Hamilton-Jacobi framework for Regge-Teitelboim gravity

In higher co-dimension, we discuss the Hamilton-Jacobi formalism for brane gravity described by the Regge-Teitelboim model, in higher co-dimension. Considering that it is originally a second-order in derivatives singular theory, we analyze its constraint structure by identifying the complete set of Hamilton-Jacobi equations, under Carathéodory's equivalent Lagrangians method, which goes hand in hand with the study of the integrability for this type of gravity. Besides, we calculate the characteristic equations, including the one that satisfies the Hamilton principal function S. We find the presence of involutive and non-involutive constraints so that by properly defining a generalized bracket, the non-involutive constraints that originally arise in our framework are removed while the set of parameters related to time evolution and gauge transformations is identified. A detailed comparison is also made with a recent Ostrogradsky-Hamilton approach for constrained systems, developed for this brane gravity. Some facts about the gauge symmetries behind this theory are discussed.