  A host of elastic systems consisting of active components exhibit
path-dependent elastic behaviors not found in classical elasticity, which is
known as odd elasticity. Odd elasticity is characterized by antisymmetric (odd)
elastic modulus tensor. Here, from the perspective of geometry, we construct
the Hamiltonian formalism to show the origin of the antisymmetry of the elastic
moduli. Furthermore, both non-conservative stress and the associated nonlinear
constitutive relation naturally arise. This work also opens the promising
possibility of exploring the physics of odd elasticity in dynamical regime by
Hamiltonian formalism.
