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Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation

  • Melvin LeokEmail author
Article
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Abstract

We describe a systematic mathematical approach to the geometric discretization of gauge field theories that is based on Dirac and multi-Dirac mechanics and geometry, which provide a unified mathematical framework for describing Lagrangian and Hamiltonian mechanics and field theories, as well as degenerate, interconnected, and nonholonomic systems. Variational integrators yield geometric structure-preserving numerical methods that automatically preserve the symplectic form and momentum maps, and exhibit excellent long-time energy stability. The construction of momentum-preserving variational integrators relies on the use of group-equivariant function spaces, and we describe a general construction for functions taking values in symmetric spaces. This is motivated by the geometric discretization of general relativity, which is a second-order covariant gauge field theory on the symmetric space of Lorentzian metrics.

Keywords

Geometric numerical integration Variational integrators Symplectic integrators Hamiltonian field theories Manifold-valued data Gauge field theories Numerical relativity 

Mathematics Subject Classification

37M15 53C35 65D05 65M70 65P10 70H25 

Notes

Acknowledgements

The author would like to thank Evan Gawlik for helpful discussions and comments. The author has been supported in part by the National Science Foundation under Grants DMS-1010687, CMMI-1029445, DMS-1065972, CMMI-1334759, DMS-1411792, DMS-1345013, DMS-1813635.

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Copyright information

© SFoCM 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSan DiegoUSA

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