Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation

Leok, Melvin

doi:10.1007/s10208-019-09420-4

Cited by 8 publications

(3 citation statements)

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“…It may also be of interest to explore the potential union between the algorithm defined here and other structurepreserving discretizations suitable for numerical relativity. For example, it may be useful to explore the relationship between our holonomy-centric approach and the recently developed technique of group-equivariant interpolation in symmetric spaces [49,50]. It may also be useful to compare our effort with finite element cochain complexes suitable for applications in numerical relativity [51].…”

Section: Discussionmentioning

confidence: 99%

Discrete Gravity with Local Lorentz Invariance

Kur¹,

Glasser²

2022

Preprint

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Section: Discussionmentioning

confidence: 99%

Discrete Gravity with Local Lorentz Invariance

Kur¹,

Glasser²

2022

Preprint

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“…Discrete reduction. The construction of discrete theories mimicking the continuous ones has been a fruitful approach to obtain variational and geometric integrators for the dynamical equations of the systems (see, for example, [43] for Mechanics or [21,34] for field theories). In the same vein, reduction by symmetries has also been analyzed in the discrete setting for both mechanics [5,7,38,39,46] and field theories [50].…”

Section: Data Availability Statementmentioning

confidence: 99%

Gauge reduction in covariant field theory

Castrillón López,

Rodríguez Abella

2024

J. Phys. A: Math. Theor.

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In this work, we develop a Lagrangian reduction theory for covariant field theorieswith gauge symmetries. These symmetries are modeled by a Lie group fiber bundleacting fiberwisely on a configuration bundle. In order to reduce the variational principle,we utilize generalized principal connections, a type of Ehresmann connections that areequivariant by the fiberwise action. After obtaining the reduced equations, we give thereconstruction condition and we relate the vertical reduced equation with the Noethertheorem. Lastly, we illustrate the theory with several examples, including the classicalcase (Lagrange–Poincaré reduction), Electromagnetism, symmetry-breaking and non-Abelian gauge theories.

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“…(with the caveat that, in the continuum case, G acts on the restricted dual jet bundle, whereas in the discrete case, G acts on the discrete restricted dual jet bundle). One can obtain such a G-invariant R d via G-equivariant interpolation (see Leok and Zhang [24] and Leok [22]), in which case, the discrete Noether theorem is precisely quadrature applied to Noether's theorem.…”

Section: Proofmentioning

confidence: 99%

Multisymplectic Hamiltonian Variational Integrators

Tran,

Leok

2021

Preprint

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Variational integrators have traditionally been constructed from the perspective of Lagrangian mechanics, but there have been recent efforts to adopt discrete variational approaches to the symplectic discretization of Hamiltonian mechanics using Hamiltonian variational integrators. In this paper, we will extend these results to the setting of Hamiltonian multisymplectic field theories. We demonstrate that one can use the notion of Type II generating functionals for Hamiltonian partial differential equations as the basis for systematically constructing Galerkin Hamiltonian variational integrators that automatically satisfy a discrete multisymplectic conservation law, and establish a discrete Noether's theorem for discretizations that are invariant under a Lie group action on the discrete dual jet bundle. In addition, we demonstrate that for spacetime tensor product discretizations, one can recover the multisymplectic integrators of Bridges and Reich, and show that a variational multisymplectic discretization of a Hamiltonian multisymplectic field theory using spacetime tensor product Runge-Kutta discretizations is well-defined if and only if the partitioned Runge-Kutta methods are symplectic in space and time.

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Variational Discretizations of Gauge Field Theories Using Group-Equivariant Interpolation

Cited by 8 publications

References 50 publications

Discrete Gravity with Local Lorentz Invariance

Discrete Gravity with Local Lorentz Invariance

Gauge reduction in covariant field theory

Multisymplectic Hamiltonian Variational Integrators

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