Adiabatic Principle and Action-Angle Variables

hi, guys here, i'm going to introdused all the theories of the great scientist "ALBERT EINSTEIN" 1)Theory of… by akki

An adiabatic invariant is a property of a physical system that stays constant when changes occur slowly.

We analyze a very simple variant of the Lorentz pendulum, in which the length is varied exponentially, instead of uniformly, as it is assumed in the standard case. We establish quantitative criteria for the condition of adiabatic changes in both pendula and put in evidence their substantially different physical behavior with regard to adiabatic invariance.

Imagine a particle in one dimension oscillating back and forth in some potential. The potential doesn't have to be harmonic, but it must be such as to trap the particle, which is executing periodic motion with period.

Adiabatic invariants, denoted by I, are approximate constants (invariants) of motion of a given dynamical system (not necessarily Hamiltonian), which are approximately preserved (constant) during a process of very slow change of the system’s parameters (denoted by λ), on a time scale T, which is supposed to be much larger than any typical dynamical time scale such as traversal time or the period of the shortest periodic orbits etc.

Adiabatic principle and action-angle variables

Invariance in a slow process