=0

p1

= p1d

d

=0

q1 + q1d

d

=0

p1 p0d

d

=0

q0 q1d

d

=0

p1

= p1d

d

=0

exp()(q1) p0d

d

=0

exp()(q0)

= p1 Q(q1) p0 Q(q0),

where we used the discrete right Hamiltons equations (49) in going from the second to the third line, andq0 = exp()(q0) and q

1 = exp()(q1) are used in going from the third to the fourth line. Then, by the

definition of FH+dand J, (50) states that F

H+d

J = J.

G-invariant generalized discrete Lagrangians from G-equivariant interpolants. We now provide

a systematic means of constructing a discrete Hamiltonian, so that the generalized discrete LagrangianRd(q0, q1, p1) = p1q1 H

+d (q0, p1) is G-invariant, provided that the generalized Lagrangian R(q, q, p) =

pq H(q, p) is G-invariant.Our construction will be based on an interpolatory function : Qr C2([0, h], Q), that is parameterized

by r + 1 internal points q Q, defined at the times 0 = d0h < d1h < < drh h, i.e., (dh; {q}r=0) =q. We also use a numerical quadrature formula given by quadrature weights bi and quadrature points ci.We denote the momentum at the time cih by p

i. Then, we construct the following discrete Hamiltonian,

(51) H+d (q0, p1) = extqQ,piQ

q0=q0

p1 (h; {q

}r=0) s

i=1

biR

T (cih; {q}r=0) , p

i

,

where R(q, q, p) = pq H(q, p). An interpolatory function is G-equivariant if

(t; {gq}r=0) = g(t; {q}r=0).

Then, a G-invariant discrete Hamiltonian can be obtained if we use G-equivariant interpolatory functions.

Lemma 3. LetG be a Lie group acting on Q, such that gQ=Q, for all g G. If the interpolatory function(t; {gq}r=0) is G-equivariant, and the generalized Lagrangian R : T Q T

Q R,

R(q, q, p) = pq H(q, p),

is G-invariant, then the generalized discrete Lagrangian Rd : Q TQ R, given by

Rd(q0, q1, p1) = p1q1 H+d (q0, p1),

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where,

H+d (q0, p1) = extqQ,piQ

q0=q0

p1 (h; {q

}r=0) s

i=1

biR

T (cih; {q}r=0) , p

i

,

is G-invariant.

Proof. To streamline the notation, we denote the cotangent lifted action ofG on Q by TQ

g(q, p) = (gq,gp).

First, we note that

Rd(q0, q1, p1) = p1q1 extqQ,piQ

q0=q0

p1 (h; {q

}r=0) s

i=1

biR

T (cih; {q}r=0) , p

i

= extqQ,piQ

q0=q0

s

i=1

biR

T (cih; {q}r=0) , p

i

.

Then,

Rd(gq0, gq1, gp1) = extqQ,piQ

q0=gq0

s

i=1

biR

T (cih; {q}r=0) , p

i

= extqg1Q,pig1Q

gq0=gq0

s

i=1

biR

T (cih; {gq}r=0) , gp

i

= extqQ,piQ

q0=q0

s

i=1

biR

T Lg T (cih; {q}r=0) , gp

i

= extqQ,piQ

q0=q0

s

i=1

biR

T (cih; {q}r=0) , p

i

= Rd(q0, q1, p1),

where we used the identification q = gq in the second equality, the G-equivariance of the interpolatory

function and the property that gQ = Q in the third equality, and the G-invariance of the generalizedLagrangian in the fourth equality.

In view of Theorem 4, and the above lemma, if we use a G-equivariant interpolatory function to constructa discrete Hamiltonian as given in (51), then the discrete flow given by the discrete right Hamiltons equa-tions will preserve the momentum map J : TQ g.

Natural Charts and G-equivariant interpolants. Following the construction in [18], we use the groupexponential map at the identity, expe : g G, to construct a G-equivariant interpolatory function, and ahigher-order discrete Lagrangian. As shown in Lemma 3, this construction yields a G-invariant generalizeddiscrete Lagrangian if the generalized Lagrangian itself is G-invariant.

In a finite-dimensional Lie group G, expe is a local diffeomorphism, and thus there is an open neighborhoodU G of e such that exp1e : U u g. When the group acts on the left, we obtain a chart g : LgU u

at g G byg = exp

1e Lg1.

Lemma 4. The interpolatory function given by

(g ; h) = 1g0s

=0g0(g

)l,s()

,

is G-equivariant.

Proof.

(gg ; h) = 1(gg0)

s=0

gg0(gg)l,s()

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= Lgg0 expe

s=0

exp1e ((gg0)1(gg))l,s()

= LgLg0 expe

s=0

exp1e ((g0)1g1gg)l,s()

= Lg

1g0

s=0

exp1e L(g0)1(g)l,s()

= Lg

1g0 s

=0

g0(g)l,s()

= Lg(g ; h).

This G-equivariant interpolatory function based on natural charts allows one to construct discrete Liegroup Hamiltonian variational integrators that preserve the momentum map.

6. Conclusions and Future Directions

In this paper, we provided a variational characterization of the Type II generating function that generatesthe exact flow of Hamiltons equations, and show how this is a Type II analogue of Jacobis solution ofthe HamiltonJacobi equation. This corresponds to the exact discrete Hamiltonian for discrete Hamiltonianmechanics, and Galerkin approximations of this lead to computable discrete Hamiltonians. In addition, weintroduced a discrete Type II HamiltonJacobi equation, which can be viewed as a composition theorem fordiscrete Hamiltonians.

We introduced generalized Galerkin variational integrators from both the Hamiltonian and Lagrangianapproach, and when the Hamiltonian is hyperregular, these two approach are equivalent. Furthermore, wedemonstrated how these methods can be implemented as symplectic partitioned RungeKutta methods,and derived several examples using this framework. Finally we characterized the invariance properties of adiscrete Hamiltonian which ensure that the discrete Hamiltonian flow preserves the momentum map.

We are interested in the following topics for future work:

LiePoisson Reduction and Connections to the HamiltonPontryagin principle. Since we provideda method for constructing discrete Hamiltonians that yields a numerical method that is momentumpreserving, it is natural to consider discrete analogues of Lie-Poisson reduction. In particular, theconstrained variational formulation of continuous Lie-Poisson reduction [6] appears to be relatedto the HamiltonPontryagin variational principle [27]. It would be interesting to develop discreteLiePoisson reduction [18] from the Hamiltonian perspective, in the context of the discrete HamiltonPontryagin principle [16, 25].

Extensions to Multisymplectic Hamiltonian PDEs. Multisymplectic integrators have been developedin the setting of Lagrangian variational integrators [17], and Hamiltonian multisymplectic integra-tors [4]. In the paper [20], the Lagrangian formulation of multisymplectic field theory is relatedto Hamiltonian multisymplectic field theory [3]. It would be interesting to construct Hamiltonianvariational integrators for multisymplectic PDEs by generalizing the variational characterization ofdiscrete Hamiltonian mechanics, and the generalized Galerkin construction for computable discreteHamiltonians, to the setting of Hamiltonian multisymplectic field theories.

Acknowledgments

The research of ML was supported by NSF grants DMS-0726263, and CAREER Award DMS-0747659.The research of JZ was conducted in the mathematics departments at Purdue University and the Universityof California, San Diego, supported by a fellowship from the Academy of Mathematics and System Science,

Chinese Academy of Sciences. We would like to thank Jerrold Marsden, Tomoki Ohsawa, Joris Vankerschaver,and Matthew West for helpful discussions and comments.

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Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California

920930112.

E-mail address: [email protected]

LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.

E-mail address: [email protected]

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