Non-Linear Semi-Quantum Hamiltonians and Its Associated Lie Algebras

We show that the non-linear semi-quantum Hamiltonians which may be expressed as ( ) ( ) ˆ ˆ ∑ N j j j p H a q p O V q m 2 1 , 2 = = + + (where ˆ j O is the set of generators of some Lie algebra and ( ) q p , are the classical conjugated canonical variables) always close a partial semi Lie algebra under commutation and, because of this, it is always possible to integrate the mean values of the quantum degrees of freedom of the semi-quantum non-linear system in the fashion: ( ) ( ) { } ˆ ˆ ˆ j j O t Tr t O = ρ (where ρ̂ is the Maximum Entropy Principle density operator) and, so, these kind of Hamiltonians always have associated dynamic invariants which are expressed in terms of the quantum degrees of freedom’s mean values. Those invariants are useful to characterize the kind of dynamics (regular or irregular) the system displays given that they can be fixed by means of the initial conditions imposed on the semi-quantum non-linear system.