Mullhaupt, Ph.

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Previous Up Next ArticleCitationsFrom References: 0From Reviews: 0MR2245201 (Review) 70E55 (70F20 70F25 70G45 70H03 70H45)Casey, James (1-CA-ME); O’Reilly, OliverM. (1-CA-ME)Geometrical derivation of Lagrange’s equations for a system of rigid bodies. (Englishsummary)Math.Mech. Solids 11 (2006), no. 4, 401–422.In this paper, based on the assumption that Euler’s two balance laws for a rigid body hold,Lagrange’s equations for a system of rigid bodies subject to general holonomic and non-holonomicconstraints are derived. One advantage of this assumption is that it extends directly to rigid continua(i.e. infinite-dimensional systems [J. Casey, Z. Angew.Math. Phys. 46 (1995), Special Issue, S805–S847; MR1359345 (96g:70024)]).The authors introduce a representation of the rigid bodies in a well-suited coordinate systemwhich consists of the center of mass xi (expressed in the current configuration) of each rigidbody i = 1, . . . , N , and 9 quantities for each rotation tensor Qi that is associated with a rigidbody i = 1, . . . , N . This leads to a 12N -dimensional vector space C2N representing the system.From the formulation of the two Euler balance laws in tensor form, and using the aforementionedcoordinates, a simple formulaΦ= mv˙ is obtained representing the dynamics. Of course, althoughv =(x˙1, . . . , x˙N, Q˙1, . . . , Q˙N)is of a simple structure, the termΦ is quite complicated:Φ= m(F1m1, . . . ,FNmN,12(M̂1 +W1sym)Q1(E10)−1, . . . ,12(M̂N +WNsym)QN(EN0 )−1),where Fi are the external forces, M̂i are the external torques andWisym is the symmetric part ofthe tensorWi = 2..QiEi0(Qi)T,withEi0 =∫ρi0Πi⊗ΠidV ithe Euler tensors (Πi = Xi−Xi is the relative position to the center of mass Xi expressed in afixed occupiable configuration, as opposed to xi). Based on this Newton-like formula Φ = mv˙,and using some kinematical arguments concerning the holonomic and nonholonomic constraints,the Lagrange equationsddt(∂T∂q˙γ)− ∂T∂qγ= QγwithQγ = [Φ,aγ] and aγ = ∂vqγare derived. The coordinate space is embedded in C12N and is 6N -dimensional. It is also shown tobe a Riemannian space with the length of a line-element written as ds2 = aαβ q˙αq˙βdt2. The paperthen gives explicit expressions ofQγ based on the expressions of the holonomic and nonholonomicconstraints.In the literature, there are both alternate derivations of Lagrange’s equations and expositions ofthe dynamics of rigid bodies using the Riemannian manifold setting. For example, regarding thewhole system as a simple point moving in an abstract space appeared in M. M. G. Ricci and T.Levi-Civita’s work [Math. Ann. 54 (1900), no. 1-2, 125–201; MR1511109; JFM 31.0297.01]. J. L.Synge [Philos. Trans. Roy. Soc. London Ser. A 226 (1926), 31–106; JFM 52.0798.05] presenteda similar exposition based on the opposite progression of thought as in the present paper, namelytaking the Lagrange equations as an assumption (E. T. Whittaker’s derivation [A treatise on theanalytical dynamics of particles and rigid bodies: With an introduction to the problem of threebodies, Cambridge Univ. Press, New York, 1959; MR0103613 (21 #2381)]) and concluding thatthe dynamics can be expressed as geodesics in a suitable Riemannian space (i.e. the metric shouldinvolve the kinetic energy). H. Hertz [The principles of mechanics, Translation by D. E. Jones andJ. T.Walley, Dover, NewYork, 1956;MR0077293 (17,1017f)] tried to bypass altogether the notionof force and energy so as to only rely on time, space, and mass. However, he admitted that themotion is along a stationary solution of the minimization problem for a metric. He admitted that tojustify this would simply involve the acceptance of both the law of inertia and Gauss’s principle ofleast constraint [H. Hertz, op. cit. (p. 28)] (the latter being a formulation of d’Alembert’s principleleading to a minimum problem [C. La´nczos, The variational principles of mechanics, Fourthedition, Univ. Toronto Press, Toronto, Ont., 1970; MR0431821 (55 #4815)]). Synge and A. Schildpresented an almost purely tensorial derivation of dynamics. However, once we look more closelyat it, in [J. L. Synge and A. Schild, Tensor Calculus, Univ. Toronto Press, Toronto, Ont., 1949;MR0033165 (11,400f); reprint, Dover, New York, 1978; MR0521984 (80b:53001)], Lagrange’sequations (5.521) (or (5.531)) are explicitly stated as a consequence of (2.431) and (2.438) which,in turn, follow from the equations of geodesics. The geodesic equations are a consequence ofthe calculus of variations presented earlier in the book, which is simply applied for solving thestationary problem involving the kinetic-energy Riemannian metric.Finally, there is the original derivation of Lagrange himself, which rests on the principle ofequilibrium stated as the principle of virtual speed (i.e. the principle of virtual displacement ofJohann Bernouilli). A system is in equilibrium whenever the forces are in inverse correspondencewith their potential speed (i.e. the speed that the system would gain should the equilibrium cease).D’Alembert rendered this static equilibrium principle to be extendable to the dynamic case,which then allowed Lagrange to find an appropriate analytical treatment unfolded in his famoustreatise [J. L. Lagrange,Me´chanique analitique, Desaint, Paris, 1788; reprint, ?d. Jacques Gabay,Paris, 1989]. Incidentally, d’Alembert presented the dynamics in his book [J. R. d’Alembert,Traite´ de dynamique, Reprint of the 1758 edition, E´d. Jacques Gabay, Sceaux, 1990; MR1451137(98a:01011)] without resorting to the origin of the concept of force based on three principles: (1)the law of inertia, (2) the composition of movements, and (3) the equilibrium principle.Hence, all derivations must start from unprovable statements that can only be justified by theiragreement with experiments (here, Euler’s balance equations are admitted). All these points ofview complement and enlighten each other, forging a deeper understanding of mechanics. Thispaper contributes nicely to this mission. From the computational point of view, the pros and consmust be addressed based on the specific problem at hand.Reviewed byPhilippe A.Mu¨llhauptc© Copyright American Mathematical Society 2007

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Review of "Geometrical derivation of Lagrange's equations for a system of rigid bodies."

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