Contact geometric mechanics: the Tulczyjew triples

We propose a generalization of the classical Tulczyjew triple as a geometric tool in Hamiltonian and Lagrangian formalisms which serves for contact manifolds. The rôle of the canonical symplectic structures on cotangent bundles in Tulczyjew’s case is played by the canonical contact structures on the bundles $J^{1}L$ of first jets of sections of line bundles $L\to M$. Contact Hamiltonians and contact Lagrangians are understood as sections of certain line bundles, and they determine (generally implicit) dynamics on the contact phase space $J^{1}L$. We also study a contact analog of the Legendre map and the Legendre transformation of generating objects in both contact formalisms. Several explicit examples are offered.