Equivalence between the Lagrangian and Hamiltonian formalism for constrained systems.

*(English)*Zbl 0613.70012In this paper the authors claim that they give an explicit and complete proof of the equivalence between the Lagrangian and Hamiltonian formalisms for constrained systems. They construct an implicit inverse relation between velocities and momenta, i.e., the inverse Legendre transformation, using that they deduce the Hamiltonian-Dirac equations from Euler-Lagrange equations. Neither is a set of normal differential equations, therefore, the uniqueness and existence theorem cannot be applied. This means that, at most, they will only have solutions in a submanifold of the respective spaces and in general these solutions will not be unique.

A careful analysis shows that given a solution of the Euler-Lagrange equations they can construct a solution of the Hamilton-Dirac equations and vice-versa. Then the authors look for the appropriate submanifold of the tangent bundle and a submanifold of the cotangent bundle where the solutions exist. These submanifolds are constructed through an iterative procedure. In a given local chart they are characterized by a set of functions that are called constraints.

The Hamiltonian formalisms as developed in this paper differs from the usual development. The first class primary constraints play a privileged role. Other constraints are either first or second class with respect to them. These constraints that are first class with respect to the primary first class constraints can be associated with Lagrangian constraints. It is shown that all constraints other than the primary constraints have either a symmetric or antisymmetric Poisson bracket structure with the first class primary constraints.

A careful analysis shows that given a solution of the Euler-Lagrange equations they can construct a solution of the Hamilton-Dirac equations and vice-versa. Then the authors look for the appropriate submanifold of the tangent bundle and a submanifold of the cotangent bundle where the solutions exist. These submanifolds are constructed through an iterative procedure. In a given local chart they are characterized by a set of functions that are called constraints.

The Hamiltonian formalisms as developed in this paper differs from the usual development. The first class primary constraints play a privileged role. Other constraints are either first or second class with respect to them. These constraints that are first class with respect to the primary first class constraints can be associated with Lagrangian constraints. It is shown that all constraints other than the primary constraints have either a symmetric or antisymmetric Poisson bracket structure with the first class primary constraints.

Reviewer: Y. Kozai

##### MSC:

70F20 | Holonomic systems related to the dynamics of a system of particles |

70H45 | Constrained dynamics, Dirac’s theory of constraints |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

##### Keywords:

constrained systems; implicit inverse relation; inverse Legendre transformation; Euler-Lagrange equations; normal differential equations; uniqueness; existence theorem; solutions in a submanifold; Hamilton-Dirac equations; tangent bundle; cotangent bundle; iterative procedure
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