Applied Numerical Mathematics 10 (1992) 397-413 North-Holland 397 APNUM 349 QDAE methods for the numerical solution of Euler-Lagrange equations Licda R. Petzold zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Department of Computer Science, University of Minnesotti, Mivuzsapolk,MN 55455, USA Florian A. Potra Department of M athema ti ~5, lkcersity of Iowa, Iowa City, IA 52242, USA Abstract Petzold, L.R. and F.A. Potra, ODAE methods for the numerical solution of Euler-Lagrange equations, Applied Numerical Mathematics 10 (1992) 397-413. In a series of recent papers [5-71 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 extendmg 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. Keywords. Euler-Lagrange equations; differential-algebraic equations; overdetermined differential-algebraic equations; multibody systems. 1. Introduction In the last years we have witnessed a renewed interest in the field of kinematics and dynamics of multibody systems. Several methods for the automatic generation of the equations of motion of very complex systems have been developed. Some of them are already integrated in general-purpose computer codes (see e.g. [ll]). Also real-time simulators for mechanical systems have recently been built [4]. These developments have increased the need for fast and reliable numerical methods for solving the equations of motion of constrained mechanical systems. These equations, often called the Euler-Lagrange equations, are a classical example Correspondence to: L.R. Petzold, Department of Computer Science, University of Minnesota, 200 Union Street SE, Rm. 4-192, Minneapolis, MN 55455, USA. E-mail: petzold@cs.umn.edu. 016%9274/92/$05.00 0 1992 - Elsevier Science Publishers B.V. All rights reserved