THE GEOMETRY OF MIXED FIRST AND SECOND-ORDER DIFFERENTIAL EQUATIONS
WITH APPLICATIONS TO NON-HOLONOMIC MECHANICS

W Sarlet

Key words: Mixed first and second-order equations,
        connections, symmetries, non-holonomic mechanics.

1991 Math. Subject Classification: 58F35, 70F25.

Abstract: A geometrical framework is presented for a class of
dynamical systems, which are modelled by a mixed system of first and
second-order ordinary differential equations.
The starting point for the model is a bundle $\pi:E\rightarrow M$, where
both $E$ and $M$ are fibred over $\R$. We show that there are two
connections associated to systems of this kind: the first one comes from
the `constraint equations' (the first-order equations); the second one is
related to the full system on the constraint manifold. Among other
things, we discuss the concepts of symmetry and adjoint symmetry for such
systems and identify for that purpose an appropriate notion of `dynamical
covariant derivative' and `Jacobi endomorphism'.
The curvature tensors of the two connections play an important
role in defining these concepts. The present approach in particular is
relevant for modelling certain mechanical systems with non-holonomic
constraints.