Study of the Perihelion shift of Mercury in the Framework of Non-Commutative Classical Mechanics

Abstract

In this thesis, we have investigated the non-commutative Poisson algebra of classical observables Aα = (C∞ (R2d),*α ‚{,}a) obtained by a-deformation of the usual Poisson algebra A。 = (C° (R²d), ".",{}p) on Classical Phase Space (CPS). This a- deformation is characterized by two non-commutative parameters: 0 on the space sector of the CPS and ẞ on the momentum sector of the CPS. The resulting phase space becomes non-commutative and it is said Non-Commutative Classical Phase Space (NCCPS). It is shown that the two parameters 0 and ẞ are very tied, ẞ~0. In this context, all the apparatus of Hamilton formulation of Classical Mechanics (CM) is generalized towards a Hamiltonian Non-Commutative Classical Mechanics (NCCM). Furthermore, we propose another alternative way to obtain exactly the same non-commutative canonical equations of motion, by using some special transformation operated on the usual CPS without any a-deformation. In this framework, we explore the classical non-commutative effects resulting from the study of some basic examples, such as a free particle (U = 0), a harmonic oscillator (U=kx²) and a particle moving in a gravitational central potential (U = −).In particular, we have studied the possible perihelion shift per revolution due to non- commutativity, and compared it with that predicted by General Relativity with Schwarzschild metric, for the case of Mercury planet.