1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

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

energy. However, this time L is right invariant and the reduced Lagrangian is

defined on the Lie algebra g, t he divergence free vector fields on B , by

l(u) = ~ j llull^2 dμ,

where II · 112 denotes the Riemannian norm on B and dμ is the associated volume

form. Note again l is quadratic, so we get geodesics. The Euler-Poincare equations

are the ideal fluid equations

au

-+V at u U=-\lp

where the pressure p is determined implicitly by the constraint div u = 0. Again

the reader should work out the reduced variational principle and verify that these

equations are of Euler-Poincare form. Relevant references are Arnold [1966a], Ebin

and Marsden [1970], Marsden and Ratiu [1998].

Example 3. The dynamics of a rigid body in a fluid are often modeled by the

classical Kirchhoff equations in which the fluid is assumed to be potential flow,

responding to the motion of the body. (For underwater vehicle dynamics one needs

to include buoyancy effects.)^1 Here we choose G = SE(3), the group of Euclidean

motions of IR?.^3 and the Lagrangia n is the total energy of the body-fluid system.

Recall that the Lie algebra of 80(3) is .se(3) = JR?.^3 x JR?.^3 with the bracket

[(n, u), (I:, v)] = (n x I: , n xv - L; x u).

The reduced Lagrangian is again quadratic, so has the form

l(Sl, v ) = ~nTJn + n T Dv +~VT Mv.

The Euler-Poincare equations are computed to b e

Il=IIxn+Pxv }

F=Pxn

where II= az;an =Jn+ Dv the "angular momentum" and P = az/av = Mv +

DTn, the "linear momentum".

Again, we suggest that the reader work out the reduced variational principle.

Relevant references are Lamb [1932], Leonard [1996], Leonard and Marsden [1997],

and Holmes, Jenkins and Leonard [1997].

Example 4. Following Ovsienko and Khesin [1987], we will now indicate how the

KdV equations may be recast as Euler-Poincare equations. The KdV equation is

the following equation for a scalar function u(x, t) of the real variables x and t:

Ut + 6uux + Uxxx = 0.

We let g be the Lie algebra of vector fields u on the circle (of length 1) with the
standard bracket

[u, v] = u'v - v'u.

(^1) This model may be viewed inside the larger model of an elastic-fluid interacting system with
the constraint of rigidity imposed on the elastic body and with the reduced space for the fluid
variables (potential flow is simply reduction at zero for fluids).