variational (cf. Theorem 2.2.10 and Theorem 2.2.11).
More information on variational integrators can be found in [41].
is defined on the set of all smooth curves C(Q) = {q∶ [a, b] Ð→Q}, that is,S ∶ C(Q) Ð→R. The action functional SN ∶ C(N) Ð→ R for the Lagrangian LN is given by an analogous formula. SinceN ⊂QandT N⊂T Qare submanifolds, we haveSN =S∣C(N). The dynamics of the constrained system is given by Hamilton’s principle, i.e., extremizingSN. To consider this dynamics in the embedding space, we further introduce the augmented configuration manifold Q×Rd and the augmented LagrangianLC∶T(Q×Rd) Ð→R defined by
LC(q, λ,q,˙ λ˙) =L(q,q˙) − ⟨λ, g(q)⟩, (2.5.2) where λ ∈ Rd denotes the vector of Lagrange multipliers and ⟨., .⟩ is the standard scalar product onRd. The corresponding augmented action functional SC ∶ C(Q) × C(Rd) Ð→Ris given by
SC[q(t), λ(t)] = ∫abLC(q(t), λ(t),q˙(t),λ˙(t))dt=S[q(t)] − ∫ab⟨λ(t), g(q(t))⟩dt. (2.5.3) The relation between the dynamics of the restricted system onN and the augmented system on Q×Rd is given by the following theorem (see [38]).
Theorem 2.5.1 (Lagrange multiplier theorem). The following statements are equiva- lent:
1. The curve q∶RÐ→N ⊂Qis an extremum of SN;
2. The pair of curves q∶ [a, b] Ð→Q andλ∶ [a, b] Ð→Rd is an extremum of SC.
The curves extremizing the augmented action functional SC satisfy the Euler-Lagrange equations associated withLC, the so-called constrained Euler-Lagrange equations,
∂L
∂qµ− d dt
∂L
∂q˙µ = ⟨λ, ∂g
∂qµ⟩,
g(q) =0. (2.5.4)
Deriving the equations of motion for constrained Hamiltonian systems is a little more cumbersome. Suppose we have a Hamiltonian system with the HamiltonianH∶T∗QÐ→R,
but we restrict the dynamics to the constraint submanifold N =g−1(0) ⊂Q as before. The main complication is the fact that there is no canonical way of embedding T∗N in T∗Q, and it is not obvious how to define an appropriate Hamiltonian HN on T∗N. However, if the Hamiltonian H is hyperregular, that is, there exists a corresponding hyperregular Lagrangian L on T Q, as in Theorem 2.3.3, then one can realize T∗N as a symplectic submanifold of T∗Q by defining the embeddingη∶T∗NÐ→T∗Qasη=FL○T i○ (FLN)−1. Using canonical coordinates on T∗Q, we have the following characterization
η(T∗N) = {(q, p) ∈T∗Q∣g(q) =0 andDg(q)∂H
∂p =0}. (2.5.5) It is straightforward to show that the canonical symplectic form ΩN on T∗N is then com- patible with the canonical symplectic form Ω on T∗Q, that is, ΩN =η∗Ω. We can further take HN =H○η. The Hamiltonian equations onT∗N will be defined by (2.1.4), and the theory discussed in Section 2.1 directly applies. However, it is again useful to consider the dynamics of this constrained system in the embedding space. We have the natural embed- ding T η ∶T T∗N Ð→T T∗Q, so the Hamiltonian vector field ZN on T∗N can be regarded as a vector field on η(T∗N). This leads to the following constrained equations on motion on T∗Q:
q˙µ= ∂H
∂pµ
, p˙µ= −∂H
∂qµ− ⟨λ, ∂g
∂qµ⟩,
g(q) =0, (2.5.6)
where λ again denotes Lagrange multipliers. If the Lagrangian L is hyperregular, then (2.5.6) and (2.5.4) are equivalent, as in Theorem 2.3.3.
For more information on the geometry of constrained Lagrangian and Hamiltonian sys- tems we refer the reader to [38].
2.5.2 Variational integrators for constrained systems
Variational integration of constrained Lagrangian systems follows the main ideas presented in Section 2.4. We start with a regular discrete Lagrangian Ld∶Q×QÐ→R. The discrete
dynamics is restricted to the constraint submanifold N×N ⊂Q×Q, whereN =g−1(0) ⊂Q as before. We consider the augmented discrete state space (Q×Rd) × (Q×Rd) and the augmented discrete LagrangianLCd ∶ (Q×Rd) × (Q×Rd) Ð→R,
LCd(q, λ,q,¯λ¯) =Ld(q,q¯) − ⟨λ, g(q)⟩. (2.5.7) The discrete Euler-Lagrange equations (2.4.2) forLCd yield
D2Ld(qk−1, qk) +D1Ld(qk, qk+1) =Dg(qk)Tλk,
g(qk+1) =0, (2.5.8)
whereDgdenotes the Jacobi matrix of the constraint functiong, andλkdenotes the column vector of Lagrange multipliers. If qk−1, qk are known, then (2.5.8) can be solved for qk+1 and λk.
The discrete Lagrangian Ld is chosen to approximate the exact discrete Lagrangian for L. However, the exact discrete Lagrangian for a constrained system is not simply the standard exact discrete Lagrangian (2.4.9) restricted to the constraint submanifoldN×N, as that would be the action along an unconstrained trajectory. Instead, the constrained exact discrete LagrangianLN,Ed ∶N×N Ð→Ris defined by
LN,Ed (q,q¯) = ∫0hLN(qE(t),q˙E(t))dt, (2.5.9) where qE(t) is the solution to the constrained Euler-Lagrange equations (2.5.4) satisfying the boundary conditionsqE(0) =q and qE(h) =q.¯
Lobatto IIIA-IIIB pair. Higher-order variational integrators for constrained systems can be constructed as in Section 2.4.3: a variational Runge-Kutta method for a constrained Lagrangian system described by a regular Lagrangian L is equivalent to a symplectic par- titioned Runge-Kutta method applied to the constrained Hamiltonian equations (2.5.6).
Consider a Hamiltonian system H∶T∗QÐ→Rwith the holonomic constraint g∶QÐ→
Rd, and two Runge-Kutta methods with the coefficientsaij,bi and ¯aij, ¯bi, respectively. An s-stageconstrained partitioned Runge-Kutta method is the map
Fh∶η(T∗N) ∋ (q, p) Ð→ (q,¯p¯) ∈η(T∗N), (2.5.10) implicitly defined by the system of equations
Q˙i=∂H
∂p(Qi, Pi), i=1, . . . , s, (2.5.11a) P˙i= −∂H
∂q (Qi, Pi) −Dg(Qi)TΛi, i=1, . . . , s, (2.5.11b)
0=g(Qi), i=1, . . . , s, (2.5.11c)
Qi=q+h
s
∑
j=1
aijQ˙j, i=1, . . . , s, (2.5.11d) Pi=p+h
s
∑
j=1
¯aijP˙j, i=1, . . . , s, (2.5.11e) q¯=q+h
s
∑
i=1
biQ˙i, (2.5.11f)
p¯=p+h
s
∑
i=1
¯biP˙i, (2.5.11g)
0=Dg(q)¯ ∂H
∂p(q,¯ p),¯ (2.5.11h)
where Qi, ˙Qi, Pi, ˙Pi and Λi are the internal stages of the method. However, this system is not solvable for an arbitrary choice of the Runge-Kutta methods—note we have only (4s+2)n+sdunknowns (the internal stages and ¯q, ¯p), but(4s+2)n+(s+1)dequations. It can be shown that (2.5.11) is solvable if one chooses the coefficients of thes-stage Lobatto IIIA- IIIB method (see Section 2.2.2). For the Lobatto IIIA-IIIB schemes we have a1j =0 for j=1, . . . , s(cf. Table 2.4, Table 2.5, and Table 2.6), therefore in (2.5.11d) we have Q1=q.
If we assume g(q) = 0, i.e., the initial position is consistent with the constraint, then d equations in (2.5.11c) for i=1 are automatically satisfied, and the whole system becomes solvable. Further, the Lobatto IIIA-IIIB schemes satisfy asj =bj for j = 1, . . . , s, so from (2.5.11d) and (2.5.11f) we have ¯q=Qs, and consequently, by (2.5.11c), we also haveg(q¯) =0, i.e., the new position of the system is consistent with the constraint. This result, together with (2.5.11h), means that(q,¯ p¯) ∈η(T∗N), that is, (2.5.11) indeed defines an integrator for the constrained Hamiltonian system. The following theorem can be proved (see [31], [23]).
Theorem 2.5.2. The s-stage constrained Lobatto IIIA-IIIB scheme (2.5.11)is symplectic
onη(T∗N) and convergent of order 2s−2.
If one defines the discrete Lagrangian Ld∶Q×QÐ→Ras
Ld(q,q¯) =h
s
∑
i=1
biL(Qi,Q˙i), (2.5.12) whereQiand ˙Qi satisfy (2.5.11), then the resulting variational integrator will be equivalent to the constrained Lobatto IIIA-IIIB scheme, whereLandHare related as in Theorem 2.3.3.
More information on variational and symplectic integration of constrained mechanical systems can be found in [23], [26], and [41].