Symplectic vs Contact vs Jet analogy

All three geometries follow the same architecture: a differential form defines an "empty" stratum of admissible integral manifolds (potential solutions), and an independent function (or its zero-level set) cuts out the specific sub-stratum that corresponds to actual solutions.

The three-way table

Jet-space mechanics ( $J^{2}$ )	Contact Hamiltonian ( $T^{*} (Q \times R)$ )	Symplectic ( $T^{*} Q$ )
Contact system $β_{1}, β_{2}$ gives universal kinematics	Tautological 1-form $θ = p_{i} d q^{i} + p_{t} d t$ gives universal kinematics	symplectic form $ω = d θ = d p^{i} \land d q_{i}$ gives universal phase-space kinematics
Hypersurface $Ξ : \ddot{q} = f$ encodes the law	Hypersurface $Σ : p_{t} + H = 0$ encodes the Hamiltonian	Energy level set $H^{- 1} (E)$ (or the function $H$ itself) encodes the specific system
Solutions = curves tangent to $Ξ$ + contact system	Solutions = Reeb/Hamiltonian flow tangent to $Σ$	Solutions =Lagrangian submanifold $L \subset H^{- 1} (E)$ (or graphs of $d S$ in H–J theory)

Abstract pattern

A geometric stratum (symplectic/contact/jet-contact) provides universal kinematics.
A function (Hamiltonian/PDE/constraint) specifies the physical system.

Symplectic $ω$ → Lagrangian submanifolds (potential states)
Contact $α$ → Legendrian submanifolds (potential dynamical trajectories)
Jet-contact ${β_{i}}$ → prolongations (kinematically admissible curves)

Each "waits" for a function to become a specific theory.

See Hamiltonian systems in contact geometry#11 Analogy with Second-Order ODEs for the original jet-space ↔ contact table that this note extends.