316 research outputs found
Well-posed infinite horizon variational problems on a compact manifold
We give an effective sufficient condition for a variational problem with
infinite horizon on a compact Riemannian manifold M to admit a smooth optimal
synthesis, i. e. a smooth dynamical system on M whose positive
semi-trajectories are solutions to the problem. To realize the synthesis we
construct a well-projected to M invariant Lagrange submanifold of the
extremals' flow in the cotangent bundle T*M. The construction uses the
curvature of the flow in the cotangent bundle and some ideas of hyperbolic
dynamics
Rolling balls and Octonions
In this semi-expository paper we disclose hidden symmetries of a classical
nonholonomic kinematic model and try to explain geometric meaning of basic
invariants of vector distributions
Invariant Lagrange submanifolds of dissipative systems
We study solutions of modified Hamilton-Jacobi equations H(du/dq,q) + cu(q) =
0, q \in M, on a compact manifold M
Nonholonomic tangent spaces: intrinsic construction and rigid dimensions
A nonholonomic space is a smooth manifold equipped with a bracket generating family of vector fields. Its infinitesimal version is a homogeneous space of a nilpotent Lie group endowed with a dilation which measures the anisotropy of the space. We give an intrinsic construction of these infinitesimal objects and classify all rigid (i.e. not deformable) cases
On the Hausdorff volume in sub-Riemannian geometry
For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative
of the spherical Hausdorff measure with respect to a smooth volume. We prove
that this is the volume of the unit ball in the nilpotent approximation and it
is always a continuous function. We then prove that up to dimension 4 it is
smooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4
on every smooth curve) but in general not C^5. These results answer to a
question addressed by Montgomery about the relation between two intrinsic
volumes that can be defined in a sub-Riemannian manifold, namely the Popp and
the Hausdorff volume. If the nilpotent approximation depends on the point (that
may happen starting from dimension 5), then they are not proportional, in
general.Comment: Accepted on Calculus and Variations and PD
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