Contact Geometry

Contact structures

A contact structure on a $(2n+1)$-dimensional manifold $M$ is a completely non-integrable hyperplane distribution given locally by the kernel of a 1-form $\theta$ such that:

This condition ensures that the structure is genuinely contact—i.e., not integrable and "maximally twisted."

Such a form defines a contact manifold $(M,\theta )$, where $\theta$ is only determined up to multiplication by a nowhere-zero function.

Two flavors of contact coordinates

Contact geometry in mechanics appears in two main flavors, depending on the formulation of the system:

1. Jet bundle coordinates $(q^i,v^i,t)$

• These coordinates live on the first-order jet bundle $J^1(Q)$ of a configuration manifold $Q$. The original bundle is $\mathbb{R}\times Q\to \mathbb{R}$.
• Local coordinates:
  • $q^i$: configuration variables
  • $v^i$: generalized velocities (first derivatives of $q^i$ with respect to $t$)
  • $t$: time (or a base parameter)
• The jet bundle encodes the geometry of first-order differential constraints. A canonical contact structure arises from the Cartan form:

• This formulation is natural for Lagrangian mechanics, where the equations of motion are derived from a variational principle on curves in $Q$.

2. Extended phase space coordinates $(q^i,p_i,z)$

• These coordinates live on an extended phase space, often denoted $\mathcal{P}$ or $\mathcal{C}$. They can also be thought as a first-order jet bundle, in this case with original bundle $Q\times \mathbb{R}\to Q$.
• Local coordinates:
  • $q^i$: configuration variables
  • $p_i$: momenta conjugate to $q^i$
  • $z$: additional scalar coordinate, often representing internal energy, action, or Hamilton’s principal function
• The canonical contact form here is:

• This setting supports contact Hamiltonian mechanics, a generalization of Hamiltonian mechanics that naturally includes time-dependence and dissipative effects.

Example: encompassing time-dependent Hamiltonian

A primary motivation for this structure comes from describing a standard mechanical system with a time-dependent Hamiltonian $H(q^i,t,p_i)$. Here's how the contact structure arises naturally:

1. We start with a classical Hamiltonian system with coordinates $(q^i,p_i)$, extended to a bigger symplectic manifold, the cotangent bundle $T^{\ast }(Q\times \mathbb{R})$, which has coordinates $(q^i,t,p_i,p_0)$, and its corresponding canonical tautological 1-form $\mathrm{\Theta }=p_{i}d{q}^i+p_{0}dt$.
2. We consider here a inherited Hamiltonian: $K(q^i,t,p_i,p_0)=H(q^i,t,p_i)+p_0$.
3. To describe the system's evolution, we restrict our attention to the $(2n+1)$-dimensional submanifold given by $K=0$, that is, $p_0=-H(q,p,t)$. This is our manifold of interest.
4. The contact structure is then given by the pullback (restriction) of the tautological form $\mathrm{\Theta }$ to this submanifold. Since $\iota :(q^i,t,p_i)\mapsto (q^i,t,p_i,-H)$, we have

This form $\alpha$ defines a valid contact structure. While it is not in the simple canonical form, Pfaff-Darboux theorem guarantees that we can always find local coordinates $(q^{\prime i},p_i^{\prime },z)$ in which this structure can be written as $\theta =dz-p_i^{\prime }d{q}^{\prime i}$. This justifies using the simpler canonical form as the general definition.

This approach let us recast Hamiltonian systems as an EDSs.

Relation between the two: Legendre transform

The two formulations are interconnected via the Legendre transform, which maps the Lagrangian picture to the Hamiltonian one.