Abstract
The existence and uniqueness of solutions to multivalued stochastic differential equations of the second order on Riemannian manifolds are proved. The class of problem is motivated by rigid body and multibody dynamics with friction and an application to the spherical pendulum with friction is presented.
Highlights
In the theory of rigid bodies mechanics, a system is described by the configuration manifold M, which is generally the set of isometries of the three-dimensional space; the movement of the body is a curve on this manifold
If a stochastic term is involved in the forces acting on the system, stochastic differential equations must be used and we refer to Ikeda & Watanabe (1989), Norris (1992) and Emery (1989) for information
In order to obtain the existence of a solution we will use the results of Cepa (1998), which ensure the existence of solutions to the multivalued stochastic differential equations (MSDEs) of the following type: dXt + B(Xt) dt α(Xt) dt + β(Xt) dWt, where B is a maximal monotone operator on E and α and β verify the classical Lipschitz and linear growth conditions
Summary
In the theory of rigid bodies mechanics, a system is described by the configuration manifold M , which is generally the set of isometries of the three-dimensional space; the movement of the body is a curve on this manifold. The difference x1 −x2 cannot be assigned an intrinsic meaning and, the main principle of differential geometry, that is, chart independence, cannot be upheld This objection fails if, instead of working on a manifold M , we work on its tangent space T M , which is the space of classical mechanics, our present setting. Working in a local chart is equivalent to working in an open subset of Rd; in § 3 we extend the coefficients of the equation to all of Rd and we will denote these coefficientsb, ν, σand A. They are only locally Lipschitz continuous with respect to the velocity v. Throughout this article, the Einstein summation convention is used, the time derivative of a map f is denoted by fand the value of f at the point t is denoted indifferently by f (t) or ft
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