Exact Results in Quantum Theory & Gravity

A Lie system is a non-autonomous first-order ordinary differential equation admitting a (generally nonlinear) superposition rule, i.e. a function allowing us to describe the general solution of the Lie system in terms of a finite-family of particular solutions and a set of constants. Non-autonomous systems of first-order ordinary linear differential equations and matrix Riccati equations are the most simple examples of Lie systems.In the context of the Floquet theory, using a variation of parameter argument, we show that the logarithm of the monodromy of a real periodic Lie system with appropriate properties admits a splitting into two parts called dynamic and geometric phases. The dynamic phase is intrinsic and linked to the Hamiltonian of a periodic linear Euler system on the co-algebra. The geometric phase is represented as a surface integral of the symplectic form of a co-adjoint orbit. Results can be applied to the study of geometric phases of relevant physical models such as Winternitz-Smorodinsky oscillators.