The Explicit Gibbs-appell Equation and Generalized Inverse Forms

This paper develops an extended form of the Gibbs-Appell equation and shows that it is equivalent to the generalized inverse equation of motion. Both equations are shown to follow from Gauss's principle. An example to highlight the two equivalent, though different, equations of motion is provided. Conceptual differences between the equations, and differences in their practical application to physical situations are discussed. We also present in this paper a more general explicit generalized inverse equation of motion than has been hereto obtained. It is shown that many different forms of the generalized inverse equation of motion exist, all of which nonetheless are equivalent and uniquely determine the accelerations of a constrained mechanical system. The generalized inverse equation of motion retains its structure in any coordinate system. Introduction. The equations of motion, which are today commonly referred to as the Gibbs-Appell equations, were discovered independently by Gibbs (1879) and Appell (1899). The ability of the equations to describe the evolution of both holonomically and nonholonomically constrained systems without the use of Lagrange multipliers was considered to be a major leap in the development of analytical mechanics. These equations are considered by many to represent the simplest and most comprehensive form of the equations of motion so far discovered (Pars, 1972). Both Gibbs and Appell used the principle of virtual work to arrive at these equations. Most treatises on analytical dynamics, such as Whittaker (1904) and Pars (1972), derive these equations in the same vein. In this paper we show that the Gibbs-Appell equations can be derived in a more straightforward and explicit manner from Gauss's principle Received May 18, 1995. 1991 Mathematics Subject Classification. Primary 70F25, 70H35, 90C20. ©1998 Brown University 277 278 F. E. UDWADIA AND R. E. KALABA (Gauss, 1829). Besides its simplicity, there is an additional advantage of some consequence that follows from such a viewpoint: the equations can be extended to systems in which the nonholonomic constraints depend nonlinearly on the components of the velocities, thereby expanding the compass of their normal applicability, and the equations can be extended, rather simply, to include constraint equations which may not be independent of each other. We shall refer to these equations as the explicit extended Gibbs-Appell equations of motion. The unifying approach provided by Gauss's principle enables us to show that these extended Gibbs-Appell equations and the generalized inverse equations of motion (Udwadia and Kalaba, 1992), though different in form and appearance, are in strict equivalence. In this paper we also present the various generalized inverse forms of the equation of motion for constrained systems thereby generalizing some previous results. We provide the most general form of the generalized inverse equation of motion for constrained mechanical systems. We show that the Moore-Penrose inverse hereto used in the equation (Udwadia and Kalaba, 1992) is overly restrictive. What is needed is simply any {1,4}generalized inverse1 (Udwadia and Kalaba, 1996). Consider a system of n particles of masses m, > 0, i = 1,2, ...,n, in an inertial Cartesian coordinate space. The masses of these n particles will be assumed constant and known. In most of this paper, for expository purposes, we shall restrict ourselves to Cartesian coordinates. We shall see that all the underlying ideas can be illustrated without the unnecessary complications created by generalized coordinates. At the end, we will show how the results can be easily transcribed to apply to generalized coordinates. Let the 3n Cartesian coordinates of the n particles be described by the column vector x = [x\xi ■ ■ ■x3n}T. Rather than consider k given independent nonholonomic constraints, as is usual in deriving the Gibbs-Appell equation (see, for example, Pars (1972), Whittaker (1904), Neimark and Fufaev (1972)) of the form 3 n y]cjj(x,t)xj = bj(x,t), i = 1,2,... ,k, (1) j=l where each equation represents a linear constraint on the components of the velocities ij of the system, we shall allow more general constraints of the form <Pi{x, x, t) — 0, i 1,2,..., fc, (2) where each function ipj is considered to be at least C3 in its arguments. Furthermore, we do not restrict the k equations to be independent. Differentiating each constraint equation in the set (2) with respect to time once (twice, if the constraint is holonomic and ipj does not contain x), we obtain the set of equations A(x, x, t)x(t) = b(x, x, t) (3) 1 Given a real matrix X, its Moore-Penrose generalized inverse is the matrix Y that satisfies the conditions: (1) XYX — X, (2) YXY = Y, (3) XY is symmetric, and (4) YX is symmetric. Any matrix U that satisfies the first and the fourth of these four conditions is referred to as a {1,4}-generalized inverse of X\ similarly any matrix V that satisfies the first, the second, and the fourth of these conditions is called a {1, 2, 4}-inverse of X, etc. GIBBS-APPELL EQUATION AND GENERALIZED INVERSE FORMS 279 where the matrix A is k by 3n. When we say that a mechanical system is subjected to the constraint (3), we shall mean that the elements of the matrix A and the components of the vector b are known functions of their arguments. We note that the form of the constraint provided by (3) is more general than that usually used in the development of the equations of motion for constrained systems as may be found in Neimark and Fufaev (1972), Whittaker (1904), and Pars (1972). Let us say that at some time t, we know the position x and the velocity x of the constrained system. We now conceive of this n-particle constrained mechanical system in two steps. We start with an n-particle unconstrained system, subjected to a known impressed force F(x, x, t) = [F\F2 ■ ■ ■ F3n\l. By known force we mean that the functional dependence of each of the 3n components of the impressed force F on x, x, and t is explicitly known. By unconstrained we mean that the number of coordinates describing the system equals the number of degrees of freedom of the system. It is then a simple matter to obtain the acceleration a(t) = [a^ ■ • -asnp of the unconstrained system at time t by writing down Newton's law for this unconstrained system as Ma(t) = F(x,x,t), (4) where the 3n by 3n matrix M is a diagonal matrix in which the masses m, of the n particles appear in sets of threes along the diagonal. Since x and x are assumed known at time t, and the functional dependence of F on x, x, and t is assumed known, the right-hand side of Eq. (4) can be explicitly determined, and hence, a(t) = M~l F(x,x,t). (5) We next impose the constraint equation (3) on this unconstrained system. Thus the constrained system (at time t) may be thought of as being completely specified by x and x at time t, along with the four quantities M, F, A, and b (also evaluated at time t). We inquire how the acceleration x(t) of the resulting constrained system differs (at time t) from that of the known acceleration a(t) of the unconstrained system. Our aim is to determine the acceleration x at time t explicitly in terms of the four quantities M, F, A, and b that describe the constrained mechanical system at that time. We pursue this line of thinking by invoking Gauss's principle (Gauss, 1829) which states that the acceleration x(t) of the constrained system at each instant of time t is such as to minimize, at time t, the Gaussian G(x) = ^(x — a)TM(x — a) (6) over all possible 3n-vectors that satisfy the constraint set (3) at time t. Note that as with the vector F at time t, the elements of the vector b and those of the matrix A are known functions of x, x, and t; hence they are known at time t. In what follows, for the sake of brevity, we drop the arguments of the various vectors and matrices, unless their presence becomes conceptually helpful. We shall now develop the explicit extended Gibbs-Appell equations starting from this point. The explicit extended Gibbs-Appell equation. The basic idea that we shall follow is to convert the constrained minimization problem of Gauss into an unconstrained 280 F. E. UDWADIA and R. E. KALABA minimization problem by eliminating the dependent components of the acceleration vector x(t). Since the constrained acceleration vector x(t) at time t satisfies the equation set (3), the components of this acceleration vector are obviously not all linearly independent. Let us assume that the rank of the matrix A at time t is r < k. We shall assume that the first r columns of the matrix A are independent; if not, we can always re-label the components of the vector x(t) appropriately so that this occurs. We then partition the matrix A and the vector x(t) appropriately, so as to express the constraint equation (3) at time t as Ax = [Ae A\\ = b (7) where the matrices Ae and A\ are k by r and k by (3n — r) respectively. The subvectors xe and x\ have r and (3n — r) components respectively. The subscript "e" refers to the subvector that we shall eliminate, as we shall see below, and the subscript "I" refers to the subvector whose components may be taken to be independent. Equation (7) can be solved for the vector xe to yield xe = A+{b-AiXi), (8) where A* = the superscript "+" denoting the Moore-Penrose (MP) inverse of the matrix Ae. Note that the subvector x\ contains the components of x that are independent. We may likewise partition the matrix M = diag[Mee, Mu], where Mee and Mii are each r by r and (3n — r) by (3n — r) diagonal matrices respectively, and the vector a = [aj, aj ]T. The Gaussian G in expression (6) can now be written as G(x) — \{xlMeexe + xjMuXi) — a], Meexe — ajM\\X\ + |aTMa. (9) For convenience, we denote the first term on the right-hand side with the brackets—the so-called "kinetic energy of accelerations"—by the Gibbs function S(x), because it is a function of both the subvectors x\ and xe. Thus, Eq. (9) can be alternately expressed as G(x) — 7jS(x) — ajMeexe — ajM\\X\ + ^a} Ma. (10) Using Eq. (8), the subvector xe may be eliminated from Eq. (9) to yield G(xi) = \({b Aixi)1 A+l MeeA+ (b A\x\) + xjMux\) — ajMeeAg (b — A\X\) — ajM\\x\ + \aTMa. We again recognize the first member on the right-hand side with brackets as the "kinetic energy of accelerations", except that now it is expressed in terms of only the vector of independent accelerations, xi. We shall denote this quantity by <S(xi), the "script S" indicating that it is the quantity S = |iTMi expressed in terms of the independent vector xi. Equation (11) now becomes G(x i) = iS(ah) — MeeA^(b — A\X\) — aj M\\X\ + ^a1 Ma. (12) We have thus converted the constrained minimization problem stated in Gauss's principle to an unconstrained minimization problem. A necessary condition for the extremum of GIBBS-APPELL EQUATION AND GENERALIZED INVERSE FORMS 281 (11) with respect to the independent acceleration vector x\ is JS= 0. This yields (Mii + RrMeeR)x i — Rt MeeA^b = M\ia\ — R1 Meeae, (13) where we have denoted R = A+A\ ■ Noting our definition of <S(a5i), Eq. (13) can also be stated as g = Fl RTF, := P (14) where we have partitioned the known, impressed force vector F — [f„t f,t ]t of Eq. (4) into two subvectors F\ = M\\a\ and Fe = Meeae. The vectors Fe and Fy have r and (3n —r) components respectively. Equation (14) is the core that forms the Gibbs-Appell equation of motion for the constrained mechanical system. It results from enforcing the necessary condition for the expression in (12) to have an extremum. We have shown that Eq. (14) is true when the constraints are of the general form given by Eqs. (2) or (3); furthermore, these constraints need not be independent. To understand more fully the right-hand side of Eq. (14) (which we have defined as P), we use the extended D'Alembert principle (Udwadia and Kalaba, 1995) which says that a virtual displacement vector v compatible with the constraints (3) is any vector (Udwadia and Kalaba, 1995) that satisfies the relation Av = [Ae Ai] = 0, (15) fe yi. where we have, as before, partitioned the matrix A; the subvectors ve and v\ have r and (3n — r) components, respectively. This yields the relation ue = -A+A\v\ = — Ru\. (16) The virtual work done by the given impressed force then becomes ujFi + uerFe = vJ(F, RTFe) = vJP. (17) We note that the term in brackets in the above equation also shows up on the right-hand side of Eq. (14). Hence this right-hand side is obtained by determining the work done, vJP, by the impressed forces under the (independent) virtual displacements f\ that are compatible with the constraints. Yet Eq. (14) cannot, in general, stand alone, for though the expression G in Eq. (11) does not contain any components of the vector xe (since this vector was eliminated using Eq. (8)), it does contain, in general, components of the vectors xe and xe. To complete the system of differential equations one would therefore then need to append the equation of constraint (3), so that the complete set of equations would be formed by Eq. (14) and Eq. (3).2 Using Eqs. (13) and (3), this becomes Ae Aj 0 (Mu + RTMeeR)_ xi b Fi RTFe + RTMeeA+b (18) where the matrix R — /!+A[. Equation (18) may then be thought of as the explicit extended Gibbs-Appell equation in Cartesian coordinates, applicable to constraints (i) 2 Actually, it would suffice to include any r independent rows of the equation set (3) that will make the system of equations given in (18) complete. 282 F. E. UDWADIA and R. E. KALABA that are nonholonomic and nonlinear in the velocity components, and (ii) that are not necessarily independent. The generalized inverse forms of the equation of motion. An alternative approach to minimizing the Gaussian G (at time t) subject to the constraint (3) at time t is to directly solve the constrained minimization problem without first converting it to an unconstrained minimization problem. To do this, it would be more convenient to write the Gaussian in the form Gs(xs) \{xs as)T(xs as) (19) where the subscript s denotes the scaled quantities xs = M1/2i and as = M'/2a. The constraint equation (3) at time t can then be expressed as