3,641 research outputs found

    On the multiplicity of non-iterated periodic billiard trajectories

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    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

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    We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the dd-dimensional torus with singular pp-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

    Critical manifolds and stability in Hamiltonian systems with non-holonomic constraints

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    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

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    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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