A Basic Power Decomposition in Lagrangian Mechanics

Abstract

In Lagrangian mechanics, under certain conditions, the Jacobi energy integral exists and plays a fundamental role (see 123456). More generally, when Jacobi’s integral does not exist, it is still possible to gain useful engineering information from a consideration of power versus rate-of-energy relations. In the present note, we are concerned with a system of N (≥1) particles subject to general holonomic and non-holonomic constraints. The unconstrained physical system may be represented by an abstract particle P in a 3N-dimensional Euclidean configuration space. In the presence of holonomic constraints, the motion of P is confined to a submanifold M whose dimension is equal to the number of generalized coordinates needed to describe the system. In general, M moves through configuration space and may also change its shape with time.1 Now, the velocity v of P can always be expressed as the vector sum of two components v ′ and v ″ such that v ″ is the velocity of the point A (say) of M that P occupies at time t, and v ′ is the velocity of P relative to A. It will be shown that when this decomposition is employed, the corresponding portions P ′ and P ″ of the total power P of the forces acting on the particles, can be expressed as time derivatives (partial and total) of portions of the kinetic energy.2 These expressions furnish a convenient means for calculating the power expended in moving the manifold M, and in moving P relative to M. This is particularly useful in the former case, because the constraint forces that move M would have been eliminated from the Lagrangian analysis.