1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

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

g

pO 3

center of buoyancy

center of gravity

~

Figure 2.4. A rising, spinnin g vehicle.

momentum map J : P --) g*. Vve also have a G-invariant Hamiltonia n H that de-

scribes t he dynamics of interest. The associated Hamiltonian vector field is denoted

XH and the dynamical equations of motion are i = XH(z ). Noether 's theorem of

co urse states that J is a constant of t he motion.

Relative equilibria and energy-momentum method.

Consider a relative equilibrium Ze E P: t here is a Lie algebra element ~e E g

such that exp(~et)ze is a dynamic orbit. Let μe = J(ze). Assume t hat ze is a regular

point of J (equivalent to no infiinitesmimal symmetries). We need not assume that

μe is a generic point. As mentioned earlier t he augmented energy Ht.e = H - (J, ~e)

has a crit ical point at Ze· Next, calculate the second derivative 82 Ht.e (ze) of Ht,

at Ze· Note that ker DJ(ze) is the tangent space to the level set J(z) = μ e at the

point Ze. The tangent space to the Gμe-orbit of Ze at Zeis given by the vector sp ace

consisting of infinitesimal generators ~p(ze) as~ ranges over 9μe' t he Lie algebra of

Gμe·

Now choose a vector subspace E 2 e C kerDJ(ze) t hat complements t he tangent

space to the Gμe-orbit of Ze; i.e., kerDJ(ze) = E ze EBTzeGμJze), as in Figure 2.1.

Assume that t he second derivative of 82 Ht.e (ze) restricted to E 2 e is definite.

Patrick's theorem.

Assume the restriction of the coadjoint action of G μ 0 on g* is proper and there

is an inner product on g* that is invariant under this action; for example, these

both hold if G μ e is compact (and this in turn holds if G is. compact). Then the

relative equilibrium Ze is stable modulo G μ e.