---
bibliography:
- 'Literature.bib'
---

=1

Introduction
============

Relativistic field theories of gauge symmetry type can be formulated as constrained Hamiltonian systems. Using a $ (1 + 3) $-decomposition of spacetime, the four dimensional field equations usually split into hyperbolic evolution equations and elliptic constraint equations for the Cauchy data. A remarkable observation is that the constraints often can be formulated in terms of the momentum map associated to the action of the symmetry group of the theory. This is usually verified by a fairly routine calculation in each example separately (see, , [@ArmsMarsdenEtAl1981; @Arms1981] for pure Yang–Mills theory and [@FischerMarsden1979; @ArmsFischerMarsden1975] for general relativity). Going beyond a case-by-case study, the general philosophy that the subset of the phase space cut out by the constraints can be identified with the zero set of a momentum map seems to be true in a large number of models. In this paper, we will argue that the relationship between constraints and momentum maps is not a lucky coincidence but is rooted in a special feature of the four dimensional physical action. This special feature is the main subject of the present paper.

Inspired by the Clebsch optimal control problem [@Gay-BalmazRatiu2011], we study a 