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In this paper, structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton–Pontryagin variational principle. From this principle, one can derive a novel class of variational partitioned Runge– Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and(More)
This note is to point out an error in the theory part of the publication [1]. We will follow the notations and definitions of [1] unless stated otherwise. Contrary to what is claimed in Section 2.2 of [1], the modified Metropolis-Hastings acceptance criterion (eqn. (6) in [1]) does not satisfy a modified detailed balance condition for the choice of a linear(More)
This paper presents a Lie–Trotter splitting for inertial Langevin equations (geometric Langevin algorithm) and analyzes its long-time statistical properties. The splitting is defined as a composition of a variational integrator with an Ornstein–Uhlenbeck flow. Assuming that the exact solution and the splitting are geometrically ergodic, the paper proves the(More)
By treating tippe top inversion as a dissipation-induced instability, we explain tippe top inversion through a system we call the modified Maxwell–Bloch equations. We revisit previous work done on this problem and follow Or's mathematical model [SIAM J. A linear analysis of the equations of motion reveals that only the equilibrium points correspond to the(More)
In this paper, we numerically examine the stability of a standing cantilever conveying fluid in a multiparameter space. Based on nonlinear beam theory, our mathematical model turns out to be replete with exciting behavior, some of which was totally unexpected and novel, and some of which confirm our intuition as well as the work of others. The numerical(More)
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(More)
Keywords: Molecular dynamics Metropolis–Hastings Verlet RATTLE RESPA a b s t r a c t This paper extends the results in [8] to stochastic differential equations (SDEs) arising in molecular dynamics. It implements a patch to explicit integrators that consists of a Metropolis–Hastings step. The 'patched integrator' preserves the SDE's equilibrium distribution(More)
We present explicit methods for simulating diffusions whose generator is self-adjoint with respect to a known (but possibly not normalizable) density. These methods exploit this property and combine an optimized Runge-Kutta algorithm with a Metropolis-Hastings Monte-Carlo scheme. The resulting numerical integration scheme is shown to be weakly accurate at(More)
This paper demontrates that the conditions for the existence of a dissipation-induced heteroclinic orbit between the inverted and nonin-verted states of a tippe top are determined by a complex version of the equations for a simple harmonic oscillator: the modified Maxwell– Bloch equations. A standard linear analysis reveals that the modified Maxwell–Bloch(More)
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