NDLI: A Hamiltonian formulation of causal variational principles

Content Provider	Springer Nature Link
Author	Finster, Felix Kleiner, Johannes
Copyright Year	2017
Abstract	Causal variational principles, which are the analytic core of the physical theory of causal fermion systems, are found to have an underlying Hamiltonian structure, giving a formulation of the dynamics in terms of physical fields in space-time. After generalizing causal variational principles to a class of lower semi-continuous Lagrangians on a smooth, possibly non-compact manifold, the corresponding Euler–Lagrange equations are derived. In the first part, it is shown under additional smoothness assumptions that the space of solutions of the Euler–Lagrange equations has the structure of a symplectic Fréchet manifold. The symplectic form is constructed as a surface layer integral which is shown to be invariant under the time evolution. In the second part, the results and methods are extended to the non-smooth setting. The physical fields correspond to variations of the universal measure described infinitesimally by one-jets. Evaluating the Euler–Lagrange equations weakly, we derive linearized field equations for these jets. In the final part, our constructions and results are illustrated in a detailed example on $$\mathbb {R}^{1,1} \times S^1$$ where a local minimizer is given by a measure supported on a two-dimensional lattice.
Ending Page	33
Page Count	33
Starting Page	1
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	3
Volume Number	56
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2017-04-27
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Variational problems in a geometric measure-theoretic setting Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Symplectic structures of moduli spaces Analysis Theoretical, Mathematical and Computational Physics Integration on manifolds; measures on manifolds Methods of quantum field theory Problems involving relations other than differential equations Problems in abstract spaces Variational principles of physics Applications to physics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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