# ADIABATIC INVARIANTS AND SYMPLECTIC GEOMETRY

The idea of adiabatic invariants dates back at least as far as 1905, when, before the discovery of quantum mechanics, some people (e.g., Einstein) attempted to explain why the ratio of energy to frequency of atom’s radiation is always (Plank’s) constant. The explanation went like this: an electron in an atom (then thought of as a miniature solar system held together by electric forces) is orbiting around the nucleus so fast that the ambient EM forces seem to change slowly. Loosely speaking, the atom is similar to a pendulum whose string is slowly changes length. In general, the energy and the frequency in a pendulum are unrelated. It was discovered by Einstein that if the length of a pendulum changes slowly, the frequency becomes tied to the energy; in fact, the ration of the two becomes almost constant. This ratio is an example of what became known as an adiabatic invariant. Adiabatic invariants are ubiquitous in Hamiltonian systems.

The class of linear Hamiltonian systems has been essentially fully analyzed (to the extent is seems reasonable to hope for). The foundation of this analysis was laid by Lyapunov, Poincare, Krein, Gelfand, Lidskii and Moser. The report [1] is a survey of the work of Krein, Gelfand, Lidskii and Moser on the subject.

In [5] I had established a connection between adiabatic invariants and the Gelfand--Lidskii signature of a symplectic matrix. The paper deals with linear Hamiltonian systems whose Hamiltonians have both periodic and slow time--dependence: $\dot z = JH(t, \epsilon t)z$. One question was: are there adiabatic invariants, and how many? The answer, in terms of the Gelfand--Lidskii signature, is this: if the periodic system $\dot z= JH(t, \tau)z$ (with $\tau$ frozen) is strongly stable for all $\tau$ then the number of adiabatic invariants of the system is not less than the number of clusters of the same sign in the Gelfand--Lidskii signature of the Floquet matrix.