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