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.
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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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Communicated by Wolfgang Dahmen.
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Leok, M. Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation. Found Comput Math 19, 965–989 (2019). https://doi.org/10.1007/s10208-019-09420-4
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DOI: https://doi.org/10.1007/s10208-019-09420-4
Keywords
- Geometric numerical integration
- Variational integrators
- Symplectic integrators
- Hamiltonian field theories
- Manifold-valued data
- Gauge field theories
- Numerical relativity