Progress in classical and quantum variational principles

Abstract

We review the development and practical uses of a generalized Maupertuis least action principle in classical mechanics, in which the action is varied under the constraint of fixed mean energy for the trial trajectory. The original Maupertuis (Euler-Lagrange) principle constrains the energy at every point along the trajectory. The generalized Maupertuis principle is equivalent to Hamilton's principle. Reciprocal principles are also derived for both the generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis Principle is the classical limit of Schrödinger's variational principle of wave mechanics, and is also very useful to solve practical problems in both classical and semiclassical mechanics, in complete analogy with the quantum Rayleigh-Ritz method. Classical, semiclassical and quantum variational calculations are carried out for a number of systems, and the results are compared. Pedagogical as well as research problems are used as examples, which include nonconservative as well as relativistic systems.

Stochastic modelling of the eradication of the HIV-1 infection by stimulation of latently infected cells in patients under highly active anti-retroviral therapy

“…Moreover, the classical Hamilton's principle can be seen as the classical limit of the quantum stationary phase condition for constructive interference. We exploit these relations below, see Goldstein (1980), Landau and Lifshitz (1976), Gray and Taylor (2007) and Gray et al (2004).…”

The Extrema of an Action Principle for Dissipative Mechanical Systems

“…This variation of dissipative energy at a previous moment may produce variation of the energy at a later moment if the conservation of energy of the total system is taken into account as a constraint of the variational calculus. However, energy conservation is not a constraint in this version of LAP using the action defined with Lagrangian [2,9], meaning that the consideration of the variation at time τ to derive the equation of motion of a later moment t is questionable.…”

Sub-Quantum Thermodynamics as a Basis of Emergent Quantum Mechanics

“…Gray et al, in an extensive survey of variational principles [38], provide a so-called -unconstrained Maupertius principle‖ (UMP) for nonconservative systems, which relates the variations of a mean energy E , of action S , and of the travel time t , such that the Lagrange multipliers are the true travel time, and the difference between energy and mean energy of the true trajectory at time t , the latter being    …”