Geometric Hamilton–Jacobi theory for systems with external forces

@article{deLen2021GeometricHT,
  title={Geometric Hamilton–Jacobi theory for systems with external forces},
  author={Manuel de Le{\'o}n and Manuel Lainz and Asier L'opez-Gord'on},
  journal={Journal of Mathematical Physics},
  year={2021}
}
In this paper, we develop a Hamilton-Jacobi theory for forced Hamiltonian and Lagrangian systems. We study the complete solutions, particularize for Rayleigh systems and present some examples. Additionally, we present a method for the reduction and reconstruction of the Hamilton-Jacobi problem for forced Hamiltonian systems with symmetry. Furthermore, we consider the reduction of the Hamilton-Jacobi problem for a Čaplygin system to the Hamilton-Jacobi problem for a forced Lagrangian system. 
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8 Citations
Background Citations
3
Methods Citations
1

8 Citations

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References

SHOWING 1-10 OF 72 REFERENCES

The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

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…

Discrete Hamilton–Jacobi theory for systems with external forces

A discrete version of systems with Rayleigh-type forces is introduced, the equations of motion are obtained and the equivalence is characterized, and a Hamilton-Jacobi theory for forced discrete Hamiltonian systems is developed.

Geometric Hamilton–Jacobi theory for higher-order autonomous systems

The geometric framework for the Hamilton–Jacobi theory is used to study this theory in the background of higher-order mechanical systems, in both the Lagrangian and Hamiltonian formalisms. Thus, we…

Geometric Hamilton-Jacobi theory

The Hamilton–Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in…

A Hamilton-Jacobi theory for implicit differential systems

In this paper, we propose a geometric Hamilton-Jacobi theory for systems of implicit differential equations. In particular, we are interested in implicit Hamiltonian systems, described in terms of…

Geometric Hamilton-Jacobi theory for nonholonomic dynamical systems

The geometric formulation of Hamilton–Jacobi theory for systems with nonholonomic constraints is developed, following the ideas of the authors in previous papers. The relation between the solutions…

A Hamilton-Jacobi Theory for Singular Lagrangian Systems in the Skinner and Rusk Setting

We develop a Hamilton-Jacobi theory for singular lagrangian systems in the Skinner-Rusk formalism. Comparisons with the Hamilton-Jacobi problem in the lagrangian and hamiltonian settings are…

Hamilton-Jacobi theory for Hamiltonian systems with non-canonical symplectic structures

Hamilton-Jacobi theory and the evolution operator

We present a new setting of the geometric Hamilton-Jacobi theory by using the so-called time-evolution operator K. This new approach unifies both the Lagrangian and the Hamiltonian formulation of…

Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems

A generalization of the Hamilton-Jacobi theory to arbitrary dynamical systems, including non-Hamiltonian ones, is considered. The generalized Hamilton-Jacobi theory is constructed as a theory of…
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