On Jourdain's principle

Abstract This paper reexamines the effect of the validity, or the lack thereof, of the commutation rule for kinematically possible, or admissible, and virtual variations i.e. dδ(…) vs δd(…), on the form of Jourdain's principle (JP), in general quasivariables. It is shown that JP is independent of any such rule. A similar independence holds in the derivation of the equations of motion from Lagrange's principle.


INTRODUCTION
In recent years the diferential variational principle of Jourdain (JP), originally formulated by him in 1909 [1,2] as an "inte~ediatc" principle between those of Lagrange and Gauss/Gib~, and for long regarded by some as nothing more than an academic curiosity, has reemerged as a very versatile and powerful tool for the treatment of both finite and impulsive motion of constrained mechanical systems [3-81. Unfortunately, and inexplicably, this principle seems to be very little known among English-speaking engineers; its brief statement in Kane and Levinson [9] does not correspond to the classical principle as encountered in the literature.
The formulation of JP assumes, explicitly or implicitly, the satisfaction of the commlatation rule d(6r) -S(dr) = 0, (la) or (lb) where r = inertial position vector of a typical particle P, of the system under discussion S, v = inertial velocity of P, d(* -a) and 6(* --) = kine~aficalLy possible and virtual inertial differentials of (* --)_ One solidly gets the impression that equations (1) The following question then naturally arises: is JP also independent of commutation rules, like (l)? The purpose of this paper is to reexamine the theoretical foundations of JP and establish the necessity, or lack thereof, of the commutation rule (1). Section 2 summarizes the theory of LP and JP, while Section 3 investigates in detail the Jourdain form of the commutation rule.
The results are summarized in the concluding Section 4.

THE PRINCIPLES OF LAGRANGE AND JOURDAIN
Consider a material system S subject to constraints that are bilateral, ideal, and linear in the (particle and/or system) velocities; these constraints may be genuinely nonintegrable (nonholonomic), or they may be integrable (holonomic) disguised in kinematical, or ES 30:s8 135 differential, or velocity form. A typical particle P, of S, obeys Newton's "second law" where dm = mass of P, a = inertial acceleration of P, i.e. d*r/dt', r = inertial position vector and t = time; dF = total impressed force on P, and dR = total constraint reaction force on P.
The derivation of reactionless equations of motion is based on Lagrange's Principle (LP): (3), takes the customary Lagrangean form: where 6r represents the (inertial) we find: This leads to the following differential variational principle: the reactionless (or purely kinetic) equations of motion of S follow from the differential variational equation under the (fixed time and position) constraints  (7) and (9) coincide, under the set of assumptions (8). The question is: can this equivalence happen under all circumstances, i.e. without any special assumptions like (lo), that is, even if 6*r = (6r)' -6(i) # 0.
The answer to this requires utilization of the most general expression for 6*r. We now turn our attention to this problem.

THE TRANSITIVITY (OR COMMUTATIVITY) RELATIONS
In terms of the fundamental, generally nonholonomic, particle and system basis vectors {ai, i=m+l,..., n; m = number of velocity constraints, II = number of (initially independent) coordinates q-see e.g. [lo-12]}, the particle virtual displacement Sr is expressed as the following linear combination: If we are to obtain something nontriviaE from (6-8), then we should not assume that from (12) it follows that (dr)' = 0, and/or (S&)' = 0. Let's now take for algebraic simplicity, but without loss of generality, the case of scleronomic systems, i.e. stationary constraints. Then Ui [last of (ll) despite (12).