Symmetries and reduction Part II — Lagrangian and Hamilton–Jacobi picture

Marmo, Giuseppe; Schiavone, Luca; Zampini, Alessandro

doi:10.1142/s0219887821300063

Cited by 2 publications

(2 citation statements)

References 62 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we review the geometrical theory of symmetries that can be constructed for dynamical systems, admitting a symplectic or presymplectic formulation. We refer to [6,7,21], and references therein, for a more detailed exposition.…”

Section: Symmetries For Presymplectic Dynamicsmentioning

confidence: 99%

“…These aspects, and many more, have been extensively analyzed in the last two centuries, both within the mathematical and theoretical physics literature: a (far from exhaustive) list of contributions where the previous ideas have been developed include [1][2][3][4][5][6][7][8][9][10][11][12][13][14], where Noether's theorems (that is the relation between symmetries and the so called conservation laws) are also elucidated. In particular, in recent references where classical field theories were analyzed within the setting of jet bundles and their duals, conserved currents associated with symmetries of an action functional are modeled as (m − 1)-forms on a fiber bundle underlying the theory (with m the dimension of the space-time in which the theory is developed, see, for instance, [8,[15][16][17]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds

et al. 2022

Self Cite

View full text Add to dashboard Cite

As the space of solutions of the first-order Hamiltonian field theory has a presymplectic structure, we describe a class of conserved charges associated with the momentum map, determined by a symmetry group of transformations. A gauge theory is dealt with by using a symplectic regularization based on an application of Gotay’s coisotropic embedding theorem. An analysis of electrodynamics and of the Klein–Gordon theory illustrate the main results of the theory as well as the emergence of the energy–momentum tensor algebra of conserved currents.

show abstract

Section: Symmetries For Presymplectic Dynamicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds

et al. 2022

Self Cite

View full text Add to dashboard Cite

show abstract

Reviewing the geometric Hamilton–Jacobi theory concerning Jacobi and Leibniz identities

Esen¹,

León²,

Valcázar³

et al. 2022

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

In this survey, we review the classical Hamilton–Jacobi theory from a geometric point of view in different geometric backgrounds. We propose a Hamilton-Jacobi equation for different geometric structures attending to one particular characterization: whether they fulfill the Jacobi and Leibniz identities simultaneously, or if at least they satisfy one of them. In this regard, we review the case of time-dependent ($t$-dependent in the sequel) and dissipative physical systems as systems that fulfill the Jacobi identity but not the Leibnitz identity. Furthermore, we review the contact-evolution Hamilton-Jacobi theory as a split off the regular contact geometry, and that actually satisfies the Leibniz rule instead of Jacobi. Furthermore, we include a novel result, which is the Hamilton-Jacobi equation for conformal Hamiltonian vector fields as a generalization of the well-known Hamilton-Jacobi on a symplectic manifold, that is retrieved in the case of a zero conformal factor. The interest of a geometric Hamilton–Jacobi equation is the primordial observation that if a Hamiltonian vector field $X_H$ can be projected into a configuration manifold by means of a $1$-form $dW$, then the integral curves of the projected vector field $X_{H}^{dW}$ can be transformed into integral curves of $X_H$ provided that $W$ is a solution of the Hamilton-Jacobi equation. Geometrically, the solution of the Hamilton-Jacobi equation plays the role of a Lagrangian submanifold of a certain bundle. Exploiting these features in different geometric scenarios we propose a geometric theory for multiple physical systems depending on the fundamental identities that their dynamic satisfies. Different examples are pictured to reflect the results provided, being all of them new, except for one that is reassessment of a previously considered example.

show abstract

Symmetries and reduction Part II — Lagrangian and Hamilton–Jacobi picture

Abstract: Following the analysis we have presented in a previous paper (that we refer to as [I]), we describe a Noether theorem related to symmetries, with the associated reduction procedures, for classical dynamics within the Lagrangian and the Hamilton–Jacobi formalism.

Cited by 2 publications

References 62 publications

Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds

Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds

Reviewing the geometric Hamilton–Jacobi theory concerning Jacobi and Leibniz identities

Contact Info

Product

Resources

About