Abstract
In general, the system of 2nd-order partial differential equations made of the Euler–Lagrange equations of classical field theories are not compatible for singular Lagrangians. This is the so-called second-order problem. The first aim of this work is to develop a fully geometric constraint algorithm which allows us to find a submanifold where the Euler–Lagrange equations have solution, and split the constraints into two kinds depending on their origin. We do so using k-symplectic geometry, which is the simplest intrinsic description of classical field theories. As a second aim, the Einstein–Palatini model of General Relativity is studied using this algorithm.
Highlights
It is an established fact that symplectic geometry is the most suitable geometric framework to describe Lagrangian and Hamiltonian mechanics [1,3,31]
The name of these constraints refers to the fact that their origin has nothing to do with the second-order condition, but only with the compatibility of Lagrangian equations in general. As it happens in mechanics [5,9,23,34], the following property characterizes these kinds of constraints: Proposition 3 First generation dynamical constraints can be expressed as FL-projectable functions
The first goal of this work has been to solve the second-order problem for singular field theories, generalizing the constraint algorithm of [34] to k-presymplectic Lagrangian systems and completing the results of [27]
Summary
It is an established fact that symplectic geometry is the most suitable geometric framework to describe Lagrangian and Hamiltonian (autonomous) mechanics [1,3,31]. This problem is usually solved by applying suitable constraint algorithms which allow us to find a submanifold of the phase space of the system where the existence of solutions is assured The first of these constraint algorithms was given by P.G. Bergmann and P.A.M. Dirac, using a local coordinate language, for the Hamiltonian formalism of singular mechanics [2,18]. The emergent constraints are classified into two groups, depending on whether they are a consequence of the compatibility of field equations or the requirement that solutions verify the second order condition This algorithm is the generalization of what is presented in [34] for singular Lagrangian mechanics to k-presymplectic Lagrangian systems, and completes the algorithm presented in [27], where the second-order problem was not considered.
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