= Solution
For a finite-dimensional mechanical system let $Q$ be its <configuration space>, a smooth manifold incorporating the holonomic constraints. <Lagrangian mechanics> starts with $L:TQ\to\mathbb R$; a curve $q(t)$ makes the action stationary under fixed-endpoint variations. The resulting equations are
$$
\frac{d}{dt}\frac{\partial L}{\partial v^i}-\frac{\partial L}{\partial q^i}=0.
$$
This equation is coordinate-independent even though generalized coordinates give its familiar expression.
The fibre derivative $\mathbb FL(q,v)=(q,p)$, with $p_i=L_{v^i}$, defines the <Legendre transform in mechanics>. The <Cartan form of a regular mechanical Lagrangian> is
$$
\theta_L=p_i\,dq^i,\qquad\omega_L=-d\theta_L,\qquad E_L=p_iv^i-L.
$$
For a <regular Lagrangian>, the velocity Hessian is invertible, $\omega_L$ is symplectic, and $\iota_{\Gamma_L}\omega_L=dE_L$ determines a second-order vector field on $TQ$. In coordinates this recovers the displayed Euler–Lagrange equations: the velocity components of the vector field are $v^i$, and its acceleration components are fixed by the invertible Hessian.
<Hamiltonian mechanics> takes place on $T^*Q$. Its <tautological one-form> is $\theta=p_i\,dq^i$, and its canonical <symplectic form> is $\omega=-d\theta=dq^i\wedge dp_i$. With $\iota_{X_H}\omega=dH$,
$$
\boxed{\dot q^i=H_{p_i},\qquad\dot p_i=-H_{q^i}.}
$$
The associated <Poisson bracket> is $\{f,g\}=f_{q^i}g_{p_i}-f_{p_i}g_{q^i}$, so $X_Hf=\{f,H\}$. For a <hyperregular Lagrangian>, $\mathbb FL$ is a global diffeomorphism, $H=E_L\circ(\mathbb FL)^{-1}$ and $(\mathbb FL)^*\omega=\omega_L$. The two formulations then describe the same trajectories. Regularity gives only local equivalence; degenerate Lagrangians require constraint analysis instead. These geometric structures distinguish equations of motion from arbitrary coordinate formulae and make <symmetry> reduction precise.
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