High-order symplectic Lie group methods on $SO(n)$ using the polar decomposition

Xuefeng Shen; Khoa Tran; Melvin Leok

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arXiv · 2201.10768

High-order symplectic Lie group methods on $SO(n)$ using the polar decomposition

Xuefeng Shen·Khoa Tran·Melvin Leok

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Abstract

A variational integrator of arbitrarily high-order on the special orthogonal group $SO(n)$ is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second-order derivative of the exponential map that arises in traditional Lie group variational methods. In addition, a reduced Lie--Poisson integrator is constructed and the resulting algorithms can naturally be implemented by fixed-point iteration. The proposed methods are validated by numerical simulations on $SO(3)$ which demonstrate that they are comparable to variational Runge--Kutta--Munthe-Kaas methods in terms of computational efficiency. However, the methods we have proposed preserve the Lie group structure much more accurately and and exhibit better near energy preservation.

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Xuefeng Shen, Khoa Tran, Melvin Leok. 2022-01-26. High-order symplectic Lie group methods on $SO(n)$ using the polar decomposition. https://arxiv.org/abs/2201.10768

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