378 J.E. MARSDEN, MECHANICS, DYNAMICS, AND SYMMETRY
structure Q -> R is really a "red herring". The notion of curvature as a TqQ/Vq
valued form makes good sense and is given locally by the same expressions as above.
However, keeping in mind that we eventually want to deal with symmetries and in
that case there is a natural bundle, the Ehresmann assumption is nevertheless a
reasonable bridge to the more interesting case with symmetries.
More on the Euler-Poincare equations. There is a way to carry out reduc-
tion for a nonholonomic system. We shall focus on the Lagrangian side; for the
equivalent Poisson picture, see Koon and Marsden [1998b,c].
The idea is to pass the Lagrange d'Alembert principle to V/G; in a way that
is similar to the reduction of Hamilton's principle (e.g., giving the Routhian, etc.),
as mentioned in the first lecture. Recall that a simple example of Lagrangian
reduction is the free rigid body; the Euler equations, namely ID = JD, x n, are not
variational, but they satisfy a Lagrange d 'Alembert type of principle (i.e., there
are constraints on the allowed variations) that is obtained obtained by reducing
Hamilton's principle from 80(3).
To enhance the discussion in the first lecture on the Euler-Poincare equations,
we now present an extension of them, following Holm, Marsden and Ratiu [1998a].
In fact, this provides a Lagrangian version of the semidirect product reduction
theory discussed in the last lecture. Following this, we return to the question of
how these procedures work for nonholonomic systems.
The basic ingredients we start with are as follows. Assume there is a left
representation of Lie group G on the vector space V; then G acts on TC x V
as well. Assume L : ·re x V , JR is left G-invariant. For a 0 c V *, define
La 0 : TC , JR by
and define l : g x V* -> JR by
l(g-^1 v 9 , g -^1 a) = L(v 9 , a).
For a curve g(t) E G, let
.;(t) := g(t)-^1 g(t)and define the curve a(t) as the unique solution of the equation
a(t) = -((t)a(t)
with initial condition a(O) = a 0 ; i. e., a(t) = g(t)-^1 a 0.
Theorem 3.1. The following assertions are equivalent:
- With ao fixed, the standard Hamilton principle holds:
8jt
2
L a 0 (g(t),g(t))dt = 0
t1
for variations with fixed endpoints.- The curve g(t) satisfies the standard Euler-Lagrange equations for Lao.
- The Lagrange d'Alembert-type principle
8 jt
2l(((t), a(t))dt = 0
t1
holds on g, using variations of ( and a of the form