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Symplectic Integrator on SO(n)
Xuefeng Shen
UCSD
Abstract:
Successfully constructing symplectic variational integrator relies on intrinsic representation of manifold configuration space, since local coordinate chart representation would lead to different flow map, and thus destroying long time stability of symplectic integrator. One way is to embed manifold in Euclidean space, and treat it as constrained mechanics, another, when underlying configuration space is Lie group, we can use left trivialization to get Euclidean representation of tangent space. For Rigid body problems, rotation is SO(3), both ways have been tried, either as unit quaternion, which is an embedded unit surface in R^4, or as orthonormal matrix, which is a Lie group. We realize that unit quaternion is also a Lie group, and construct variational integrator based on this observation. The resulting method is an equivalent version of Rattle's method in unit quaternion setting. We also construct high order symplectic integrator on SO(n) via Polar decomposition, traditional Galerkin variational integrator on Lie group via exponential map involves second order derivative of exponential map, which is too complicated in practice. Here, we construct an algorithm which can be efficiently solved by fixed point iteration.
Tuesday, October 23, 2018
11:00AM AP&M 2402
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Randolph E. Bank
Philip E. Gill
Michael Holst
Administrative Contact:
[ Click to Send Email ]
Xuefeng Shen
UCSD
Abstract:
Successfully constructing symplectic variational integrator relies on intrinsic representation of manifold configuration space, since local coordinate chart representation would lead to different flow map, and thus destroying long time stability of symplectic integrator. One way is to embed manifold in Euclidean space, and treat it as constrained mechanics, another, when underlying configuration space is Lie group, we can use left trivialization to get Euclidean representation of tangent space. For Rigid body problems, rotation is SO(3), both ways have been tried, either as unit quaternion, which is an embedded unit surface in R^4, or as orthonormal matrix, which is a Lie group. We realize that unit quaternion is also a Lie group, and construct variational integrator based on this observation. The resulting method is an equivalent version of Rattle's method in unit quaternion setting. We also construct high order symplectic integrator on SO(n) via Polar decomposition, traditional Galerkin variational integrator on Lie group via exponential map involves second order derivative of exponential map, which is too complicated in practice. Here, we construct an algorithm which can be efficiently solved by fixed point iteration.
Tuesday, October 23, 2018
11:00AM AP&M 2402