1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 1. REDUCTION THEORY 341

Theorem 1.2. (Euler-Poincare reduction-part 2.) Hamilton's principle for

L on G is equivalent to the reduced variational principle:

86red = 0 J l(~(t))dt = 0

for variations of the form

8~ = ( + [~, (]

for some curve ( in g vanishing at the endpoints.

Here, 6red is the reduced action, the integral of the reduced Lagrangian l

along curves in g. Now it is easy to work out the corresponding equations as one

does in the calculus of variations and this yields a proof of part 1 of the Euler-

Poincare reduction.

History and literature. Looking back with modern notation, it is fair to say

that Lagrange [1788] realized the importance of relating the dynamics on TG to

the dynamics on g and devoted much of Volume II of Mecanique Analytique to it

for the case of the rotation group 80(3). The Euler-Poincare equations for gen-

eral Lie algebras were first written down by Poincare [190lb] who realized that

they were fundamental equations in fluid and solid mechanics, as is apparent from

Poincare [1910]. However, it seems that Poincare did not take a variational point of

view. Arnold [1966a] developed the geometry and mechanics of the Euler-Poincare

equations, including stability theory. In fluid mechanics, the constraints on the

variations appearing in reduced variational principles go under the name "Lin con-

straints" and this subject has a large and complex literature. The general theorem

presented here is due to Marsden and Scheurle [1993b] and, for general Lie groups,

to Bloch, Krishnaprasad, Marsden and Ratiu [1996]. The Euler-Poincare equations

in a general context including advected parameters are developed in Holm, Mars-

den and Ratiu [1998a], linked to semidirect product reduction theory and applied

to continuum mechanics.

Example 1. The first example is the rigid body, free to spin about its center of

mass. Here we take G = 80(3) so that g ~ JR.^3 with Lie algebra bracket given

by the cross product. The Lagrangian L is the total kinetic energy. The reduced

Lagrangian l : JR.^3 _, JR. is a quadratic function of~, which in this case is called the

body angular velocity and is written D. Thus,

l(D) = ~ (D, ID)

for a symmetric positive definite matrix I , the moment of inertia tensor. The

Euler-Poincare equations (or the rigid body equations) are readily seen to be

ID= (ID) x D.

The reader may, as an exercise, check directly that these equations come from
the reduced variational principle. Since l is quadratic, L is the kinetic energy of
a Riemannian metric on 80(3), so the Euler-Lagrange equations on 80(3) are the
geodesic equations.

Example 2. For the motion of an ideal fluid, we choose G = Diffv 0 1(B), the

group of volume preserving diffeomorphisms of a given domain B , a Riemannian
manifold in which the fluid moves. The Lagrangian L is the total fluid kinetic