13.4 Advanced Mechanics Concepts

Newton-Euler mechanics has the advantage that it starts with very basic principles, but it has frustrating restrictions that make modeling more difficult for complicated mechanical systems. One of the main limitations is that all laws must be expressed in terms of an inertial frame with orthogonal axes. This section introduces the basic ideas of Lagrangian and Hamiltonian mechanics, which remove these restrictions by reducing mechanics to finding an optimal path using any coordinate neighborhood of the C-space. The optimality criterion is expressed in terms of energy. The resulting techniques can be applied on any coordinate neighborhood of a smooth manifold. The Lagrangian formulation is usually best for determining the motions of one or more bodies. Section 13.4.1 introduces the basic Lagrangian concepts based on the calculus of variations. Section 13.4.2 presents a general form of the Euler-Lagrange equations, which is useful for determining the motions of numerous dynamical systems, including chains of bodies. The Lagrangian is also convenient for systems that involve additional differential constraints, such as friction or rolling wheels. These cases are briefly covered in Section 13.4.3. The Hamiltonian formulation in Section 13.4.4 is based on a special phase space and provides an alternative to the Lagrangian formulation. The technique generalizes to Pontryagin's minimum principle, a powerful optimal control technique that is covered in Section 15.2.3.