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)?

Answer

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.