Abstract
A novel covariant Hamilton formalism for field theories is proposed in terms of multi-contact structure which is a higher rank generalization of contact 1-form. This formulation is naturally derived from our previous work, Finsler-Kawaguchi Lagrange formulation. This new covariant Hamilton formulation has a strong covariance so that we can take not only time but also spacetime parameter arbitrarily. In other words, our formulation preliminarily includes the concept of spacetime and we should think of the solution submanifold as our spacetime.
Highlights
A novel covariant Hamilton formalism for field theories is proposed in terms of multi-contact structure which is a higher rank generalization of contact 1-form
This formulation is naturally derived from our previous work, Finsler-Kawaguchi Lagrange formulation
It is known that there is a geometric formulation of Lagrange systems using Finsler geometry [1,2,3]
Summary
We usually start Lagrange formalism from (Q, L) where Q is a configuration manifold and L is a Lagrangian of a system, and in general (Q, L) cannot have a (metric) geometry. In this formulation, an oriented curve c represents an orbital of a system and Finsler length A[c] corresponds to the action of the system, so the variational principle and Euler-Lagrange equation become geometric and parameterisation invariant (covariant) Using this covariant formulation, we can take parameter for convenience of calculation, and we think that there is theoretical deep importance. This Kawaguchi 2-form satisfies the following homogeneity condition, K(xμ, λdxμν) = λK(xμ, dxμν), λ > 0 By this property, Kawaguchi 2-form K can define an geometric area A[σ] = K(x, dx) for an σ oriented surface σ ⊂ M independently of parameterisation. Same way of Finsler-Lagrange formulation, oriented surfaces σ represent field configuration and dynamics and Kawaguchi area A[σ] corresponds to a geometric action of field Lagrange system Using this formulation, we can define an energy-momentum of gravity from Nöther theorem and reformulate field path integral. We should construct a Hamiltonian version of these new Lagrange formalism
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