Elementary Hyper-Trigonometry of the Particle Triangle

摘要

In the hyperspherical treatment of three-body rotational states we introduced earlier two special subsets of body-fixed hyperspherical harmonics. With their help elementary trigonometry-like identities are derived, interconnecting the familiar geometric angles of the triangle, the hyperspherical angles which intermix the geometric angles and the particle masses, and the so-called angles of the kinematic rotation which are purely mass-dependent. One set represents stringent constraints to be observed in any calculation using at the same time hyperspherical coordinates belonging to different fragmentations of the particles. The other one supports the conjecture of the nonexistence of simple coordinate transformation laws between hyperspherical angles belonging to different fragmentations.