Lagrangian submanifolds and the Hamilton–Jacobi equation

Abstract

Lagrangian submanifolds are becoming a very essential tool to generalize and geometrically understand results and procedures in the area of mathematical physics. Here we use general Lagrangian submanifolds to provide a geometric version of the Hamilton-Jacobi equation. This interpretation allows us to study some interesting applications of Hamilton-Jacobi equation in holonomic, nonholonomic and time-dependent dynamics from a geometrical point of view.

A Hamilton–Jacobi theory for implicit differential systems

“…If the fiber variable µ is frozen, then the exterior derivative dW | µ of the Morse family W becomes a differentiable mapping from Q to T * Q and its tangent mapping T (dW | µ ) goes from T Q to T T * Q. We remark that this last comment is a generalization of the Hamilton-Jacobi theory for the Lagrangian submanifolds studied in [4] as well.…”

“…This implies that there exists a Hamiltonian function H depending on (q, p) satisfying dH = φ. As a result, the system (39) can be written as in form of the Hamilton's equations (4). A stronger result follows from the generalized Poincaré lemma [7,27,38,52,55].…”

“…It is immediate to see from equations [4] and [5] that γ 1 = γ 2 . If γ is closed and γ 3 = 0, one obtains that γ 1 and γ 2 are independent of q 3 .…”

A Hamilton–Jacobi formalism for higher order implicit Lagrangians

“…Therefore, a HJ theory finds solutions on the lower dimensional manifold Q and retrieves them on the higher dimensional manifold T * Q by the existence of a section γ of the cotangent bundle which is the solution γ of the Hamilton-Jacobi equation (1). This picture (3) can be devised in different situations, as it is the case of nonholonomic systems [14,23,32,37,39,50,51], geometric mechanics on Lie algebroids [5] and almost-Poisson manifolds, singular systems [41], Nambu-Poisson framework [44], control theory [7], classical field theories [38,40,45], partial differential equations in general [64], the geometric discretization of the Hamilton-Jacobi equation [43,52], and others [6,13].…”

Hamilton–Jacobi theory for gauge field theories

“…Its geometrical formulation was investigated in many papes, e.g. in [16,3,4,5,6,13,14,15,17,18,28,32,37,7] . Recently a Hamilton-Jacobi for field theories version has been developed too [8,34,38,35].…”