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We consider the problem of stabilizing what we call a pendulum skate, a simple model of a figure skater developed by Gzenda and Putkaradze. By exploiting the symmetry of the system as well as taking care of the part of the symmetry broken by the gravity, the equations of motion are given as nonholonomic Euler–Poincaré equation with advected parameters. Our main interest is the stability of the sliding and spinning equilibria of the system. We show that the former is unstable and the latter is stable only under certain conditions. We use the method of Controlled Lagrangians to find a control to stabilize the sliding equilibrium. 

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 \begin{document}$ \mathfrak{so}(2,1)^{*} $\end{document} and \begin{document}$ (\mathfrak{se}(3) \ltimes \mathbb{R}^{3})^{*} $\end{document}, respectively.

 

The article considers smooth optimization of functions on Lie groups. By generalizing NAG variational principle in vector space (Wibisono et al., 2016) to general Lie groups, continuous Lie-NAG dynamics which are guaranteed to converge to local optimum are obtained. They correspond to momentum versions of gradient flow on Lie groups. A particular case of SO(𝑛) is then studied in details, with objective functions corresponding to leading Generalized EigenValue problems: the Lie-NAG dynamics are first made explicit in coordinates, and then discretized in structure preserving fashions, resulting in optimization algorithms with faithful energy behavior (due to conformal symplecticity) and exactly remaining on the Lie group. Stochastic gradient versions are also investigated. Numerical experiments on both synthetic data and practical problem (LDA for MNIST) demonstrate the effectiveness of the proposed methods as optimization algorithms (\emph{not} as a classification method).