Geometrical Derivation of Lagrange's Equations for a System of Rigid Bodies. (english Summary)

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-holonomic constraints 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 system which consists of the center of mass xi (expressed in the current configuration) of each rigid body i = 1, . . . , N , and 9 quantities for each rotation tensor Qi that is associated with a rigid body 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 aforementioned coordinates, a simple formulaΦ= mv̇ is obtained representing the dynamics. Of course, although v = ( ẋ1, . . . , ẋN, Q̇1, . . . , Q̇N ) is of a simple structure, the termΦ is quite complicated: