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**Reduction, symmetry, and phases in mechanics.**
*(English)*
Zbl 0713.58052

Mem. Am. Math. Soc. 436, 110 p. (1990).

Various holonomy phenomena are shown to be instances of the reconstruction procedure for mechanical systems with symmetry. This point of view is explored for fixed systems (for example with controls on the interval, or reduced, variables) and for slowly moving systems in an adiabatic context. One of the crucial new ingredients in the present paper is the introduction of a connection that is associated to the movement of a classical system that one terms the Cartan connection.

Another ingredient is the systematic use of symmetry and reduction, which is the key concept needed to generalize to the nonintegrable case. It is through the reconstruction process that the holonomy enters.

This way allows to treat in a natural way examples like the ball in the rotating hoop and nonintegrable mechanical systems.

Another ingredient is the systematic use of symmetry and reduction, which is the key concept needed to generalize to the nonintegrable case. It is through the reconstruction process that the holonomy enters.

This way allows to treat in a natural way examples like the ball in the rotating hoop and nonintegrable mechanical systems.

Reviewer: P.Khmelevskaja

### MSC:

37-XX | Dynamical systems and ergodic theory |

70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |