Integrability of second-order Lagrangians admitting a first-order Hamiltonian formalism

Abstract

Second-order Lagrangian densities admitting a first-order Hamiltonian formalism are studied; namely, i) necessary and sufficient conditions for the Poincaré-Cartan form of a second-order Lagrangian on an arbitrary fibred manifold p: E → N to be projectable onto J 1 E are explicitly determined; ii) for each of such Lagrangians, a first-order Hamiltonian formalism is developed and a new notion of regularity is introduced; iii) the variational problems of this class defined by regular Lagrangians are proved to be involutive.

New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity

“…(Geometrically, they are a consequence of the fact that Ω L Ep is π 1 -projectable [7,24,35,36,41,42]). In this way, the connection and metric equations become semiholonomic constraints, which are called connection and metric constrains, respectively.…”

“…It is proved [7,42] that there are first-order (regular) Lagrangians in J 1 π Σ which are equivalent to the the Hilbert-Einstein Lagrangian and that allow us a description of the Einstein-Hilbert model in J 1 π Σ (with coordinates (x µ , g αβ , g αβ,µ )). The first-order Lagrangian density proposed in [41]…”

“…Thus, in [3,8,9,10,21,25,26,27,35,36,43,44,46] general aspects of the theory are studied in this way, meanwhile other papers are devoted to consider several particular problems. For instance, in [7,24,41,42] the reduction and projectability of higher-order theories (such as the Hilbert-Einstein model) is analized, in [47] the vielbein models of General Relativity are studied using the multisymplectic formulation and in [32,33,34] interesting contributions to the problem of the precanonical quantization of gravity are done.…”

Multisymplectic unified formalism for Einstein-Hilbert gravity

“…The geometrisation of the theories of gravitation (General Relativity) and, in particular the multisymplectic framework, allows us to do a covariant description of these theories, considering and understanding several inherent characteristics of it, and it has been studied by different authors. For instance, relevant references devoted to develop geometrically general aspects of the theory are [2,5,6,7,15,22,26,40], the reduction of the order of the theory and the projectability of the Poincaré-Cartan form associated with the Hilbert-Einstein action is explained in [4,38,39], meanwhile in [32,33] different aspects of the theory are studied using Lepage-Cartan forms, and in [45,46] a multisymplectic analysis of the vielbein formalism of General Relativity is done. Finally, some general features of the gravitational theory following the polysymplectic version of the multisymplectic formalism are described in [21,41], including the problem of its precanonical quantization [27,28,29].…”

“…As a consequence of the singularity of the Lagrangian, the Einstein-Hilbert model exhibits gauge freedom and it can be reduced to a first-order field theory [4,32,33,38,39]. Then, related to this topic, we analyse also a first-order theory equivalent to Einstein-Hilbert (without matter-energy sources), which is described by a first-order regular Lagrangian, showing, in this way, that General Relativity can be realised as a regular multisymplectic field theory (without constraints).…”