Variational principles and symmetries on fibered multisymplectic manifolds

Journal of Geometric Mechanics

2020

Self Cite

A complete geometric classification of symmetries of autonomous Hamiltonian mechanical systems is established; explaining how to obtain their associated conserved quantities in all cases. In particular, first we review well-known results and properties about the symmetries of the Hamiltonian and of the symplectic form and then some new kinds of non-symplectic symmetries and their conserved quantities are introduced and studied.

Section: Discussionmentioning

confidence: 68%

A summary on symmetries and conserved quantities of autonomous Hamiltonian systems

Journal of Geometric Mechanics

2020

Self Cite

“…The variational problem [23,40] associated to the system (J 1 π, Ω L EP ) consists in finding holonomic sections ψ L = j 1 φ ∈ Γ(π 1 ) (with φ ∈ Γ(π)) which are solutions to the equation…”

Section: Consider a Natural System Of Coordinatesmentioning

confidence: 99%

“…In this way we have constructed the Hamiltonian system (P, Ω H ), which is associated with the almost-regular Lagrangian system . Then, the variational problem associated with this system [23,40] consists in finding sections ψ H : M → P which are solutions to the equation or, what is equivalent, which are integral sections of a multivector field contained in a class of τ Ptransverse integrable multivector fields {X H } ⊂ X 4 (P) such that…”

Section: Lagrangian Symmetries Of the Einstein-palatini Modelmentioning

confidence: 99%

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

Gaset

Journal of Geometric Mechanics

2019

Self Cite

We present a covariant multisymplectic formulation for the Einstein-Palatini (or Metric-Affine) model of General Relativity (without energy-matter sources). As it is described by a first-order affine Lagrangian (in the derivatives of the fields), it is singular and, hence, this is a gauge field theory with constraints. These constraints are obtained after applying a constraint algorithm to the field equations, both in the Lagrangian and the Hamiltonian formalisms. In order to do this, the covariant field equations must be written in a suitable geometrical way, using integrable distributions which are represented by multivector fields of a certain type. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalism. The gauge symmetries of the model are discussed in both formalisms and, from them, the equivalence with the Einstein-Hilbert model is established.

“…For the Lagrangian-Hamiltonian unified formalism, we have to consider the symmetric higher-order jet multimomentum bundles W = J 3 π × J 1 π J 2 π † and W r = J 3 π × J 1 π J 2 π ‡ (see [35,36] for details), which have as natural local coordinates (x µ , g αβ , g αβ,µ , g αβ,µν , g αβ,µνλ , p, p αβ,µ , p αβ,µν ) and (x µ , g αβ , g αβ,µ , g αβ,µν , g αβ,µνλ , p αβ,µ , p αβ,µν ), (0 ≤ α ≤ β ≤ 3; 0 ≤ µ ≤ ν ≤ 3). These bundles are endowed with the canonical projections…”

Section: Lagrangian-hamiltonian Unified Formalism 221 the Higher-ormentioning

confidence: 99%

“…Another interesting aspect of the theory is the study of its symmetries. An analysis of this subject is also given, stating the basic definitions and properties of symmetries and conservation laws in the Lagrangian formalism (including the corresponding version of Noether's theorem) [12,19], and extending these concepts and properties to the unified formalism. The application of these results to the Einstein-Hilbert model is briefly analyzed, recovering some previous results [32,38].…”

Section: Introductionmentioning

confidence: 99%

Multisymplectic unified formalism for Einstein-Hilbert gravity

Gaset

Journal of Mathematical Physics

2018

Self Cite

We present a covariant multisymplectic formulation for the Einstein-Hilbert model of General Relativity. As it is described by a second-order singular Lagrangian, this is a gauge field theory with constraints. The use of the unified Lagrangian-Hamiltonian formalism is particularly interesting when it is applied to these kinds of theories, since it simplifies the treatment of them; in particular, the implementation of the constraint algorithm, the retrieval of the Lagrangian description, and the construction of the covariant Hamiltonian formalism. In order to apply this algorithm to the covariant field equations, they must be written in a suitable geometrical way, which consists of using integrable distributions, represented by multivector fields of a certain type. We apply all these tools to the Einstein-Hilbert model without and with energy-matter sources. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalisms in both cases. As a consequence of the gauge freedom and the constraint algorithm, we see how this model is equivalent to a first-order regular theory, without gauge freedom. In the case of presence of energy-matter sources, we show how some relevant geometrical and physical characteristics of the theory depend on the type of source. In all the cases, we obtain explicitly multivector fields which are solutions to the gravitational field equations. Finally, a brief study of symmetries and conservation laws is done in this context.