A methodology for formulating dynamical equations in analytical mechanics based on the principle of energy conservation

TL;DR

This paper introduces a unified methodology for deriving dynamical equations in mechanics based on energy conservation, simplifying the understanding of classical principles and extending to complex systems like dissipative and multi-physical interactions.

Contribution

It presents a new energy-based framework that derives classical and modern mechanical equations, including D'Alembert's principle, from the fundamental law of energy conservation.

Findings

Unified derivation of Lagrange, Hamilton, and Hamilton-Jacobi equations from energy conservation

Application of the methodology to fluid mechanics for deriving Cauchy's first law

Enhanced understanding of the physical basis of mechanical principles

Abstract

In this work, a methodology is proposed for formulating general dynamical equations in mechanics under the umbrella of the principle of energy conservation. It is shown that Lagrange's equation, Hamilton's canonical equations, and Hamilton-Jacobi's equation are all formulated based on the principle of energy conservation with a simple energy conservation equation, i.e., the rate of kinetic and potential energy with time is equal to the rate of work with time done by external forces; while D'Alembert's principle is a special case of the law of the conservation of energy, with either the virtual displacements ('frozen' time) or the virtual displacement ('frozen' generalized coordinates). It is argued that all of the formulations for characterizing the dynamical behaviors of a system can be derived from the principle of energy conservation, and the principle of energy conservation is an underlying guide for constructing mechanics in a broad sense. The proposed methodology provides an efficient way to tackle the dynamical problems in general mechanics, including dissipation continuum systems, especially for those with multi-physical field interactions and couplings. It is pointed out that, on the contrary to the classical analytical mechanics, especially to existing Hamiltonian mechanics, the physics essences of Hamilton's variational principle, Lagrange's equation, and the Newtonian second law of motion, including their derivatives such as momentum and angular momentum conservations, are the consequences of the law of conservation of energy. In addition, our proposed methodology is easier to understand with clear physical meanings and can be used for explaining the existing mechanical principles or theorems. Finally, as an application example, the methodology is applied in fluid mechanics to derive Cauchy's first law of motion.