3,641 research outputs found
On the multiplicity of non-iterated periodic billiard trajectories
We introduce the iteration theory for periodic billiard trajectories in a
compact and convex domain of the Euclidean space, and we apply it to establish
a multiplicity result for non-iterated trajectories.Comment: 21 pages, 2 figures; v3: final version, as publishe
Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions
We prove existence and uniqueness of solutions to a nonlinear stochastic
evolution equation on the -dimensional torus with singular -Laplace-type
or total variation flow-type drift with general sublinear doubling
nonlinearities and Gaussian gradient Stratonovich noise with divergence-free
coefficients. Assuming a weak defective commutator bound and a
curvature-dimension condition, the well-posedness result is obtained in a
stochastic variational inequality setup by using resolvent and Dirichlet form
methods and an approximative It\^{o}-formula.Comment: 26 pages, 58 references. Essential changes to Version 4: Examples
revised. Accepted for publication in Stochastic Processes and their
Application
The type numbers of closed geodesics
A short survey on the type numbers of closed geodesics, on applications of
the Morse theory to proving the existence of closed geodesics and on the recent
progress in applying variational methods to the periodic problem for Finsler
and magnetic geodesicsComment: 29 pages, an appendix to the Russian translation of "The calculus of
variations in the large" by M. Mors
Critical manifolds and stability in Hamiltonian systems with non-holonomic constraints
We explore a particular approach to the analysis of dynamical and geometrical
properties of autonomous, Pfaffian non-holonomic systems in classical
mechanics. The method is based on the construction of a certain auxiliary
constrained Hamiltonian system, which comprises the non-holonomic mechanical
system as a dynamical subsystem on an invariant manifold. The embedding system
possesses a completely natural structure in the context of symplectic geometry,
and using it in order to understand properties of the subsystem has compelling
advantages. We discuss generic geometric and topological properties of the
critical sets of both embedding and physical system, using Conley-Zehnder
theory and by relating the Morse-Witten complexes of the 'free' and constrained
system to one another. Furthermore, we give a qualitative discussion of the
stability of motion in the vicinity of the critical set. We point out key
relations to sub-Riemannian geometry, and a potential computational
application.Comment: LaTeX, 52 pages. Sections 2 and 3 improved, Section 5 adde
On a Gromoll-Meyer type theorem in globally hyperbolic stationary spacetimes
Following the lines of the celebrated Riemannian result of Gromoll and Meyer,
we use infinite dimensional equivariant Morse theory to establish the existence
of infinitely many geometrically distinct closed geodesics in a class of
globally hyperbolic stationary Lorentzian manifolds.Comment: 39 pages, LaTeX2e, amsar
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