Covariant Hamiltonian first order field theories with constraints on manifolds with boundary: the case of Hamiltonian dynamics

TL;DR

This paper develops a covariant Hamiltonian field theory framework for systems with boundaries, analyzing solution spaces and constraints inspired by Yang-Mills and gravity theories, with implications for boundary conditions and constraints in field theories.

Contribution

It introduces a covariant Hamiltonian formulation for boundary-involved field theories, providing conditions for boundary solution spaces to form Lagrangian submanifolds and modeling gravity constraints.

Findings

Solution spaces at the boundary form Lagrangian submanifolds under certain conditions.

A new theory of constraints mimicking Palatini's gravity constraints is proposed.

The variational principle is precisely formulated in the covariant boundary context.

Abstract

Inspired by problems arising in the geometrical treatment of Yang-Mills theories and Palatini's gravity, the covariant formulation of Hamiltonian dynamical systems as a Hamiltonian field theory of dimension $1+0$ on a manifold with boundary is discussed. After a precise statement of Hamilton's variational principle in this context, the geometrical properties of the space of solutions of the Euler-Lagrange equations of the theory are analyzed. A sufficient condition is obtained that guarantees that the set of solutions of the Euler-Lagrange equations at the boundary of the manifold, fill a Lagrangian submanifold of the space of fields at the boundary. Finally a theory of constraints is introduced that mimics the constraints arising in Palatini's gravity.