φ-Lagrangian and Hamiltonian mechanics on SO(2)

Purpose: In this paper, the canonical formalism for plane rotation group SO(2) is presented.Methods: The φ-Lagrangian method is applied in order to find a Lagrangian that the Euler-Lagrange equations would correspondingly coincide with the Lie equations.Results: The canonical formalism for plane rotation group SO(2) is presented. The Lagrangian and Hamiltonian are explicitly constructed.Conclusions: We introduce a family of Lagrange functions on the plane rotation group and verify that Euler-Lagrange and Hamilton equations associated to each of these functions must coincide with the Lie equations in the Lie transformation group.