  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.
