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.

References in corpus (6)

• Raman Spectroscopic Investigation of H2, HD, and D2 Physisorption on Ropes of Single-Walled, Carbon Nanotubes
• Chaos in a Relativistic 3-body Self-Gravitating System
• Hindered rotation of H2 adsorbed interstitially in nanotube bundles
• Exploring a rheonomic system
• Symmetry Decomposition of Potentials with Channels
• The Linear Baryon

Cited by in corpus (8)

• The Vacuum Fluctuation Theorem: Exact Schroedinger Equation via Nonequilibrium Thermodynamics
• Variational Principle of Bogoliubov and Generalized Mean Fields in Many-Particle Interacting Systems
• NVU dynamics. I. Geodesic motion on the constant-potential-energy hypersurface
• The extrema of an action principle for dissipative mechanical systems
• Developing an Action Concept Inventory
• Path probability distribution of stochastic motion of non dissipative systems: a classical analog of Feynman factor of path integral
• A Generalization of Fermat's Principle for Classical and Quantum Systems
• When action is not least for systems with action-dependent Lagrangians