Energy Conservation Equations of Motion

TL;DR

This paper introduces an alternative method for deriving equations of motion in mechanics that explicitly conserves energy, using a function-based approach without relying on traditional variational principles.

Contribution

It presents a novel derivation of motion equations that maintain energy conservation through a function-based approach, incorporating gyroscopic forces and avoiding variational principles.

Findings

Derived generalized equations of motion conserving energy.

Extended Lagrange and Hamilton equations without variational principles.

Applied new technique to specific mechanical systems.

Abstract

A conventional derivation of motion equations in mechanics and field equations in field theory is based on the principle of least action with a proper Lagrangian. With a time-independent Lagrangian, a function of coordinates and velocities that is called energy is constant. This paper presents an alternative approach, namely derivation of a general form of equations of motion that keep the system energy, expressed as a function of generalized coordinates and corresponding velocities, constant. These are Lagrange equations with addition of gyroscopic forces. The important fact, that the energy is defined as the function on the tangent bundle of configuration manifold, is used explicitly for the derivation. The Lagrangian is derived from a known energy function. A development of generalized Hamilton and Lagrange equations without the use of variational principles is proposed. The use of new technique is applied to derivation of some equations.