2.4.1 Discrete Mechanics
As indicated in Section 2.2, the main purpose of symplectic integration is to preserve at the discrete level the symplectic structure underlying continuous Hamiltonian systems. Simi- larly, the central idea of variational integration is to preserve the variational structures of Lagrangian systems. This leads to so-called Discrete Mechanics and the underlying idea of discretization due to Veselov. For a Veselov-type discretization we consider the discrete state spaceQ×Q, which serves as a discrete approximation of the tangent bundle (see [41]). We define a discrete LagrangianLdas a smooth mapLd∶Q×QÐ→Rand the corresponding
discrete action S= N−1 ∑ k=0 Ld(qk, qk+1). (2.4.1)
The variational principle now seeks a sequenceq0,q1,...,qN that extremizesSfor variations
holding the endpointsq0 and qN fixed. The Discrete Euler-Lagrange equations follow
D2Ld(qk−1, qk) +D1Ld(qk, qk+1) =0. (2.4.2)
Assuming that these equations can be solved for qk+1, i.e., Ld is non-degenerate, they
implicitly define the discrete Lagrangian map FLd∶Q×QÐ→Q×Q such that
FLd(qk−1, qk) = (qk, qk+1). (2.4.3)
Let(qµ,q¯µ) denote local coordinates onQ×Q. We can define the discrete Legendre trans- formsFL+
d,FL −
d∶Q×QÐ→T
∗Q, which in local coordinates onQ×Qand T∗Qare respec-
tively given by
F+Ld(q,q¯) = (q, D¯ 2Ld(q,q¯)),
F−Ld(q,q¯) = (q,−D1Ld(q,q¯)), (2.4.4)
where q = (q1, . . . , qn) and ¯q = (q¯1, . . . ,q¯n). The Discrete Euler-Lagrange equations (2.4.2) can be equivalently written as
F+Ld(qk−1, qk) =F −
Ld(qk, qk+1). (2.4.5)
Using either of the transforms, one can define the discrete Lagrange two-form onQ×Q by
ωLd= (F ±L
d)∗Ω, which in coordinates gives˜
ωLd=
∂2Ld ∂qµ∂q¯νdq
µ∧dq¯ν. (2.4.6)
It then follows that the discrete flow FLd is symplectic, i.e., F ∗
LdωLd=ωLd.
Using the Legendre transforms we can pass to the cotangent bundle and define the discrete Hamiltonian map ˜FLd∶T
∗QÐ→T∗Qby ˜ FLd =F±Ld○FLd○ (F ± Ld)−1 . (2.4.7)
This map is also symplectic, i.e., ˜F∗
LdΩ = Ω, where Ω is the canonical symplectic form
on T∗Q. Using (2.4.4), the discrete Euler-Lagrange equations (2.4.2) can be equivalently
rewritten in the position-momentum formulation onT∗Q as
pk= −D1Ld(qk, qk+1),
pk+1=D2Ld(qk, qk+1). (2.4.8)
This system implicitly defines the discrete Hamiltonian map: given (qk, pk), one can solve
(2.4.8) for (qk+1, pk+1).
2.4.2 Correspondence between discrete and continuous systems
To relate discrete and continuous mechanics it is necessary to introduce a timesteph∈R. If
the continuous LagrangianLis non-degenerate, it is possible to define a particular choice of discrete Lagrangian which gives an exact correspondence between discrete and continuous systems (see [41]), the so-calledexact discrete Lagrangian.
Definition 2.4.1. The exact discrete Lagrangian LEd for the non-degenerate Lagrangian
LEd(q,q¯) = ∫ h
0 L(qE(t),
˙
qE(t))dt, (2.4.9)
for sufficiently small h and q¯sufficiently close to q, where qE(t) is the solution to (2.3.3)
that satisfies the boundary conditions qE(0) =q and qE(h) =q¯. The discrete Legendre transforms F±LE
d associated with LEd can be related to the con-
tinuous Legendre transformFL.
Theorem 2.4.2. A regular Lagrangian L and the corresponding exact discrete Lagrangian
LEd have Legendre transforms related by
F+LEd(q,q¯) =FL(qE(h),q˙E(h)),
F−LEd(q,q¯) =FL(qE(0),q˙E(0)), (2.4.10)
for sufficiently small h and q¯sufficiently close to q, where qE(t) is the solution to (2.3.3) that satisfies the boundary conditions qE(0) =q and qE(h) =q¯.
Solving the discrete Euler-Lagrange equations (2.4.2) associated with LEd yields the dis- crete trajectoryq0,q1,. . . such that qk coincides with the exact solution of the continuous
Euler-Lagrange equations (2.3.3) at timetk=kh. This important property can be summa- rized in the following theorem, which we cite after [41].
Theorem 2.4.3. Consider a regular Lagrangian L, its corresponding exact discrete La- grangian LEd, and the pushforward of both the continuous and discrete systems to T∗Q,
yielding a Hamiltonian system with Hamiltonian H and a discrete Hamiltonian map F˜LE d,
respectively. Then, for a sufficiently small timestep h, the flow of the Hamiltonian system equals the discrete Hamiltonian map, that is,
FhH =F˜LE
d. (2.4.11)
For a given continuous system described by the regular Lagrangian L, a variational integrator is constructed by choosing a discrete Lagrangian Ld which approximates the
exact discrete Lagrangian LEd. The precision of this approximation can be measured by defining the order of accuracy of the discrete Lagrangian.
Definition 2.4.4. A discrete Lagrangian Ld∶Q×QÐ→R is of order r if there exists an
open subset U ⊂T Q with compact closure and constantsC>0 and ¯h>0 such that
∣Ld(q(0), q(h)) −LdE(q(0), q(h))∣ ≤Chr+1
(2.4.12) for all solutionsq(t)of the Euler-Lagrange equations(2.3.3)with initial conditions(q(0),q˙(0))∈
U and for all h≤¯h.
As discussed in Section 2.4.1, the discrete LagrangianLddefines the discrete Hamiltonian
map ˜FLdon the cotangent bundleT∗Q. This map is a numerical scheme for the Hamiltonian
system corresponding to the LagrangianL(cf. Theorem 2.3.3), and so Definition 2.2.1 and Definition 2.2.2 apply. If the Lagrangian L is regular, then one can show that the discrete Lagrangian Ld is of order r if and only if the associated discrete Hamiltonian map ˜FLd is
of orderr (see [41]).
Example: Symplectic Euler scheme revisited. Consider a regular LagrangianLand the following discrete Lagrangian
Ld(q,q¯) =hL(q,¯
¯
q−q
h ). (2.4.13)
One can check that this discrete Lagrangian is first-order. A variational numerical integrator is obtained by forming the discrete Euler-Lagrange equations (2.4.8). It is straightforward to verify that these equations are equivalent to (2.2.5), i.e., the symplectic Euler scheme, where the Hamiltonian H and the Lagrangian Lare related as in Theorem 2.3.3.
2.4.3 Variational partitioned Runge-Kutta methods
To construct higher-order variational integrators, one may consider a class of partitioned Runge-Kutta methods similar to partitioned Runge-Kutta methods for Hamiltonian sys- tems. We will construct an s-stage variational partitioned Runge-Kutta integrator for the regular LagrangianL by considering the discrete Lagrangian
Ld(q,q¯) =h s
∑
i=1
biL(Qi,Q˙i), (2.4.14)
Qi=q+h s
∑
j=1
aijQ˙j, (2.4.15)
and are chosen so that the right-hand side of (2.4.14) is extremized under the constraint
¯ q=q+h s ∑ i=1 biQi.˙ (2.4.16)
A variational integrator is then obtained by forming the corresponding discrete Euler- Lagrange equations (2.4.8). We have the following result:
Theorem 2.4.5. The s-stage variational partitioned Runge-Kutta method based on the discrete Lagrangian (2.4.14) with the coefficients aij and bi is equivalent to the following
partitioned Runge-Kutta method applied to the implicit Hamiltonian equations (2.3.12):
Pi= ∂L ∂q˙(Qi, ˙ Qi), i=1, . . . , s, ˙ Pi= ∂L ∂q(Qi, ˙ Qi), i=1, . . . , s, Qi=q+h s ∑ j=1 aijQj˙ , i=1, . . . , s, Pi=p+h s ∑ j=1 ¯ aijPj˙ , i=1, . . . , s, ¯ q=q+h s ∑ j=1 bjQj˙ , ¯ p=p+h s ∑ j=1 bjP˙j, (2.4.17)
where the coefficients satisfy the condition
bi¯aij+bjaji=bibj, ∀i, j=1, . . . , s, (2.4.18) and(q, p)denote the current values of position and momentum, (q,¯ p¯) denote the respective values at the next time step, and Qi, Q˙i, Pi,P˙i are the internal stages.
If the Lagrangian Lis regular, then (2.4.17) is equivalent to the symplectic partitioned Runge-Kutta method (2.2.6), where H and L are related as in Theorem 2.3.3. We there- fore have that the Gauss and Lobatto IIIA-IIIB methods discussed in Section 2.2.2 are
variational (cf. Theorem 2.2.10 and Theorem 2.2.11).
More information on variational integrators can be found in [41].