4 LAGRANGIAN REDUCTION BY STAGES
of doing Poisson reduction by stages, but keeping track of the local structure of
Poisson manifolds, as in the Lie-Weinstein theorem (see Weinstein [1983a]).
The theory we establish may be viewed as the Lagrangian analogue of the bun-
dle picture on the Hamiltonian side developed by Montgomery [1986] and Mont-
gomery, Marsden and Ratiu [1984]. This bundle picture was in turn influenced by
work on Wong's equations for a particle in a Yang-Mills field, as studied by Stern-
berg [1977], Weinstein [1978], Montgomery [1984], and Koon and Marsden [1997a].
As we shall see in §3.3, this theory has a very beautiful Lagrangian analogue. In
§6 we give a number of additional examples. In future works we plan to establish
additional links with the Hamiltonian side.
1.3. Future Work and Related Issues.
Our theory naturally suggests a number of additional things that warrant fur-
ther investigation.
Geometric Phases. The development of the theory of geometric phases in the
Lagrangian context is natural to develop given the relatively large amount of activ-
ity from the symplectic and Poisson point of view (see, eg, Marsden, Montgomery
and Ratiu [1990], Marsden [1992], Blaom [2000] and references therein).
The paper of Marsden, Ratiu and Scheurle [2000] gives geometric phase for-
mulas in the context of Routh reduction. The development of geometric phases by
stages would be of interest. In fact, the Lagrangian setting provides natural connec-
tions and also a natural setting for averaging which is one of the basic ingredients
in geometric phases.
Nonholonomic Mechanics. As mentioned above, the work of Cendra, Mars-
den and Ratiu [2000] extends the notion of Lagrange-Poincare bundles to those of
Lagrange-d'Alembert-Poincare bundles that are appropriate for nonholonomic me-
chanics. This extension may be regarded as the Lagrangian analogue of the notion
of an almost Poisson manifold (in which Jacobi's identity can fail), as in Koon and
Marsden [1998] and Cannas da Silva and Weinstein [1999]. Furthering the links
with almost Poisson manifolds and also developing a nonholonomic reduction by
stages theory would of course be of interest.
Further Relations with the Hamiltonian Side. It would also be significant to
investigate the precise relationship of the work here with the Hamiltonian reduction
by stages theory in more detail, in particular, the relation with symplectic reduc-
tion by stages applied to cotangent bundles. This requires the nonabelian Routh
reduction analogue of the work here, namely that of Marsden, Ratiu and Scheurle
[2000], which extends the work of Marsden and Scheurle [1993a] and Jalnapurkar
and Marsden [2000].
Relations with Poisson Geometry. The Lie-Weinstein theorem states that a
Poisson manifold is locally the product of a symplectic manifold and the dual of a
Lie algebra. This paper develops a Lagrangian category that locally looks like the
dual of this local structure for Poisson manifolds. Our bundles actually have more
structure than this, which is important for carrying out the reduction, namely we
also carry along a connection and a two-form that helps keep track of curvature (or
magnetic terms). This structure is also very useful for writing covariant versions of
the reduced equations, that is, the Lagrange-Poincare equations.
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