Jacobi multipliers and Hamel’s formalism

Abstract

In this work we establish the relation between the Jacobi last multiplier, which is a geometrical tool in the solution of problems in mechanics and that provides Lagrangian descriptions and constants of motion for second-order ordinary differential equations, and nonholonomic Lagrangian mechanics where the dynamics is determined by Hamel’s equations.

Jacobi multipliers and Hojman symmetry

“…i(Γ)ω L = dE L , which implies L Γ ω L = 0 (see e.g [16]), we know (see e.g. [15] and references therein) that a particular Jacobi multiplier for Γ with respect to the volume form Ω is given by the determinant of the Hessian matrix W in the velocities, with elements W ij = ∂ 2 L/∂v i ∂v j . Actually, from L Γ ω L = 0, we see that (ω L ) ∧n is an invariant volume under Γ, and the proportionality factor of (ω L ) ∧n and Ω is det W .…”

“…and therefore the condition for R to be a Jacobi multiplier (see [15] and references therein for more details) becomes…”

“…i(Γ)ω L = dE L (see e.g [16]), we know (see e.g. [15] and references therein) that a particular Jacobi multiplier for given by the determinant of the Hessian matrix in the velocities, with elements W ij = ∂ 2 L/∂v i ∂v j . Therefore, the constant of motion obtained in [14] is just a particular case of the expression (2.5) for R equal to the determinant of the Hessian matrix W .…”

Jacobi Multipliers in Integrability and the Inverse Problem of Mechanics

“…We can give intrinsic proof of the direct part of Theorem 3 (see, e.g., [55]). Recall first that in the geometric approach to Lagrangian mechanics, we have summarised in Section 5 when one starts with a 1-dimensional configuration space, Q ≡ R, and hence, TQ ≡ R × R, and the expressions of S and θ L in local coordinates (q, v) are…”

“…Our emphasis in the theory of Jacobi multipliers was in its applications to the inverse problem of mechanics, mainly for the interest in the Lagrangian formulation as a first step in the process of obtaining an appropriate Hamiltonian formulation for its quantisation, but we do not forget that the concept of the Jacobi (last) multiplier was introduced to integrate a given system by quadratures. More specifically, the result obtained by Jacobi can be summarised as follows (see [55] for a recent geometric presentation):…”