Abstract
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.
Highlights
The classical formulation [1, 2, 26] of the Hamilton-Jacobi problem for a Hamiltonian system on T ∗Q consists in looking for a function S on Q × R, called the principal function, such that ∂S +H ∂t qi, ∂S ∂qi = 0, (1)where H : T ∗Q → R is the Hamiltonian function
Despite the difficulties to solve a partial differential equation instead of a system of ordinary differential equations, i.e., to solve the Hamilton-Jacobi equation instead of Hamilton equations, the Hamilton-Jacobi theory provides a remarkably powerful method to integrate the dynamics of many Hamiltonian systems
4 Hamilton-Jacobi theory for Lagrangian systems with external forces. As it has been seen in the previous sections, the natural framework for the Hamilton-Jacobi theory is the Hamiltonian formalism on the cotangent bundle
Summary
The classical formulation [1, 2, 26] of the Hamilton-Jacobi problem for a Hamiltonian system on T ∗Q consists in looking for a function S on Q × R, called the principal function ( known as the generating function), such that. In this paper we develop a Hamilton-Jacobi theory for systems with external forces. Mechanical systems with external forces (so-called forced systems) appear commonly in engineering and describe certain physical systems with dissipation [21, 43, 52] They emerge after a process of reduction in a nonholonomic Čaplygin system [5, 8, 24, 31, 33, 56].
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