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In this work, a methodology is proposed for formulating general dynamicalequations in mechanics under the umbrella of the principle of energyconservation. It is shown that Lagrange's equation, Hamilton's canonicalequations, and Hamilton-Jacobi's equation are all formulated based on theprinciple of energy conservation with a simple energy conservation equation,i.e., the rate of kinetic and potential energy with time is equal to the rateof work with time done by external forces; while D'Alembert's principle is aspecial case of the law of the conservation of energy, with either the virtualdisplacements ('frozen' time) or the virtual displacement ('frozen' generalizedcoordinates). It is argued that all of the formulations for characterizing thedynamical behaviors of a system can be derived from the principle of energyconservation, and the principle of energy conservation is an underlying guidefor constructing mechanics in a broad sense. The proposed methodology providesan efficient way to tackle the dynamical problems in general mechanics,including dissipation continuum systems, especially for those withmulti-physical field interactions and couplings. It is pointed out that, on thecontrary to the classical analytical mechanics, especially to existingHamiltonian mechanics, the physics essences of Hamilton's variationalprinciple, Lagrange's equation, and the Newtonian second law of motion,including their derivatives such as momentum and angular momentumconservations, are the consequences of the law of conservation of energy. Inaddition, our proposed methodology is easier to understand with clear physicalmeanings and can be used for explaining the existing mechanical principles ortheorems. Finally, as an application example, the methodology is applied influid mechanics to derive Cauchy's first law of motion.

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