1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

340 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY

bundle TG. Let L : TG -> IR be a G-invariant Lagrangian. Being invariant, L

is completely determined by its restriction to the tangent space at the identity e.

Identifying T eG, the tangent space to G at e with the Lie algebra g of G, we let

l : g -> IR be defined by l = LITeG. Alternatively, we may identify the quotient

space TG/G with g and l is the map induced by Lon the quotient space.

The velocity of the system is given by g( t), thought of as a tangent vector to

G at g(t). The body velocity is defined by ~(t) = g(t)-^1 g(t), the left translation

of g to the identity.

Theorem 1.1. (Euler-Poincare reduction-part 1.) A curve g(t) in G satisfies

the Euler-Lagrange equations for L iff ~(t) satisfies the Euler-Poincare equa-

tions for l:

d al *al

dt a~= ad~ a(

Let me explain the r10tation: al I a~ means the differential of l, so it is an

element of g* (and it is understood that al/a~ is evaluated at the point ~(t)). The

ad map is defined by Lie algebra bracketing:

ad~ : g-> g; 'TJ ~ [~, TJ]

and ad~ : g* -> g* means its dual map.

There are many ways to prove this theorem, but the technique of reduction of

variational principles is very efficient as well as carrying an important message. It

proceeds by asking if we can write Hamilton's principle entirely in terms of l and

~- By left invariance,

L(g, g) = 1(0,

so this much is easy. Variations are a little trickier but to help matters, we shall
assume G is a matrix group. Take variations of the reconstruction equation (so
called because it allows one to determine g from ~):

~ = g-19

to give

As in the calculus of variations, a variation such as 8~ or 8g is nothing more than
the derivative of a parameterized family of curves with fixed endpoints, with respect
to this parameter. Let

( = g-18g;

i.e., ( is the variation of g left translated to the identity. We refer to ( as the body

representation of the variation.

Since the time derivative of g-^1 is -g-^1 gg-^1 , we get

( = -g-lgg-18g + g-1(8g) ·'

so that

8~ = -(~ + ~( + ( = ( + [~, (].

Thus, variations of~ induced by variations of g must have this form for some curve
( E g vanishing at the endpoints. This calculation shows t hat