Various holonomy phenomena are shown to be instances of the reconstruction procedure
for mechanical systems with symmetry. We systematically exploit this point of view for fixed
systems (for example with controls on the internal, or reduced, variables) and for slowly moving
systems in an adiabatic context. For the latter, we obtain the phases as the holonomy for a
connection which synthesizes the Cartan connection for moving mechanical systems with the
Hannay-Berry connection for integrable systems. This synthesis allows one to treat in a natural
way examples like the ball in the slowly rotating hoop and also non-integrable mechanical systems.

Reduction, symmetry, and phases in mechanics

The basic adiabatic theorems of classical and quantum mechanics are over-viewed and an adiabatic theorem in quantum mechanics without a gap condition is described.

An Adiabatic Theorem without a Gap Condition

Analytic functions are introduced, which are analogous to the Fredholm determinant, but may have only finite radius of convergence. These functions are associated with operators of the form e μ(dω) ℒω, where ℒω φ(x) = ϕω(x). φ(ψω x), , φ belongs to a space of Holder or C
 r
 functions, ϕω is Holder or C
 r
 , and ψω is a contraction or a C
 r
 contraction. The results obtained extend earlier results by Haydn, Pollicott, Tangerman and the author on zeta functions of expanding maps.

An Extension of the Theory of Fredholm Determinants

We review the development and practical uses of a generalized Maupertuis least action principle in classical mechanics, in which the action is varied under the constraint of fixed mean energy for the trial trajectory. The original Maupertuis (Euler-Lagrange) principle constrains the energy at every point along the trajectory. The generalized Maupertuis principle is equivalent to Hamilton's principle. Reciprocal principles are also derived for both the generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis Principle is the classical limit of Schrodinger's variational principle of wave mechanics, and is also very useful to solve practical problems in both classical and semiclassical mechanics, in complete analogy with the quantum Rayleigh-Ritz method. Classical, semiclassical and quantum variational calculations are carried out for a number of systems, and the results are compared. Pedagogical as well as research problems are used as examples, which include nonconservative as well as relativistic systems.

Progress in Classical and Quantum Variational Principles

We introduce a cycle-expansion (fully deterministic) technique to compute the asymptotic behavior of arbitrary order transport moments. The theory is applied to different kinds of one-dimensional intermittent maps and Lorentz gas with an infinite horizon, confirming the typical appearance of phase transitions in the transport spectrum.

Anomalous Transport: A Deterministic Approach

We shall take the point of view that dynamical zeta functions are useful objects to describe the spectrum of transfer operators. To define a transfer operator, we use two ingredients: 


a map f : X → X of a topological or metric space X to itself (the dynamical system), with the property that f -1 (x) is an at most countable set for each x ∈ X;


a weight g : X → C.

A Brief Introduction to Dynamical Zeta Functions

In this paper we present results on the relations between wave scattering in the so-called modular domain, analytic number theory, and classical statistical mechanics. The motion in the modular domain is an example of so-called arithmetic chaos. In this paper we shortly review results from [5, 6, 7], but concentrate on previously unpublished results.

Irregular Scattering, Number Theory, and Statistical Mechanics

We report on previous and recent work in classical perturbation theory related to the Hannay angles. They are a means of measuring an anholonomy effect in classical mechanics closely corresponding to the Berry phase in quantum mechanics.

Hannay Angles and Classical Perturbation Theory

The notion of ergodicity was introduced by L Boltzmann in connection with Foundations of Statistical Mechanics Now its role for Statistical Mechanics is not so much clear but it is very important for the theory of dynamical systems and deterministic chaos

Ergodicity and Mixing

The Hannay angles were introduced by Hannay as a means of measuring a holonomy effect in classical mechanics closely corresponding to the Berry phase in quantum mechanics. Using parameter-dependent momentum mappings we show that the Hannay angles are the holonomy of a natural connection. We generalize this effect to non-Abelian group actions and discuss non-integrable Hamiltonian systems. We prove an averaging theorem for phase space functions in the case of general multi-frequency dynamical systems which allows us to establish the almost adiabatic invariance of the Hannay angles. We conclude by giving an application to celestial mechanics.

/pdf/the-hannay-angles-geometry-adiabaticity-and-an-example-bd4w00xkmt.pdf

The Hannay angles: geometry, adiabaticity, and an example

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