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)?
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You want mathematically equivalent objects only in the context of Classical mechanics? If not, then you can remember all the different formulations of QM – DLV Jun 28 '16 at 23:09
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I didn't mean that. I thought your question was about other examples of reformulations of the same thing. As an example I mentioned all the different formulations of QM. – DLV Jun 28 '16 at 23:14
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1If your question is solely about CM then you might like this post https://www.quora.com/Are-there-any-alternative-formulations-of-mechanics-besides-the-usual-Newtonian-and-Lagrangian-ones-Ideally-speaking-should-anymore-be-possible – DLV Jun 28 '16 at 23:15
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1If you like this question you may also enjoy reading this Phys.SE post. – Qmechanic Jun 29 '16 at 14:48
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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.
Sir Cumference
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@QuantumBrick This answer, like QuantumBrick's, is good thanks. I was hoping for greater insight into why these few objects (Hamiltonian/Lagrangian,Routhian, Hamilton-Jacobi) are imbued with magic dust for generating eq. of m. Maybe its really as simple as a single object that can be cast in different forms via Legendre transform, and thats it. – anon01 Jul 05 '16 at 22:13
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It is not at all that simple and it has really nothing to do with Legendre transforms. If you want I can update my answer with an outline of the three main formalisms of mechanics. – QuantumBrick Jul 06 '16 at 01:36
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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.
QuantumBrick
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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.
