High-order symplectic Lie group methods on \(SO(n)\) using the polar decomposition
The authors construct a variational integrator of arbitrarily high-order on the special orthogonal group \(\mathrm{SO}(n)\) using the polar decomposition and the constrained Galerkin method. \(\mathrm{SO}(n)\) is embedded naturally in the space \(\mathrm{GL}_{+}(n)=\{A\in \mathbb{R}^{n\times n} | \mathrm{det}(A) > 0\}\), an open subset of \(\mathbb{R}^{n\times n}\). It has the advantage of avoiding the second order derivative of the exponential map that arises in traditional Lie group variational methods. The Lie algebra of \(\mathrm{SO}(n)\) is the set Skew(\(n\)) \(= \{\Omega \in \mathbb{R}^{n\times n} | \Omega^{T} = -\Omega\}\), with the matrix commutator as the Lie bracket. The authors introduce the polar decomposition and the retraction on \(\mathrm{SO}(n)\). The polar decomposition of \(A\) is provided according to the following: Given \(A\in \mathrm{GL}(n)\), this decomposes uniquely as \(A\rightarrow UP\), \(U\in \mathrm{O}(n)\), \(P\in \mathrm{Sym}^{+}(n)\), where \(\mathrm{Sym}^{+}(n)\) is the set of \(n \times n\) symmetric positive-definite matrices. The authors also mention the Lagrangian variational integrators on the rotation group \(\mathrm{SO}(n)\) in details. For a system with rotational symmetry, the authors obtain a simpler integrator using Lie-Poisson reduction on the Hamiltonian side. Further, they construct a reduced Lie-Poisson integrator and the resulting algorithms can naturally be implemented by fixed-point iteration. The authors validate the proposed methods by numerical simulations on \(\mathrm{SO}(3)\) showing they are comparable to variational Runge-Kutta-Munthe-Kaas methods in terms of computational efficiency. They provide the following methods for the comparison tests: second-order Gauss-Legendre method, third-order Runge-Kutta method, fourth-order Gauss-Legendre method and sixth-order Gauss-Legendre method. However, these proposed methods preserve the Lie group structure much more accurately and exhibit better near energy preservation. At the end, the authors also give the long-term energy behaviors for their methods.
scientific article; zbMATH DE number 7613237