HERNAN CENDRA, JERROL D E. MARSDEN AND TUDO R S. RATIU 3

the Lagrangian analogue of reduction for central extensions and, as in the case of

symplectic reduction by stages, cocycles and curvatures enter in this context in a

natural way.

Nonholonomic Mechanics. The ideas of geometric mechanics and Lagrangian

reduction have had a significant impact on the theory of nonholonomic systems

(such as mechanical systems with rolling constraints), as in Bloch, Krishnaprasad,

Marsden and Murray [1996] and Koon and Marsden [1997b, c, 1998], Marsden,

Ratiu, and Weinstein [1998, 1994] Bloch and Crouch [1999] and Lewis [1996, 2000].

These references also develop Lagrangian reduction methods in the context of non-

holonomic mechanics with symmetry. These methods have been quite useful in

many control problems and in robotics. The techniques of the present paper can

be used to give an intrinsic geometric meaning to the reduction of the Lagrange-

d'Alembert equations of nonholonomic mechanics. This is the subject of the work

Cendra, Marsden and Ratiu [2000].

Control Theory. Geometric mechanics and Lagrangian reduction theory has also

had a significant impact on control theory, including stabilization (as in Bloch,

Leonard and Marsden [2000] and Jalnapurkar and Marsden [1999] as well as on op-

timal control theory; see Vershik and Gershkovich [1988, 1994], Bloch and Crouch

[1993, 1994, 1995] Montgomery [1990, 1993], Koon and Marsden [1997a] and refer-

ences therein.

1.2. The Main Results of This Paper.

We now give a few more details concerning the main results. The first of these,

given in §5, develops the theory of Lagrange-Poincare bundles, which enable one

to perform Lagrangian reduction in stages. Lagrange-Poincare bundles may be

regarded as the Lagrangian analogue of a Poisson manifold in symplectic geometry.

Lagrange-Poincare bundles include, of course, the case of reduced tangent bundles

(TQ)/G in which we take the quotient of the tangent bundle of the configuration

space Q by the action of a Lie group G on Q. This in turn, includes important

examples such as Euler-Poincare reduction for the special case Q = (7, a Lie group,

in which case, (TQ)/G — 0, the Lie algebra of G. Euler-Poincare reduction is now

a textbook topic that can be found in Marsden and Ratiu [1999]. We show that

when a general tangent bundle is reduced by a group action, one ends up in the

category of Lagrange-Poincare bundles.

We mention that Lagrange-Poincare bundles are in particular, Lie algebroids

but carry additional structure. We will not use any theory of groupoids or algebroids

in this work, but we will comment on part of the literature in the body of the work.

The Lagrange-Poincare equations are expressed using connections and curva-

ture. These equations are obtained using the idea of reducing variational principles.

The Lagrange-Poincare category is stable under reduction and the structure carried

by Lagrange-Poincare bundles is exactly what is needed to write the Lagrange-

Poincare equations in a covariant form.

In §5.3 and §5.4 we show that if the symmetry group has a normal subgroup

(i.e., one has a group extension), then reducing by the whole group is shown to be

isomorphic to what one gets by reducing in stages, first by the normal subgroup

followed by reduction by the quotient group. This result is a Lagrangian analogue