The discrete Euler-Poincare and Lie-Poisson algo- rithms

Discrete Reduction on Lie Groups

3.2 The discrete Euler-Poincare and Lie-Poisson algo- rithms

In this section we develop the discrete Euler-Poincare reduction of a Lagrangian system on TG. The discrete reduction of a right-invariant system proceeds as fol- lows. The induced group action on G X G is simply right multiplication in each component:

for all

g,

^{9k, 9k+l E} G. Then the quotient map is given by

7rd: G x G -+ (G x G)jG ~ G, (3.9)

We note that one may alternatively use 9k+l9-,;1 instead of 9k9-';~1. The projection map (3.9) defines the reduced discrete Lagrangian £ : G -+ IR for any G-invariant IL by £ ⁰ trd = IL, so that

and the reduced action sum is given by

N-l

S =

L

^£(Jkk+l),

k=O

where ikk+l

==

9k9-';~1 denote points in the quotient space. A reduction of the DEL equations results in the discrete Euler-Poincare (DEP) equations.

Discrete Euler-Poincare reduction

We state here a general reduction type theorem which mimics Theorem 2.2.1 of Chapter 2. Notice a similarity between the reduced variation of ~ in Theorem 2.2.1 and the variation of ikk+l below, e.g., the ad-action on 9 is replaced by the Ad-action on G.

Theorem 3.2.1. Let lL be a right invariant Lagrangian on G x G, and let £

(G x G)/G ~ G

---+

lR. be the restriction of [, to G given by £(glg2"l)

=

[,(gl,g2).

For any integer N

2:

3, let {(gk, gk+1)}f==-Ol be a sequence in G x G and define

!kk+1

==

9k9"k~1 to be the corresponding sequence in G. Then, the following are equivalent.

(1) The sequence {(gk, gk+1)}f==-01 is an extremum of the action sum § : G^N+1

---+

lR.

for arbitrary variations 6gk = (d/ dE) 10gk where for each k, E f--t gk is a smooth curve in G such that g2

=

gk; 6gk vanishes at endpoints.

(2) The sequence {(gk, gk+1)}f==-01 satisfies the discrete Euler-Lagrange equations (3.2).

(3) The sequence {!kk+1}f==ol is an extremum of the reduced action sum s : G^N+1

---+

lR. with respect to variations 6jkk+1 induced by the variations 6gk and given by

(4) The sequence {fkk+df==-ol satisfies the discrete Euler-Poincare (DEP) equations

(3.10)

for k

=

1, ... , N - 1. Here Rj and Lj are the right and left pull-backs by j, respectively, acting on variations of the form f}k = 69kg"k 1 and £' : G

---+

T*G is the differential of £ defined as follows. Let gE be a smooth curve in G such that gO

=

9 and (d/dE)IE=ogE

=

v. Then

Proof. Setting the end-point variations in (3.3) to zero immediately recovers the DEL equations (3.2) and, hence, establishes the equivalence of (1) and (2). To see

that (I) is equivalent to (3), notice that since lL = f ⁰ 7fd,

Now for (3) ~ (4), we compute

and find that

:,I,~o

^s(flk+1)

~ ~

^£'(fkk+,)

[09kgk~' ~ 9kgk~logk+1gk~,l

N-l N-l

=

L

£'(Jkk+l)09k9"k19k9"k~1 ^-

L

f'(Jr-lr)9r-19;109r9;1

k=l r=l

N-l

=

L

(£'(Jkk+l)TRikk+l - £'(ik-lk)TLfk_lk) 09k9"k1 k=l

N-l

=

L

(Rjkk+/ (Jkk+l) - L jk-lk

£'

(ik-lk)) {)k k=l

where {)k

==

09k9"k1 and we have used discrete integration by parts and the fact that 090 = 09N

=

O. Using the arbitrariness of {)k, we obtain the discrete Euler-Poincare

equations (3.1O) for all variations of this form.

o

Remark 3.2.1. In the case that lL is left invariant, the discrete Euler-Poincare equations take the form

(3.11)

where fk+lk

==

9"k~19k is in the left quotient (G x G)/G, and the operators act on variations of the form {)k

=

9"k109k.

Structure preservation and reconstruction

Below we introduce the issue of structure preservation by reduced algorithms and postpone a more detailed discussion of it until Chapter 4. The exposition here appeals to general Poisson reduction theory, which can be applied both to continuous and to discrete settings, while Chapter 4 is concerned with a hands-on derivation of the reduced structure using Legendre type transformations as well as a groupoid- algebroid formalism. Both descriptions address the symplecticity of the reduced discrete flow and are, in some sense, complementary. Recall also Remark 3.1.2 regarding the local character of the discrete symplectic and Poisson structures.

We may associate to any C¹ function F on G x G its Hamiltonian vector field XF satisfying XF -.J WIL

=

dF. The symplectic structure WIL naturally defines a Poisson structure {., ·}cxc on G x G by the relation

(3.12)

Theorem 3.2.2. If the action of G on G x G is proper, then the algorithm on G de- fined by the discrete Euler-Poincare equations (3.1 0) preserves the induced Poisson structure {-, ·}c on G given by

{j, h}c ⁰ 7rd = {j ⁰ 7r, h ⁰ 7r}cxc (3.13)

for any C¹ functions f, h: (G x G)jG ~ G -+ JR.

Proof. We saw in the previous section that the DEL algorithm preserves the sym- plectic structure WIL on G x G; hence, by (3.12), the DEL algorithm preserves the Poisson structure on G x G. Since the action of G on G x G is proper, the general Poisson reduction theorem (see, e.g., [40] and discussion in Chapter 2) states that the projection 7rd : G x G -+ G is a Poisson map.

By Theorem 3.2.1, the projection of the DEL algorithm,

is equivalent to the DEP algorithm on G, 1k-1k 1---7 Ikk+1. Therefore, as the Poisson structure on G is induced by 7rd and as 7rd is Poisson, we have proven the theorem.

o

As we shall prove in the following theorem, reconstruction of the DEP algorithm (3.10) on G reproduces the DEL algorithm on G x G.

Theorem 3.2.3. The discrete Euler-Lagrange algorithm governed by ^lL and the dis- crete Euler-Poincare algorithm governed by f are related as follows. The canonical projection of a solution of DEL gives a solution of DEP, while the reconstruction of a solution of the DEP equations results in a solution of the DEL equations.

Proof. The first assertion follows by construction. For the second assertion, using the definition Ikk+1 = 9k9k~1' the DEL algorithm can be reconstructed from DEP algorithm by

(3.14)

where Ikk+1 is the solution of (3.10). Indeed, fkk~l . 9k is precisely 9k+1. Thus, at each increment, one needs only to compute fkk~l ·9k since 9k =

rk"!lk

^{·9k-1 is}

already known.

Similarly, one shows that in the case of a left G action, the reconstruction of the DEP equations (3.11) is given by

(3.15)

o

Let us denote by 7f the quotient map 7f : TG -+ TG / G ~ 9 mapping (9, g) to

99-

^{1 E} g. In the limit as the time step b..t -+ 0, the discrete action sum converges to the action integral, and the DEL algorithm converges to the flow of the EL equations.

We denote the reconstruction of the flow of the Euler-Lagrange equations from the flow of the Euler-Poincare equations by 9tEP. Similarly, we denote the recon-

struction of the DEL algorithm from the DEP algorithm provided by Theorem 3.2.3 by ^'JtDEP. The following non-commutative diagram shows these relations.

Gx G ~o TG DEL ~a EL

r

^!REP

G DEP EP

where G x G -+ TG as tlt -+

°

in the following sense. Locally, G x G = FlL* (T*G) and as tlt -+ 0, FIL -+ FL which pulls-back T*G to TG. Thus, the DEP algorithm approximates the flow of the Euler-Poincare equations if properly interpreted by means of reconstruction.