Historical Lagrangian Dynamics

TL;DR

This paper introduces a universal, covariant Lagrangian formalism for dynamical systems that unifies time, space-time, and field theories using the concept of histories, including forms, with a generalized symplectic structure.

Contribution

It develops a comprehensive, covariant formalism for dynamics based on histories, extending to field theories and forms, with a new generalized symplectic structure.

Findings

Formalism is explicitly covariant and does not require a time function.

Includes space-time as a special case of evolution domain.

Introduces a covariant generalized symplectic form with conservation properties.

Abstract

This paper presents (in its Lagrangian version) a very general "historical" formalism for dynamical systems, including time-dynamics and field theories. It is based on the universal notion of history. Its condensed and universal formulation provides a synthesis and a generalization different approaches of dynamics. It is in our sense closer to its real essence. The formalism is by construction explicitely covariant and does not require the introduction of time, or of a time function in relativistic theories. It considers space-time (in field theories) exactly in the same manner than time in usual dynamics, with the only difference that it has 4 dimensions. Both time and space-time are considered as particular cases of the general notion of an evolution domain. In addition, the formalism encompasses the cases where histories are not functions (e.g., of time or of space-time), but forms. This applies to electromag-netism and to first order general relativity (that we treat explicitely). It has both Lagrangian and Hamiltonian versions. An interesting result is the existence of a covariant generalized symplectic form, which generalizes the usual symplectic or the multisymplectic form, and the symplectic currents. Its conservation on shell provides a genuine symplectic form on the space of solutions.