systems are abundant in field theories, this observation suggests that the Clebsch–Lagrange principle is fundamental for Cauchy problems. The two examples that will be discussed are the Yang–Mills–Higgs equations and the Einstein equation. In both cases, we will complete the following program:

1.  Using a $ (1+3)$-splitting of spacetime, formulate the field equations as a Cauchy problem and determine the constraints on the initial data. In particular, find a symplectic manifold on which the dynamics takes place.

2.  Identify the Lagrange multipliers as elements of the Lie algebra of the symmetry group and determine their action on the phase space.

3.  Calculate the Lagrangian in the $ (1+3) $-splitting and show that it is of the Clebsch–Lagrange form.

4.  Pass to the Clebsch–Hamiltonian picture using the Clebsch–Legendre transformation. In particular, phrase the constraints in terms of the momentum map.

5.  Relate the Clebsch–Hamiltonian picture to the Hamiltonian system with constraints obtained by the ordinary Legendre transformation.

6.  Discuss the symmetry reduction as a symplectic reduction by stages.

After accomplishing this program, we are left with a singular symplectic cotangent bundle reduction in infinite dimensions, which is studied in detail in a separate paper [@DiezRudolphReduction].

#### Acknowledgments {#acknowledgments .unnumbered}

We thank 