Hamiltonian Dynamics of Polygons and Integrability

We introduce a class of Hamiltonian systems generating motion of polygons in plane and define ‘‘geometrical’’ Hamilton functions depending arbitrarily on distances between the vertices and the areas of the inscribed triangles. The motion preserves the centroid and the moment of inertia if unit masses are assigned to the vertices. Therefore the 3-vertex systems corresponding to the dynamics of triangles are completely integrable. In the case of Hamiltonians depending only on the areas there is an additional integral leading to integrable dynamics of the quadrilaterals. For a particular subclass of such Hamiltonians the total area is also preserved, which results in integrable motion of pentagons. We present several numerical illustrations of the polygon dynamics.