Mathematics – Dynamical Systems
Scientific paper
2011-11-04
Mathematics
Dynamical Systems
24 pages
Scientific paper
A pentagon in the plane with fixed side-lengths has a two-dimensional shape space. Considering the pentagon as a mechanical system with point masses at the corners we answer the question of how much the pentagon can rotate with zero angular momentum. We show that the shape space of the equilateral pentagon has genus 4 and find a fundamental region by discrete symmetry reduction with respect to symmetry group D_5. The amount of rotation \Delta\theta, for a loop in shape space at zero angular momentum is interpreted as a geometric phase and is obtained as an integral of a function B over the region of shape space enclosed by the loop. With a simple variational argument we determine locally optimal loops as the zero contours of the function B. The resulting shape change is represented as a Fourier series, and the global maximum of \Delta \theta \approx 45\degree is found for a loop around the regular pentagram.
Dullin Holger R.
Tong William
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