Implicit Numerical Integration for Euler-Lagrange Equations via Tangent Space Parametrization∗

A new class of methods for solving the equations of motion of constrained mechanical system dynamics is presented. The tangent space local parametrization is used to form an index one system of mixed differential-algebraic equations (DAEs) that describes the constrained motion. Implicit numerical integration formulas are applied to the reduced ordinary differential equations (ODEs) in the tangent space. The resulting system of nonlinear equations is solved by Broyden's method. In the framework presented, the computational complexity of solving the implicit integration equations for the reduced ODE is about same as that of some explicit integration methods for the reduced ODE. In fact, in some cases the implicit method for solving the Euler-Lagrange equations can be implemented more efficiently than the explicit scheme.