= Cartan form of a regular mechanical Lagrangian
{c}
{title2=$\theta_L=L_{v^i}\,dq^i,\quad\omega_L=-d\theta_L$}
A <regular Lagrangian> $L:TQ\to\mathbb R$ defines a fibre derivative $\mathbb FL:TQ\to T^*Q$ and the Cartan one-form $\theta_L=(\mathbb FL)^*\theta$. Its two-form $\omega_L=-d\theta_L$ is nondegenerate when the velocity Hessian is invertible. The energy is $E_L=v^iL_{v^i}-L$. The equation $\iota_{\Gamma_L}\omega_L=dE_L$ determines the Euler–Lagrange vector field. A <hyperregular Lagrangian> makes the fibre derivative a global diffeomorphism and transports this dynamics to Hamiltonian phase space.
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