High-order symplectic Lie group methods on <inline-formula><tex-math id="M1">$ SO(n) $</tex-math></inline-formula> using the polar decomposition

A variational integrator of arbitrarily high-order on the special orthogonal group \begin{document}$ SO(n) $\end{document} is constructed using the polar decomposition and the constrained Galerkin method. It has the advantage of avoiding the second order derivative of the exponential map that arises in traditional Lie group variational methods. In addition, a reduced Lie–Poisson integrator is constructed and the resulting algorithms can naturally be implemented by fixed-point iteration. The proposed methods are validated by numerical simulations on \begin{document}$ SO(3) $\end{document} 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.