Beyond Hamiltonian and Lagrangian mechanics
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.
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.
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.