Beyond Hamiltonian and Lagrangian mechanics Lagrangian and Hamiltonian formulations are the bedrock of particle and field theories, produce the same equations of motion, and are related through a Legendre transform.  Are there more such mathematical objects that are equivalent, or are these two in some way unique?  If so, why are there two equivalent systems,  rather than a single (or more)?
 A: It's worth pointing out that the Hamiltonian and Lagrangian formalisms are independent, even though they're usually taught as if the former were a filtering of the latter (here enter Legendre transforms). Both formalisms are as independent as the notions of tangent and cotangent bundles in differential geometry: independent, but intrinsically connected. 
Also, there's a third formalism: the Hamilton-Jacobi one. It is as good as the other two, and carries a completelly different interpretation of the equations of motion. All those formalisms are deeply connected an each has its advantages and geometric interpretation.
As a last comment: you can think of many other interpretations of Mechanics. There are as many as you want. An example of a new, yet useful one, is the centre-chord interpretation, related to the Weyl-Wigner interpretation os quantum machanics. As long as your transformations are canonical, the sky is the limit regarding the creation of new points of view in Mechanics.
A: All the various "free energies" of thermodynamics are but a (or sometime a few) Legendre transform(s) away from the plain old energy.


*

*To get the Helmholtz free energy from the energy you perform a Legendre transformation between entropy and temperature.

*To get the enthalpy from the energy you perform a Legendre transformation between volume and pressure.

*To get the Gibbs free energy from the energy you perform two Legendre transforms, one between entropy and temperature and the other between volume and pressure.

*And so on (there are others, but they are less common in application). In particular you can exchange a description in therms of particle numbers for one in terms of chemical potentials when needed.
A: There is also the Routhian formalism of mechanics which is described as being a hybrid of Lagrangian and hamiltonian mechanics. The Routhian is defined as $$R = \sum_{i=1}^n p_i\dot{q}_i - L$$ You can learn more about it by clicking this link for Wikipedia's description of it. 
Reading more in regards to the routhian because I was bored, I realized it is defined as the partial Legendre transform of the Lagrangian and also in the language of differential geometry it is defined similarly to the Lagrangian as $$R^\mu : TM \to \mathbb{R}$$ where $$R^\mu(q, \dot{q}) = L(q, \dot{q}) - \langle A(q, \dot{q}), \mu\rangle$$ where $A$ is the mechanical connection term. You can read more about it in this pdf.
