Cartan form of a regular mechanical Lagrangian (${\theta }_L=L_{v^i}d{q}^i,{\omega }_L=-d{\theta }_L$)

A regular Lagrangian $L:TQ\to \mathbb{R}$ defines a fibre derivative $\mathbb{F}L:TQ\to T^{\ast }Q$ and the Cartan one-form ${\theta }_L=(\mathbb{F}L{)}^{\ast }\theta$. Its two-form ${\omega }_L=-d{\theta }_L$ is nondegenerate when the velocity Hessian is invertible. The energy is $E_L=v^i{L}_{v^i}-L$. The equation ${\iota }_{{\Gamma }_L}{\omega }_L=d{E}_L$ determines the Euler–Lagrange vector field. A hyperregular Lagrangian makes the fibre derivative a global diffeomorphism and transports this dynamics to Hamiltonian phase space.