1,200 research outputs found
Stochastic Variational Integrators
This paper presents a continuous and discrete Lagrangian theory for
stochastic Hamiltonian systems on manifolds. The main result is to derive
stochastic governing equations for such systems from a critical point of a
stochastic action. Using this result the paper derives Langevin-type equations
for constrained mechanical systems and implements a stochastic analog of
Lagrangian reduction. These are easy consequences of the fact that the
stochastic action is intrinsically defined. Stochastic variational integrators
(SVIs) are developed using a discretized stochastic variational principle. The
paper shows that the discrete flow of an SVI is a.s. symplectic and in the
presence of symmetry a.s. momentum-map preserving. A first-order mean-square
convergent SVI for mechanical systems on Lie groups is introduced. As an
application of the theory, SVIs are exhibited for multiple, randomly forced and
torqued rigid-bodies interacting via a potential.Comment: 21 pages, 8 figure
Ballistic Transport at Uniform Temperature
A paradigm for isothermal, mechanical rectification of stochastic
fluctuations is introduced in this paper. The central idea is to transform
energy injected by random perturbations into rigid-body rotational kinetic
energy. The prototype considered in this paper is a mechanical system
consisting of a set of rigid bodies in interaction through magnetic fields. The
system is stochastically forced by white noise and dissipative through
mechanical friction. The Gibbs-Boltzmann distribution at a specific temperature
defines the unique invariant measure under the flow of this stochastic process
and allows us to define ``the temperature'' of the system. This measure is also
ergodic and weakly mixing. Although the system does not exhibit global directed
motion, it is shown that global ballistic motion is possible (the mean-squared
displacement grows like t squared). More precisely, although work cannot be
extracted from thermal energy by the second law of thermodynamics, it is shown
that ballistic transport from thermal energy is possible. In particular, the
dynamics is characterized by a meta-stable state in which the system exhibits
directed motion over random time scales. This phenomenon is caused by
interaction of three attributes of the system: a non flat (yet bounded)
potential energy landscape, a rigid body effect (coupling translational
momentum and angular momentum through friction) and the degeneracy of the
noise/friction tensor on the momentums (the fact that noise is not applied to
all degrees of freedom).Comment: 33 page
Stochastic Variational Partitioned Runge-Kutta Integrators for Constrained Systems
Stochastic variational integrators for constrained, stochastic mechanical
systems are developed in this paper. The main results of the paper are twofold:
an equivalence is established between a stochastic Hamilton-Pontryagin (HP)
principle in generalized coordinates and constrained coordinates via Lagrange
multipliers, and variational partitioned Runge-Kutta (VPRK) integrators are
extended to this class of systems. Among these integrators are first and
second-order strongly convergent RATTLE-type integrators. We prove order of
accuracy of the methods provided. The paper also reviews the deterministic
treatment of VPRK integrators from the HP viewpoint.Comment: 26 pages, 2 figure
The Motion of the Spherical Pendulum Subjected to a D_n Symmetric Perturbation
The motion of a spherical pendulum is characterized by the fact that all trajectories are relative periodic orbits with respect to its circle group of symmetry (invariance by rotations around the vertical axis). When the rotational symmetry is broken by some mechanical effect, more complicated, possibly chaotic behavior is expected. When, in particular, the symmetry reduces to the dihedral group D_n of symmetries of a regular n-gon, n > 2, the motion itself undergoes dramatic changes even when the amplitude of oscillations is small, which we intend to explain in this paper. Numerical simulations confirm the validity of the theory and show evidence of new interesting effects when the amplitude of the oscillations is larger (symmetric chaos)
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