We construct a symplectic integrator for non-separable Hamiltonian systems combining an extended phase space approach of Pihajoki and the symmetric projection method. The resulting method is semiexplicit in the sense that the main time evolution step is explicit whereas the symmetric projection step is implicit. The symmetric projection binds potentially diverging copies of solutions, thereby remedying the main drawback of the extended phase space approach. Moreover, our semiexplicit method is symplectic in the original phase space. This is in contrast to existing extended phase space integrators, which are symplectic only in the extended phase space. We demonstrate that our method exhibits an excellent long-time preservation of invariants, and also that it tends to be as fast as Tao's explicit modified extended phase space integrator particularly with higher-order implementations and for higher-dimensional problems.

We construct a symplectic integrator for non-separable Hamiltonian systems combining an extended phase space approach of Pihajoki and the symmetric projection method. The resulting method is semiexplicit in the sense that the main time evolution step is explicit whereas the symmetric projection step is implicit. The symmetric projection binds potentially diverging copies of solutions, thereby remedying the main drawback of the extended phase space approach. Moreover, our semiexplicit method is symplectic in the original phase space. This is in contrast to existing extended phase space integrators, which are symplectic only in the extended phase space. We demonstrate that our method exhibits an excellent long-time preservation of invariants, and also that it tends to be as fast as and can be faster than Tao’s explicit modified extended phase space integrator particularly for small enough time steps and with higher-order implementations and for higher-dimensional problems.

The purpose of this study is to understand the force and energy balance in the wobble plate drive mechanism referred to as the virtual ellipse drive (VED). The moving affine constraint presented by the precessing wobble plate's contact with the stator and rotor plates offers a subtlety that is difficult to account for in a standard classical mechanics approach. This subtlety has made it difficult to account for energy conservation in simulations, even when including vibration and friction.

<p style='text-indent:20px;'>We propose a systematic procedure called the Clebsch canonization for obtaining a canonical Hamiltonian system that is related to a given Lie–Poisson equation via a momentum map. We describe both coordinate and geometric versions of the procedure, the latter apparently for the first time. We also find another momentum map so that the pair of momentum maps constitute a dual pair under a certain condition. The dual pair gives a concrete realization of what is commonly referred to as collectivization of Lie–Poisson systems. It also implies that solving the canonized system by symplectic Runge–Kutta methods yields so-called collective Lie–Poisson integrators that preserve the coadjoint orbits and hence the Casimirs exactly. We give a couple of examples, including the Kida vortex and the heavy top on a movable base with controls, which are Lie–Poisson systems on <inline-formula><tex-math id="M1">\begin{document}$ \mathfrak{so}(2,1)^{*} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ (\mathfrak{se}(3) \ltimes \mathbb{R}^{3})^{*} $\end{document}</tex-math></inline-formula>, respectively.</p>

We propose a systematic procedure called the Clebsch anti-reduction for obtaining a canonical Hamiltonian system that is related to a given Lie-Poisson equation via a Poisson map. We describe both coordinate and geometric versions of the procedure, the latter apparently for the first time. The anti-reduction procedure gives rise to a collection of basic Lie algebra questions and leads to classes of invariants of the obtained canonical Hamiltonian system. An important product of anti-reduction is a means for numerically integrating Lie-Poisson systems while preserving their invariants, by utilizing symplectic integrators on the anti-reduced system. We give several numerical examples, including the Kida vortex, a model system for the rattleback toy, and the heavy top on a movable base with controls.