We consider Riemannian manifolds endowed with a nonholonomic distribution. These structures model mechanical systems with a (positive definite) quadratic Lagrangian and nonholonomic constraints linear in velocities. We classify the left-invariant nonholonomic Riemannian structures on threedimensional simply connected Lie groups, and describe the equivalence classes in terms of some basic isometric invariants. The classification naturally splits into two cases. In the first case, it reduces to a classification of left-invariant sub-Riemannian structures. In the second case, we find a canonical frame with which to directly compare equivalence classes.

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The notion of an isometric immersion is extended to nonholonomic Riemannian geometry. Geodesically invariant distributions (i.e., distributions invariant under the geodesic flow) are characterized. A link between geodesic invariance and the curvature of nonholonomic Riemannian structures is established.

We consider equivalence, stability and integration of quadratic Hamilton-Poisson systems on the semi-Euclidean Lie-Poisson space se(1, 1) * − . The inhomogeneous positive semidefinite systems are classified (up to affine isomorphism); there are 16 normal forms. For each normal form, we compute the symmetry group and determine the Lyapunov stability nature of the equilibria. Explicit expressions for the integral curves of a subclass of the systems are found. Finally, we identify several basic invariants of quadratic Hamilton-Poisson systems.

In this paper we consider quadratic Hamilton-Poisson systems on the semi-Euclidean Lie-Poisson space se(1, 1) * − . The homogeneous positive semidefinite systems are classified; there are exactly six equivalence classes. In each case, the stability nature of the equilibrium states is determined. Explicit expressions for the integral curves are found. A characterization of the equivalence classes, in terms of the equilibria, is identified. Finally, the relation of this work to optimal control is briefly discussed.