A generalization of the Lagrange–Hamilton formalism with application to non-conservative systems and the quantum to classical transition

This work has two aims. The first is to develop a Lagrange–Hamilton framework for the analysis of multi-degree-of-freedom nonlinear systems in which non-conservative effects are included in the variational principle of least action from the outset. The framework is a generalization of the Bateman approach in which a set of adjoint coordinates is introduced. A function termed the M-function is introduced as the Fourier transform over the momenta of the joint probability density function (JPDF) of the displacements and momenta, and it is shown that for statistical systems, this function can be written as an expectation involving the new principle function and a general dimensional constant ℏ. This leads to a concise derivation of the Fokker–Planck–Kolmogorov equation. It is found that the equation governing the M-function can be expressed in terms of the new Hamiltonian by replacing momenta by differential operators, meaning that the function satisfies the same equation as the quantum wave function. This gives rise to the second aim of this work: to explore relations between the developed classical framework and quantum mechanics. It is shown that for an undamped linear system, the solution of the M-function equation yields the response JPDF as a sum of Wigner functions. This classical analysis leads to a number of well-known results from quantum mechanics as ℏ → 0, and the extension of this result to nonlinear systems is discussed. The quantum wave function associated with the Hamiltonian is then considered, and the relevance of this function to the physical system is discussed.