ODAE METHODS FOR THE NUMERICAL-SOLUTION OF EULER-LAGRANGE EQUATIONS

In a series of recent papers [5-7] it is shown that many of the numerical methods for solving Euler-Lagrange equations can be viewed as generalized solutions of an overdetermined differential-algebraic equation (ODAE). For a model with linear constraints, it is shown that if the ODAE is solved by a certain iteration technique in conjunction with BDF discretization, then the corresponding numerical solution coincides with the numerical solution obtained by applying the same BDF scheme to a state-space form of the original Euler-Lagrange equation. In addition, it is shown that using this iteration to solve the ODAE is equivalent to numerically solving a DAE which arises from extending the mechanical system by adding derivatives of the constraints and additional Lagrange multipliers to ensure that those constraints are satisfied. In this paper we examine these equivalences to determine to what extent they continue to hold for problems which are more general than the linear model.