On constrained motion

The general explicit equations of motion for constrained discrete dynamical systems are obtained. These new equations lead to a simple and new fundamental view of constrained motion where the forces of constraint may be ideal and/or non-ideal. 2004 Elsevier Inc. All rights reserved. The general problem of obtaining the equations of motion of a constrained discrete mechanical system is one of the central issues in analytical dynamics. While it was formulated at least as far back as Lagrange, the determination of the explicit equations of motion, even within the restricted compass of lagrangian dynamics, has been a major hurdle. The Lagrange multiplier method relies on problem specific approaches to the determination of the multipliers which are often difficult to obtain for systems with a large number of degrees of freedom and many non-integrable constraints. Formulations offered by Gibbs, Volterra, Appell, Boltzmann, and Poincare require a felicitous choice of problem specific quasi-coordinates and suffer from similar problems in dealing with systems with large numbers of degrees of freedom and many 0096-3003/$ see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.06.039 E-mail address: [email protected] 314 F.E. Udwadia / Appl. Math. Comput. 164 (2005) 313–320 non-integrable constraints. Dirac offers a formulation for hamiltonian systems with singular lagrangians where the constraints do not explicitly depend on time. The explicit equations of motion obtained by Udwadia and Kalaba [4] provide a new and different perspective on constrained motion. They introduce the notion of generalized inverses in the description of such motion and, through their use, obtain a simple and general explicit equation of motion for constrained mechanical systems without the use of, or any need for, the notion of Lagrange multipliers. These equations are applicable to general mechanical systems and include situations where the constraints may be: (1) nonlinear functions of the velocities, (2) explicitly dependent on time, and, (3) functionally dependent. However, they deal only with systems where the constraints are ideal and satisfy D Alembert s principle. This principle says that the motion of a constrained mechanical system occurs in such a way that at every instant of time the sum total of the work done under virtual displacements by the forces of constraint is zero. In this paper, we extend these results along two directions. First, we extend D Alembert s Principle to include constraints that may be, in general, non-ideal so that the forces of constraint may therefore do positive, negative, or zero work under virtual displacements at any given instant of time during the motion of the constrained system. We thus expand lagrangian mechanics to include non-ideal constraint forces within its compass. We then obtain the general, explicit equations of motion for such systems. Second, we use a different kind of generalized inverse that makes the explicit equations of motion much simpler and leads to deeper insights into the way Nature seems to work. With the help of these equations we provide a new fundamental principle governing the motion of constrained mechanical systems. Consider first an unconstrained, discrete dynamical system whose configuration is described by the n generalized coordinates q = [q1,q2, . . .,qn] . By unconstrained we mean that the components, _ qi, of the velocity of the system can be independently assigned at any given initial time, say, t = t0. Its equation of motion can be obtained, using newtonian or lagrangian mechanics, by the relation Mðq; tÞ€q 1⁄4 Qðq; _ q; tÞ; ð1Þ where the n by n matrix M is symmetric and positive definite. The matrix M (q, t) and the generalized force n-vector (n by 1 matrix), Qðq; _ q; tÞ, are known. In this paper, by known we shall mean known functions of their arguments. The generalized acceleration of the unconstrained system, which we denote by the n-vector a, is then given by €q 1⁄4 M Q 1⁄4 aðq; _ q; tÞ: ð2Þ F.E. Udwadia / Appl. Math. Comput. 164 (2005) 313–320 315 We next suppose that the system is subjected to h holonomic constraints of the form uiðq; tÞ 1⁄4 0 i 1⁄4 1; 2; . . . ; h ð3Þ and m h nonholonomic constraints of the form uiðq; _ q; tÞ 1⁄4 0; i 1⁄4 hþ 1; hþ 2; . . . ;m: ð4Þ The initial conditions q0 = q(t = t0) and _ q0 1⁄4 _ qðt 1⁄4 t0Þ are assumed to satisfy these constraints so that ui (q0, t0) = 0, i = 1,2, . . .,h, and uiðq0; _ q0; t0Þ 1⁄4 0, i = h + 1,h + 2, . . . ,m. We note that the constraints may be explicit functions of time, and the nonholonomic constraints may be nonlinear in the velocity components _ qi. Under the assumption of sufficient smoothness, we can differentiate equations (3) twice with respect to time and Eq. (4) once with respect to time to obtain the consistent equation set Aðq; _ q; tÞ€q 1⁄4 bðq; _ q; tÞ; ð5Þ where the constraint matrix, A, is a known m by n matrix and b is a known m-vector. It is important to note that for a given set of initial conditions, equation set (5) is equivalent to Eqs. (3) and (4), which can be obtained by appropriately integrating the set (5). The presence of the constraints (5) imposes additional forces of constraint on the system that alter its acceleration so that the explicit equation of motion of the constrained system becomes M€q 1⁄4 Qðq; _ q; tÞ þ Qðq; _ q; tÞ: ð6Þ The additional term, Q, on the right-hand side arises by virtue of the imposed constraints prescribed by Eq. (5). We begin by generalizing D Alembert s Principle to include forces of constraint that may do positive, negative, or zero work under virtual displacements. We assume that for any virtual displacement vector, v (t), the total work done, W = vQ, by the forces of constraint at each instant of time t, is prescribed (for the given, specific dynamical system under consideration) through the specification of a known n-vector Cðq; _ q; tÞ such that