The Theory of Adiabatic Invariants

Abstract

In this chapter, devoted to the theory of adiabatic invariants and its applications in the field of astronomy, we try to point out how procedures, seemingly different and originating in different fields, in the end derive from the same concepts. What closely relates perturbation theory, which originated from the demand for suitable solutions to the problems of planetary motion, and the theory of adiabatic invariants, which was developed to give a more solid bases to the quantization rules of early quantum mechanics, is recourse to the averaging procedure. In both cases, the average is performed over a periodic motion having a period by far shorter than the time which characterizes the evolution of the physical system one is studying. Through the application of Noether’s theorem, together with the averaging method, one then sees how also the approximate invariance of the quantities, called the adiabatic invariants, is connected with the symmetry properties of the system. It is clear that one could, as is sometimes done, overturn the argument and start from the element recognized as the central one, i.e. the averaging procedure, and apply it to the different classes of problems. But it seemed to us more effective from a didactic point of view to follow the inverse path of progressively identifying the unifying element in the different problems belonging to apparently disparate fields of research. But at this point one cannot omit the fact that what we have called the unifying element is far from being rigorously proven. Just to quote one of the most authoritative “users” of the averaging principle, “We note that this principle is neither a theorem, an axiom, nor a definition, but rather a physical proposition, i.e., a vaguely formulated and, strictly speaking, untrue assertion. Such assertions are often fruitful sources of mathematical theorems.”1 The introduction of the use of methods based on the use of the Lie transform on the one hand has emphasized the link of the adiabatic invariants theory with the traditional perturbation theory, on the other hand has enabled people to resort to automatic computations in actual applications. To conclude, we once more call the attention of the reader to the universality of the paradigm of the perturbed oscillator which, by help of a transformation of variables, can also be applied to the case of a charged particle in a magnetic field.