Lagrangian mechanics of overparameterized systems

Common methods to model the motion of mechanical systems include minimal coordinates and redundant coordinates. Classical presentations of Lagrangian mechanics have focused on these coordinate descriptions. However, sometimes an alternative type of description using overparameterized coordinates is more suitable, in order to avoid coordinate singularities or to simplify the equations of motion. Examples in astrodynamics include the Euler parameters for attitude dynamics and the orbital elements of the two-body problem. This paper investigates issues involved with these descriptions. For these system descriptions the mass matrix is singular, and Lagrange’s equations do not provide a unique solution for the generalized accelerations. A set of selective constraints are applied to determine a unique solution. One approach for selecting the constraints is to select holonomic constraints. These constraints serve a fundamentally different role from the redundant-coordinate constraints, which are applied while deriving the equations of motion. Along with these constraints, initial-condition constraints on the generalized coordinates and generalized velocities are needed.