1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

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

The idea of the energy-momentum method. The setting of the energy-

momentum method is that of phase space P with a symmetry group G , a G-

invariant Hamiltonian H and an associated conserved momentum J, an equivariant

momentum map, J : P -> g*.

A relative equilibrium (also called a steady motion) is a point Ze E P

whose dynamical orbit z (t) is coincident with a one-parameter group orbit: z(t) =

exp(tOze for some ~ E g.

Assumptions. The standard energy-momentum method assumes that the relative

equilibrium Ze is a regular point for J and that the image point μ e = J(ze) is a

generic, i.e., nonspecial point. Patrick [1992], extended this result, allowing μ e to

be nongeneric. A crucial hypothesis in Patrick's result is a compactness condition

on the isotropy subgroup G μ. , the subgroup of elements of G that fix μ e. The result

is that one gets stability modulo G μ •.

The rough idea for the energy momentum method is to first formulate the

problem directly on the unreduced space. Here, relative equilibria associated with

a Lie algebra element ~ are critical points of the augmented Hamiltonian He. :=

H - (J , 0- Next, compute the second variation of He. at a relative equilibria Ze

with momentum value μ e subjec t to the constraint J = μ e and on a space, say E z. ,

transverse to the action of Gμ. (see Figure 2.1).

J-^1 (μ) e =constant momentum surface

Gμ e ¥ z e = orbit of the equilibrium under

the symmetry group of the momentum

Figure 2 .1. The geometry of the energy-momentum method.

This, as a general method, was first given in Marsden Simo, Lewis and Posburgh

[1989] and Marsden and Simo [1990]. To extend the method to the singular case one

replaces the above figure by the corresponding symplectic slice (see Arms, Marsden

and Moncrief [1981], Guillemin and Sternberg [1984] and Ortega and Ratiu [1997b]).

Although the augmented Hamiltonian plays the role of H + C in the Arnold

method, notice that Casimir functions are not required to carry out the calculations.

Block diagonalization. The theory for carrying out this procedure was much
developed in Simo, Posbergh and Marsden [1990, 1991] and Simo, Lewis and Mars-
den [1991] as we shall briefly explain. An exposition of the method may be found,
along with additional references in Marsden [1992]. To obtain the more detailed