Geometric phases in mechanics

The note  [37] contains a formula which expresses a  continuous product of orthogonal matrices (i.e. of solution of the matrix ODE $\dot X = M(t) X$, $M^T=-M$,  $X(t)\in SO(3)$) in terms of parallel transport on the 2-sphere. It is remarkable that the cumulative angle   of rotation appears in the formula despite the fact that $M(t)$ do not commute for different $t$.

The paper  [33] gives a simple proof of the Gauss-Bonnet formula by mechanical analogy.  I stumbled upon this proof while changing a bicycle tire; holding the bike wheel made me wonder whether the wheel can be turned while held by its axle only; this led eventually to the proof of the Gauss-Bonnet theorem. Mathematically, it is a new proof of the theorem using dual cones.   The note also relates the Hannay-Berry phase with the writhing number" of the curve.

The   note  [32] is a geometrical study  of rolling of a rigid surface on the plane with some applications which include an extremely short derivation of a formula found earlier by Richard Montgomery.