Lagrangian Mechanics with Time as a Coordinate

Abstract

Abstract In traditional analytical mechanics, it is assumed that the system traces out a path (one-dimensional curved line) in a D dimensional configuration space that is defined by writing each of the generalised coordinates as a function of the absolute, Newtonian time. The extended Lagrangian theory combines the traditional Lagrange equations and generalised energy theorem into a single set of equations, and restores the symmetry of the mathematical system. The traditional Lagrangian methods are analogous to the ‘coordinate parametric method’ in the calculus of variations and to the recommended ‘general parametric method’ presented earlier. This chapter discusses Lagrangian mechanics in which time is treated as a coordinate, extended momenta, invariance under change of parameter, change of generalised coordinates, redundancy of the extended Lagrange equations, forces of constraint, and reduced Lagrangians with time as a coordinate.