Hamiltonian Mechanics from a contact–geometric perspective

Hamiltonian mechanics can be derived from a simple variational principle involving only a 1-form. The key idea is that the familiar action

S = \int (p_{a} d q^{a} - H d t)

is nothing more than the restriction of the tautological 1-form on the extended cotangent bundle ( $T^{*} (Q \times R_{t})$ ). This viewpoint unifies:

ordinary Hamiltonian mechanics,
time-dependent Hamiltonians,
parametrized/relativistic systems with constraint ( $H = 0$ ),
and contact geometry.

1. The Tautological 1-Form

Recall the tautological 1-form. For any manifold $M$ , the cotangent bundle $T^{*} M$ carries a canonical 1-form

θ \in Ω^{1} (T^{*} M), θ_{η} (v) := η (π_{*} v),

where $η \in T_{x}^{*} M$ and $π : T^{*} M \to M$ is the projection.
In coordinates $(x^{i}, p_{i})$ ,

θ = p_{i} d x^{i} .

2. Apply this to the Extended Configuration Space

Take

M = Q \times R_{t} .

A point of $T^{*} M$ has coordinates $(q^{a}, t; p_{a}, p_{t})$ , and the tautological form becomes

θ = p_{a} d q^{a} + p_{t} d t .

This is a single geometric object from which the entire Hamiltonian formalism can be recovered.

3. The Hamiltonian Constraint and the Physical 1-Form

Introduce the extended Hamiltonian constraint

H (q, p, t, p_{t}) = p_{t} + H (q, p, t) .

This condition can be interpreted as saying: "the energy, which is the momentum corresponding to $t$ , is given in this particular system by the function $H$ ".
The physical motions lie on the hypersurface

Σ := {H = 0} \subset T^{*} (Q \times R) .

Restrict the tautological 1-form to this hypersurface:

α = θ |_{Σ} = p_{a} d q^{a} - H (q, p, t) d t .

This form $α$ is the action 1-form of Hamiltonian mechanics. Its exterior derivative is

d α = d q^{a} \land d p_{a} + d t \land d H,

which restricts the symplectic structure to the hypersurface.

4. The Variational Principle ( $S = \int α$ )

Given a curve $γ (τ) \subset Σ$ , define

S [γ] = \int_{γ} α .

Varying $q^{a} (τ), p_{a} (τ), t (τ)$ produces the Euler–Lagrange equations of this action. The result is:

{\dot{q}}^{a} = \frac{\partial H}{\partial p_{a}} \dot{t}, {\dot{p}}_{a} = - \frac{\partial H}{\partial q^{a}} \dot{t} .

Choosing the parameter $τ = t$ yields the usual Hamilton equations:

{\dot{q}}^{a} = \frac{\partial H}{\partial p_{a}}, {\dot{p}}_{a} = - \frac{\partial H}{\partial q^{a}} .

Thus the entire structure of Hamiltonian mechanics comes from the line integral of the 1-form $α$ .
For the physical motivation see tautological 1-form#Physical Intuition The "Action" of a Nudge, or directly, action#What is the action, intuitively?.

5. Contact Geometry on the Constraint Surface

The hypersurface $Σ = {H = 0}$ is automatically a contact manifold, because:

$(T^{*} M, ω = d θ)$ is symplectic,
$Σ$ is a regular level set of the Hamiltonian $H$ ,
$α = θ |_{Σ}$ satisfies $$\alpha \wedge (d\alpha)^n \neq 0.$$
Thus $(Σ, α)$ is a contact manifold, and dynamics can be described in terms of its Reeb vector field.

6. The Reeb Vector Field of $α$

Given a contact manifold $(Σ, α)$ , the Reeb vector field $R$ is defined by:

α (R) = 1, ι_{R} d α = 0.

These equations uniquely determine $R$ .
Let the Reeb vector field be written as:

R = a^{i} \frac{\partial}{\partial q^{i}} + b_{i} \frac{\partial}{\partial p_{i}} + c \frac{\partial}{\partial t}

We have

d α = d p_{i} \land d q^{i} + \frac{\partial H}{\partial q^{i}} d t \land d q^{i} + \frac{\partial H}{\partial p_{i}} d t \land d p_{i}

and then

i_{R} d α = (b_{i} + c \frac{\partial H}{\partial q^{i}}) d q^{i} + (- a^{i} + c \frac{\partial H}{\partial p_{i}}) d p_{i} + (- a^{i} \frac{\partial H}{\partial q^{i}} - b_{i} \frac{\partial H}{\partial p_{i}}) d t = 0

Therefore:

a^{i} = c \frac{\partial H}{\partial p_{i}}

b_{i} = - c \frac{\partial H}{\partial q^{i}}

(Note: The $d t$ coefficient equation is automatically satisfied if these two are plugged in, confirming consistency.)

Now we substitute the expressions for $a^{i}$ and $b_{i}$ into the normalization constraint.

α (R) = p_{i} a^{i} - H c = 1

Substitute $a^{i} = c \frac{\partial H}{\partial p_{i}}$ :

p_{i} (c \frac{\partial H}{\partial p_{i}}) - H c = 1

Factor out $c$ :

c (p_{i} \frac{\partial H}{\partial p_{i}} - H) = 1

The coefficients $(a, b, c)$ are given by:

c = \frac{1}{p_{k} \frac{\partial H}{\partial p_{k}} - H}

a^{i} = \frac{\frac{\partial H}{\partial p_{i}}}{p_{k} \frac{\partial H}{\partial p_{k}} - H}

b_{i} = \frac{- \frac{\partial H}{\partial q^{i}}}{p_{k} \frac{\partial H}{\partial p_{k}} - H}

The key fact:

The Reeb flow of $α$ is precisely the Hamiltonian flow of the constraint $H = p_{t} + H$ , modulo reparametrization.

In other words:

The shape of the trajectories in $(q, p)$ is the same as in ordinary Hamiltonian mechanics.
The parametrization along the trajectory changes (Reeb time ≠ physical time).
The Reeb flow represents the canonical, parametrized flow on the constraint surface $Σ$ .

8. Relation Between the Reeb Field and the Hamiltonian Vector Field

Consider the Hamiltonian vector field $X_{H}$ on the ambient symplectic manifold:

ι_{X_{H}} ω = d H .

Properties:

$X_{H}$ is tangent to $Σ$ .
On $Σ$ , $ι_{X_{H}} d α = 0$ .
But $α (X_{H}) \neq 1$ .

Thus the Reeb field is simply a normalization:

R = \frac{X_{H}}{α (X_{H})} .

Hence:

Reeb orbits = Hamiltonian orbits
Reeb parametrization = canonical normalization determined by $α (R) = 1$

9. Summary of the Geometric Picture

Start with the extended configuration space $Q \times R_{t}$ .
Take its cotangent bundle $T^{*} (Q \times R)$ and its tautological form $θ$ .
Impose the Hamiltonian constraint $H = p_{t} + H = 0$ .
Restrict $θ$ to this hypersurface: $$\alpha = \theta|_\Sigma = p_a , dq^a - H , dt.$$
The action is the line integral $$
S=\int_\gamma\alpha.
The hypersurface is a contact manifold whose Reeb field is the normalized Hamiltonian flow.
Rovelli’s action $\int p_{a} d q^{a}$ with $H = 0$ is the projection of this contact mechanics to the $(q, p)$ -space.

10. Conceptual Takeaways

Hamiltonian mechanics is encoded entirely in the geometry of the tautological 1-form.
Traditional time-dependent Hamiltonian mechanics is simply the theory of curves in a contact manifold.
Parametrized and relativistic systems (where $H = 0$ ) appear naturally in this formulation.
The Reeb field gives a canonical parametrization of dynamical trajectories.
The usual symplectic formulation is just a projection of a more fundamental contact structure.

11. Analogy with Second-Order ODEs

A helpful way to interpret the Hamiltonian constraint

Σ = {H = p_{t} + H = 0} \subset T^{*} (Q \times R)

is to compare it with the geometric description of second-order ODEs on higher jet spaces.

1. Contact Geometry in Jet Spaces

The second jet bundle $J^{2} (Q)$ , with coordinates $(q, \dot{q}, \ddot{q})$ , carries a canonical contact system generated by forms such as

β_{1} = d q - \dot{q} d t, β_{2} = d \dot{q} - \ddot{q} d t .

These express the universal kinematic identities:

d q = \dot{q} d t, d \dot{q} = \ddot{q} d t .

Nothing dynamical has been imposed yet; these relations only ensure that a curve in $J^{2} (Q)$ arises as the 2-jet of a true function $q (t)$ .

2. Dynamics Arise from a Hypersurface in $J^{2} (Q)$

A second-order ODE

\ddot{q} = f (q, \dot{q}, t)

is encoded geometrically by introducing the dynamical hypersurface

Ξ = {\ddot{q} - f (q, \dot{q}, t) = 0} \subset J^{2} (Q) .

Integral curves tangent to both:

the contact system ${β_{1} = β_{2} = 0}$ , and
the hypersurface $Ξ$ ,
are exactly the solutions of the ODE.

Thus:

the contact structure expresses universal kinematics,
the hypersurface $Ξ$ expresses the specific physical law (force, acceleration, etc.).

3. Parallel Structure in Contact Hamiltonian Mechanics

In the extended cotangent bundle $T^{*} (Q \times R_{t})$ , the fundamental geometric object is the tautological 1-form

θ = p_{a} d q^{a} + p_{t} d t .

Restricting to the Hamiltonian constraint

Σ = {p_{t} + H (q, p, t) = 0},

one obtains the physical 1-form

α = θ |_{Σ} = p_{a} d q^{a} - H d t,

which defines a contact structure on $Σ$ .

This mirrors the jet-space picture:

Jet-space mechanics (in $J^{2}$ )	Contact Hamiltonian mechanics
Contact system $β_{1}, β_{2}$ gives universal kinematics	Tautological 1-form $θ$ gives universal $p$ - $d q$ - $d t$ kinematics
Hypersurface $Ξ : \ddot{q} = f (q, \dot{q}, t)$ encodes the law	Hypersurface $Σ : p_{t} + H = 0$ encodes the Hamiltonian
Dynamics = curves tangent to $Ξ$ and contact system	Dynamics = Reeb/Hamiltonian flow tangent to $Σ$

4. Conceptual Correspondence

Both constructions share the same architecture:

A contact structure provides universal kinematics.
A hypersurface specifies the physical system.

In jet spaces:

$(d q - \dot{q} d t) = 0$ and $(d \dot{q} - \ddot{q} d t) = 0$ encode “what velocity and acceleration mean”.
The equation $\ddot{q} = f (q, \dot{q}, t)$ is imposed as a hypersurface $Ξ$ .

In Hamiltonian mechanics:

The tautological 1-form encodes the canonical relation between momenta, coordinates, and time.
The Hamiltonian constraint $Σ$ selects the actual physical theory.

Thus, Hamiltonian dynamics stands to the tautological 1-form exactly as Newtonian dynamics stands to the jet-space contact system:
both arise from imposing a dynamical hypersurface on top of a universal geometric background.