Generalized Hamiltonian forms of dissipative mechanical systems via a unified approach

Jacobi Multipliers in Integrability and the Inverse Problem of Mechanics

“…A method for the search of a Hamiltonian formulation of a given second-order differential equation has recently been developed in [27]. Of course, such a Hamiltonian formulation is not uniquely determined.…”

“…The method proposed in [27] is an ad hoc method, and it is not fully justified because it is not clear how to use the implicit function theorem in the formula preceding (2.4) in [27] to find the differentiable function I(x, x ). However, the theory of Jacobi multipliers [22,23,29,30], which was developed to integrate a system of first-order differential equations by quadratures when some first-integrals and a Jacobi multiplier are known, can be extended to systems of second-order differential equations from a geometric perspective because if a system of first-order differential equation is replaced by a vector field in a manifold, whose integral curves are the solution of the system, in a similar way, a given system of second-order differential equations corresponds to a vector field on its tangent bundle of a special class of vector fields.…”

“…In the particular case of a second-order differential equation, the theory provides a solution for the so-called inverse problem of mechanics because it asserts the existence of a Lagrangian description for the given equation. The rôle of the function I(x, x ) appearing in [27] is that of a first-integral (see (2.2) in [27]), which may be used to determine a Jacobi multiplier and then, as a consequence, a Lagrangian formulation. This is a well-established procedure, which should replace the incomplete derivation of Section 2 in [27].…”

“…The rôle of the function I(x, x ) appearing in [27] is that of a first-integral (see (2.2) in [27]), which may be used to determine a Jacobi multiplier and then, as a consequence, a Lagrangian formulation. This is a well-established procedure, which should replace the incomplete derivation of Section 2 in [27].…”

“…The concept of Jacobi multipliers is introduced in Section 2, where the particular case of a two-dimensional autonomous system of first-order ordinary differential equations illustrates the theory. Following with these lower dimensional instances, Jacobi multipliers for the simple case of an autonomous second-order differential equation and the relation with the inverse problem is analysed in Section 3, and, as an important application in physics, the theory will be used to re-derive, in Section 4, the explicit form of a Lagrangian for a dynamics with a given constant of the motion proposed in [27,[34][35][36][37][38]. In order for the paper be self-contained, we have added, in Section 5, a short summary with the fundamental results of the theory of symplectic manifolds and its applications in the geometric approach to Hamiltonian and Lagrangian mechanics.…”