# 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.

## 8 Citations

### Discrete Hamilton–Jacobi theory for systems with external forces

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2022

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.

### Hamilton–Jacobi theory for contact systems: autonomous and non-autonomous for the in time-independent contact Hamiltonian

- Mathematics
- 2022

In this paper we obtain two Hamilton–Jacobi equations for time dependent contact Hamiltonian systems. In these systems there is a dissipation parameter and the fact of obtaining two equations reﬂects…

### Hamilton–Jacobi theory and integrability for autonomous and non-autonomous contact systems

- MathematicsJournal of Geometry and Physics
- 2023

### Geometric Hamilton-Jacobi theory and integrability for nonholonomic and forced hybrid systems

- Mathematics
- 2022

. A hybrid system is a system whose dynamics are controlled by a mixture of both continuous and discrete transitions. The geometric framework for the Hamilton–Jacobi theory is developed to study this…

### The geometry of Rayleigh dissipation

- Mathematics
- 2021

This paper is devoted to the study of mechanical systems subjected to external forces in the framework of symplectic geometry. We obtain a Noether’s theorem for Lagrangian systems with external…

### Symplectic and Cosymplectic Reduction for simple hybrid forced mechanical systems with symmetries

- Physics, MathematicsSSRN Electronic Journal
- 2022

This paper gives general conditions on whether it is possible to perform symmetry reduction for simple hybrid Hamiltonian and Lagrangian systems subject to non-conservative external forces, as well as time-dependent external forces.

### Generalized hybrid momentum maps and reduction by symmetries of forced mechanical systems with inelastic collisions

- Physics
- 2021

This paper discusses reduction by symmetries for autonomous and non-autonomous forced mechanical systems with inelastic collisions. In particular, we introduce the notion of generalized hybrid…

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

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2022

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…

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- 2022

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.

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