Routh reduction for first-order Lagrangian field theories

Abstract

We present a reduction theory for first order Lagrangian field theories which takes into account the conservation of momenta. The relation between the solutions of the original problem with a prescribed value of the momentum and the solutions of the reduced problem is established. An illustrative example is discussed in detail.

Routh Reduction of Palatini Gravity in Vacuum

“…In summary, a large part of the present article consists in repeating for Palatini gravity what was done for first order field theory in [6]; more concretely and in line with the previous discussion, we want to answer an specific problem through a particular set of techniques:…”

“…Let us indicate by G µ ⊂ G the stabilizer of the momentum µ. Then the main result in [6] established a correspondence between the extremals of the variational problem (π : E → M, L) and…”

“…From this viewpoint, it becomes interesting to find a reduction scheme relating the Lagrangian formulation of Palatini and Einstein-Hilbert gravity directly, without the detour through Hamiltonian formalism. So far, there exist two ways to implement reduction at the Lagrangian level, namely Lagrange-Poincaré reduction [8,11,12,17] and Routh reduction [4,6,14,18,31,32]. Moreover, there are physical considerations that can be said in support of this kind of reduction: They deal with not only the reduction, but also the reconstruction problem, and it is argued in [33] that reconstruction can be relevant from the physical point of view.…”

“…Therefore, it is possible to avoid this regularity condition by working in a unified setting, where variables associated to both velocities and momenta are taken into account; this approach can be used to set a Routh reduction procedure not suffering this limitation [18]. This viewpoint proves its usefulness in [6], where unified formalism was employed in the generalization of Routh reduction to the field theory realm. In fact, in order to improve the understanding of the tools and strategies used in the present work, it could be helpful to give an account of both the problem addressed in this reference and the techniques used to solve it:…”

“…• Result in [6]: For the moment, we will represent a variational problem with a pair for arbitrary variations δs. Let us suppose that there exists a Lie group acting freely and vertically on E, such that the Lagrangian density is invariant by this action, yielding to a momentum map J : J 1 π → ∧ m−1 T * J 1 π ⊗ g * .…”