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# Text 2. Geometric Phases and Locomotion

Geometric phases naturally occur in families of integrable systems depending on parameters. Consider an integrable system with action-angle variables

;

assume that the Hamiltonian H depends on a parameter . This just means that we have a Hamiltonian independent of the angular variables and we can identity the configuration space with an n-torus . Let c be a loop based at a point in M. We want to compare the angular variables in the torus over , while the system is slowly changed as the parameters traverse the circuit c. Since the dynamics in the fiber vary as we move along c, even if the actions vary by a negligible amount, there will be a shift in the angle variables due to the frequencies u>1 = dH/d P of the integrable system; correspondingly, one defines

dynamic phase=

Here we assume that the loop is contained in a neighborhood whose standard action coordinates are defined. In completing the circuit c, we return to the same torus, so a comparison between the angles makes sense. The actual shift in the angular variables during the circuit is the dynamic phase plus a correction term called the geometric phase. One of the key results is that this geometric phase is the holonomy of an appropriately constructed connection (called the Hannay-Berry connection) on the torus bundle over M that is constructed from the action-angle variables. The corresponding angular shift, computed by Harinay [1985], is called Hannay's angles, so the actual phase shift is given by

dynamic phases + Hannay’s angles.

The geometric construction of the Hannay-Berry connection for classical systems is given in terms of momentum maps and averaging in Golin, Knauf, and Marmi [1989] and Montgomery [1988], Weinstein [1990] makes precise the geometric structures that make possible a definition of the Hannay angles for a cycle in the space of Lagrangian submanifolds, even with­out the presence of an integrable system. Berry's phase is then seen as a "primitive" for the Hannay angles.

A. Are these statements true or false according to the text ?

1. Geometric phases naturally occur in families of unintegrable systems depending on parameters.

2. The dynamics in the fiber don’t vary as we move along the circuit c.

3. We can assume that the loop is contained in a neighborhood whose standard action coordinates are defined.

4. A comparison between the angles makes sense.

5. Berry's phase is seen here as a "positive" for the Hannay angles.

B. What are the key words of the text?

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