Energy Conservation Equations of Motion

The conventional derivation of the equations of motion in mechanics and the field equations in field theory is based on the principle of least action with a proper Lagrangian. For a time-independent Lagrangian, the function of coordinates and velocities, called energy, is constant. This paper presents a different approach – derivation of the general form of the equations of motion that ensure the constancy of the energy given as a function of generalized coordinates and corresponding velocities. It is shown that these are the Lagrange equations with additional gyroscopic forces. The derivation explicitly uses the important fact that the energy is given as a function on the tangent bundle of the configuration manifold. The Lagrangian is derived from a known energy function. It is proposed to develop generalized Hamilton and Lagrange equations without using variational principles. The new technique is used to derive some equations.