Equivalence between the Lagrangian and Hamiltonian formalism for constrained systems

Abstract

The equivalence between the Lagrangian and Hamiltonian formalism is studied for constraint systems. A procedure to construct the Lagrangian constraints from the Hamiltonian constraints is given. Those Hamiltonian constraints that are first class with respect to the Hamiltonian constraints produce Lagrangian constraints that are FL-projectable.

Canonical Noether symmetries and commutativity properties for gauge systems

“…But one can also consider a third evolution operator that, unlike the previous ones, is fully deterministic. 6 This is the operator K of Sec. II.…”

“…This is the Dirac's method in the Lagrangian formalism. 6 Our aim is to give a tangent space characterization of a Noether transformation ␦ L q(t;q,q ) that satisfies the property of being projectable to phase space, that is, ␦ L q is the pullback of a canonical Noether transformation ␦ H q, ␦ L qϭFL*(␦ H q). Notice at this point that we have two natural ways to define the dynamical time derivative ␦ L q in RϫTQ.…”

“…One can introduce a useful differential operator K connecting the Lagrangian and Hamiltonian formalisms, that has no ambiguity at all, and that still represents the dynamics. 6 It can be defined as a vector field along the Legendre's transformation FL, 17 and, as a differential operator, it gives the time evolution of a function h in RϫT*Q as a function K•h in RϫTQ by…”

“…However, this is not the final form of the dynamics. To get the final dynamics we must perform a stabilization algorithm: [5][6][7]21,22 consistency requirements, that is, the tangency of X H to the surface of constraints, may lead to new constraints and also to the determination of some of the Lagrangian multipliers as functions in phase space.…”

“…This is nothing but the framework first studied by Dirac to deal with gauge theories or, more generally, constrained systems. [4][5][6][7][8][9] The regular case is recovered when no Hamiltonian constraints occur.…”