Abstract
In mechanics, both classical and quantum, one studies the profound interaction between two types of energy, namely, the kinetic energy and the potential energy. The former can be organized as the kinematic metric on the configuration space while the latter can be represented by a suitable potential function, such as the Newtonian potential in celestial mechanics and the Coulomb potential in quantum mechanics of atomic and molecular physics. In this paper, the author studies the kinematic geometry of n-body systems. The main results are (i) the introduction of a canonical coordinate system which reveals the total amount of kinematic symmetry by an SO(3)× O(n-1) action in such a canonical coordinate representation; (ii) an in depth analysis of the above kinematic system both in the setting of classical invariant theory and by the technique of equivariant Riemannian geometry; (iii) a remarkably simple formula for the potential function in such a canonical coordinate system which reveals the well-fitting between the kinematic symmetry and the potential energy.
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