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New Trends in Hamilton-Jacobi Theory: Conservative and Dissipative Dynamics

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (30 April 2021) | Viewed by 12330

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Prof. Dr. Manuel De León

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Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Calle Nicolás Cabrera, 13-15, Campus Cantoblanco, UAM, 28049 Madrid, Spain
Interests: differential geometry; geometric mechanics; mathematical physics
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Dear Colleagues,

The Hamilton–Jacobi theory is a classical subject that was extensively developed in the last two centuries. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. If we are able to find a complete solution of the Hamilton–Jacobi problem, then the dynamics can be solved by quadratures.

The power of this method is that, in spite of the difficulties in solving a partial differential equation instead of an ordinary differential one, it works in many cases, representing an extremely useful tool, usually more so than Hamilton’s equations. In fact, the modern interpretation relating the Hamilton–Jacobi procedure with the theory of Lagrangian submanifolds is an important source of new results and insights.

Recently, it has been shown how to use the symmetries of the system to reduce the corresponding Hamilton–Jacobi equation as well as to reconstruct the solutions of the unreduced equation from the solutions of the reduced Hamilton–Jacobi equation.

Another important view concerns the discrete version of dynamics. This fact, combined with the geometrical properties of the Lagrangian submanifolds, permits developing new numerical methods.

Another issue is the Hamilton–Jacobi–Bellman equation, which is central to optimal control theory. The equation is a result of the theory of dynamic programming, which was pioneered in the 1950s by Richard Bellman and coworkers. The corresponding discrete-time equation is usually referred to as the Bellman equation. In continuous time, the result can be seen as an extension of earlier work in classical physics on the Hamilton–Jacobi equation by Hamilton and Jacobi.

More recently, there has been a lot of interest in the study of contact Hamiltonian systems, which lead to dissipative behaviors instead of the conservative one in symplectic dynamics. A Hamilton–Jacobi theory for this kind of dynamics is being developed, and it would have a lot of applications in thermodynamics, statistical physics, quantum mechanics, gravity, and control theory, among many other subjects.

Finally, a geometric Hamilton–Jacobi theory for classical field theories in the framework of multisymplectic theory still deserves a lot of work.

This Special Issue on Hamilton–Jacobi theory wants to bring together specialists coming from different areas of research and show how the Hamilton–Jacobi theory is so useful in their domains.

Prof. Dr. Manuel De Leòn
Guest Editor

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Keywords

Hamilton–Jacobi equation
Hamilton–Jacobi problem
Complete solution
Integrability
KAM theory
Lagrangian submanifolds
Legendrian submanifolds
Symplectic geometry
Contact geometry
Poisson and Jacobi structures
Multisymplectic field theory
Symplectic relations
Morse families
Geometric integrators
Generating functions

Published Papers (5 papers)

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Research

41 pages, 526 KiB

Open AccessArticle

Contact Dynamics: Legendrian and Lagrangian Submanifolds

by Oğul Esen, Manuel Lainz Valcázar, Manuel de León and Juan Carlos Marrero

Mathematics 2021, 9(21), 2704; https://doi.org/10.3390/math9212704 - 25 Oct 2021

Cited by 7 | Viewed by 1942

Abstract

We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics. Full article

(This article belongs to the Special Issue New Trends in Hamilton-Jacobi Theory: Conservative and Dissipative Dynamics)

24 pages, 343 KiB

Open AccessArticle

The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

by Manuel de León, Manuel Lainz and Álvaro Muñiz-Brea

Mathematics 2021, 9(16), 1993; https://doi.org/10.3390/math9161993 - 20 Aug 2021

Cited by 10 | Viewed by 2065

Abstract

The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton–Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed. Full article

(This article belongs to the Special Issue New Trends in Hamilton-Jacobi Theory: Conservative and Dissipative Dynamics)

34 pages, 476 KiB

Open AccessArticle

Extended Hamilton–Jacobi Theory, Symmetries and Integrability by Quadratures

by Sergio Grillo, Juan Carlos Marrero and Edith Padrón

Mathematics 2021, 9(12), 1357; https://doi.org/10.3390/math9121357 - 11 Jun 2021

Cited by 3 | Viewed by 1539

Abstract

In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions [...] Read more.

U \subseteq M

, such that

π (U)

has a manifold structure and

π_{|U} : U \to π (U)

, the restriction to U of the canonical projection

π : M \to M / G

, is a surjective submersion. If

X_{|U}

is not vertical with respect to

π_{|U}

, we show that such complete solutions solve the reconstruction equations related to

X_{|U}

and G, i.e., the equations that enable us to write the integral curves of

X_{|U}

in terms of those of its projection on

π (U)

. On the other hand, if

X_{|U}

is vertical, we show that such complete solutions can be used to construct (around some points of U) the integral curves of

X_{|U}

up to quadratures. To do that, we give, for some elements

ξ

of the Lie algebra

g

of G, an explicit expression up to quadratures of the exponential curve

exp (ξ t)

, different to that appearing in the literature for matrix Lie groups. In the case of compact and of semisimple Lie groups, we show that such expression of

exp (ξ t)

is valid for all

ξ

inside an open dense subset of

g

. Full article

(This article belongs to the Special Issue New Trends in Hamilton-Jacobi Theory: Conservative and Dissipative Dynamics)

19 pages, 385 KiB

Open AccessFeature PaperArticle

An Overview of the Hamilton–Jacobi Theory: the Classical and Geometrical Approaches and Some Extensions and Applications

by Narciso Román-Roy

Mathematics 2021, 9(1), 85; https://doi.org/10.3390/math9010085 - 03 Jan 2021

Cited by 4 | Viewed by 3108

Abstract

This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton–Jacobi theory. The relation with the “classical” Hamiltonian approach using canonical transformations is also analyzed. Furthermore, a more general framework for the theory is also briefly explained. It is also shown how, from this generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamical systems are recovered, as well as how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classical field theories in their multisymplectic formulation. Full article

(This article belongs to the Special Issue New Trends in Hamilton-Jacobi Theory: Conservative and Dissipative Dynamics)

19 pages, 4681 KiB

Open AccessArticle

Variational Integrators in Holonomic Mechanics

by Shumin Man, Qiang Gao and Wanxie Zhong

Mathematics 2020, 8(8), 1358; https://doi.org/10.3390/math8081358 - 13 Aug 2020

Cited by 5 | Viewed by 2523

Abstract

Variational integrators for dynamic systems with holonomic constraints are proposed based on Hamilton’s principle. The variational principle is discretized by approximating the generalized coordinates and Lagrange multipliers by Lagrange polynomials, by approximating the integrals by quadrature rules. Meanwhile, constraint points are defined in order to discrete the holonomic constraints. The functional of the variational principle is divided into two parts, i.e., the action of the unconstrained term and the constrained term and the actions of the unconstrained term and the constrained term are integrated separately using different numerical quadrature rules. The influence of interpolation points, quadrature rule and constraint points on the accuracy of the algorithms is analyzed exhaustively. Properties of the proposed algorithms are investigated using examples. Numerical results show that the proposed algorithms have arbitrary high order, satisfy the holonomic constraints with high precision and provide good performance for long-time integration. Full article

(This article belongs to the Special Issue New Trends in Hamilton-Jacobi Theory: Conservative and Dissipative Dynamics)

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New Trends in Hamilton-Jacobi Theory: Conservative and Dissipative Dynamics

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