Hyperangular momenta and energy partitions in multidimensional many-particle classical mechanics: The invariance approach to cluster dynamics

Rigorous and complete definitions of two partitions and one expansion for the kinetic energy of a general N-particle classical system are given. Our recent work, which also provides examples of applications to the molecular dynamics of nanoaggregates, based on computer programs formulated on the basis of the theory presented here, is extended to cover arbitrary physical space dimensions. The partitions and the expansion are in terms of quantities conceived to be instantaneous phase-space invariants-a far-reaching generalization of integrals of the motion. These quantities are introduced setting out as starting points the position matrix Z of the system and the time derivative of Z. In the simplest case, the matrix Z contains the mass-scaled Cartesian coordinates of the N particles. From the position matrix, the kinematic rotations naturally arise through orthogonal transformations, as a concept dual to the ordinary physical rotations. The physical meaning of each partition (expansion) term is clearly described and emphasized, and formulas for the various quantities are provided as well as inequalities among them. Proofs are presented making extensive use of the singular value decomposition (SVD) of matrices and of the signed SVD, an extended version overcoming possible singularities for particular values of N.