Hamilton’s action principle, Lagrangian mechanics, and Hamiltonian mechanics, all exploit the concept of action which is a single, invariant, quantity. These algebraic formulations of mechanics all are based on energy, which is a scalar quantity, and thus these formulations are easier to handle than the vector concept of force employed in Newtonian mechanics. Algebraic formulations provide a powerful and elegant approach to understand and develop the equations of motion of systems in nature. Chapters \(6 − 9\) applied variational principles to Hamilton’s action principle which led to the Lagrangian, and Hamiltonian formulations that simplify determination of the equations of motion for systems in classical mechanics.
A conservative force has the property that the total work done moving between two points is independent of the taken path. That is, a conservative force is time symmetric and can be expressed in terms of the gradient of a scalar potential \(V\). Hamilton’s action principle implicitly assumes that the system is conservative for those degrees of freedom that are built into the definition of the action, and the related Lagrangian, and Hamiltonian. The focus of this chapter is to discuss the origins of nonconservative motion and how it can be handled in algebraic mechanics.