Explicit Lie-Poisson Integration

Applications

5.4 Explicit Lie-Poisson Integration

In this section, we introduce a new explicit Lie-Poisson algorithm. Take a Hamiltonian system on the cotangent bundle of some Lie group. Assume that the Hamiltonian is left-invariant under the group action of G on T*G.

We saw from section 1.5 that the co-adjoint orbits on

S*,

when the phase space has been left reduced, is given by the negative Lie-Poisson bracket, i.e., if p E

G*,

then the co-adjoint orbit through p E

G*

will be (Q,,w,) where (3, = {Ad;-,(p)lg E G) and

< .,

. >is the natural pairing between the Lie algebra and its dual and [,]

is the Lie algebra bracket. From section 3.5, we know that the equation of

motion for some p E

E*

and Hamiltonian, H E

Cw(E*)

are given by

The minus arises if the Lie-Poisson system was derived by identification with right invariant vector fields and the positive arises in the left-invariant case. Therefore, for a left invariant Hamiltonian, the equations of motion for p E

E*

is given by

In order t o integrate tlzis system, we have resorted t o the construction of symplectic integrators which are based on the reduced Hamilton-Jacobi equation of Chapter 4. While this approach is applicable t o a very broad class of Lie-Poisson systems, it is only easily realizable in the subcase of regular quadratic Lie algebras, i.e., algebras endowed with an Ad-invariant, bilinear, non-degenerate form. As we have seen, tlzis property enables us t o construct a con-slngu!ar identity transfarmati~n ^=:I

G*

for the momentum preserving generating fullctions of the first kind on T*G. This was accom- plished by the application of the strict generating pair theory of Ge[13].

Armed with such an identity transformation generator, we can then form a kth-order perturbative solution to the Hamilton-Jacobi equation which is implicitly Casimir preserving and Poisson, i.e., the mapping on

G*

is sym- plectic with respect to the KAMS symplectic form on co-adjoint orbits.

We claim that it is possible to build an explicit Lie-Poisson integrator for precisely these Lie algebras without reverting to a Ruth[l4] type of approx- imation to the Hamiltonian. If

6

is regular quadratic, then it is possible t o

form a diffeomorphism between co-adjoint and adjoint orbits. For brevity, we will present a result proven in [I] for a left-invariant Poisson system.

Proposition 5.4.1 Let H : T*G ⁱ R be a left-invariant Hamiltonian, i.e.,

N

o T * L , =

N

for all g E G. O n (3, we defined the symplectic KAKS 2-form with the negative sign having been selected. If there ezists a bi- invariant 2-form (., .) o n

G,

then the adjoint orbit through

t

^E

G

given by (3( = {Ad,tlg E G) has symplectic 2-form

for ( anzd q i n

6.

As we will be using the Cartan-Killing form in our examples, let us denote (., .) by I<(., .). I< is non-degenerate, so for every p E

G*,

there exists a unique

b

^E

6

such that

<

p , t

>=

Ii'(fi,t) for all

t.

We now show that as a consequence of K being Ad-invariant, the adjoint action is skew-symmetric with respect to it.

Consider

I<(adtC, 7 ) = -lt=oI~(Adexpt[C, 7) = d dt

With this information, we return to the original equation of motion on

G*

Since is the derivative of an element of a linear space, it can be also considered an element of that space. Thus, there exists some unique

fi

in

6

such that

< b,

'I

>= ~ ( j i ,

^rl).

Also, setting

t

^{= E O ,} we deduce

-

<

p, adeq

>= li(F,

adtq) =

-

K(ad@, 7 ) . (5.21)

Therefore by the Ad-invariance of K , the system reduces t o

We can now approximate this flow by discretizing its equation of motion.

If we choose as our initial position some element of the Lie algebra, po,then for time-step, T , the time advanced element p1 will be given by

where h = is the first derivative of the Hamiltonian evaluated at the starting point. Each iteration may then be executed by replacing po by p~

and iterating through time.

The trick which makes this algorithm pratical without any neccessity t o separate the Hamiltonian into simpler pairings, is found in a formula which we have already encountered in the last chapter and proven in [24]. It can be shown that for

t

in the neigborhood of the origin in

G ,

where, just as in the final sections of the last chapter, ad< is simply a linear operator of a linear representation of (7 and thus the right-hand side of equation[1.23] can be interpreted as a power series in the operator.

The above time-stepping obviously preserves the Casimirs by definition.

However, we sliould check t o make sure that the KAKS symplectic 2-form on the adjoint leaves is preserved.

Proposition 5.4.2 Ad,-1 : O5 -i C3t preserves the symplectic 2-form, wo, i.e., (Ad,-l)*wo = wo.

To prove this, consider

By definition this equals

~O(A~,-~$)(TGA~,-~E‘B($),

T$Ads-l ?B($)). (5.26) NDW, in genera! (f )*Y ( 3 ) = (T f)-lY( f (z))? so T$.Ad,-l

t g ( $ )

= (Ad,)*JG(Ad,-I$).

So, equation(5.24) equals

where

4

= Ady-I$. We next use the fact that (Ad,)*& = ( A d , - ~ t ) ~ , t o obtain

w0(4)((Ady-1t>8(4), (Adg-l?)~(d))). (5.28) By the definition of the KAIIS bracket on adjoint orbits via the Cartan- Killing form, we find that equation(5.27) becomes

- I{(+,

[& ~ 1 ) ~

(5.29) by Ad-invariance of the Cartan-Icilling form. The transformed 2-form (5.24) now equals

WO(@>(&(+>, ??G(?It))- (5.30) Therefore, the Ad mapping preserves the KAKS 2-form.

This algorithm will be implemented in the next section for the case of the rigid body dynamics and compared to the results obtained via the implicit Hamilton- Jacobi integrator.