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A lot of physics, such as classical mechanics, General Relativity, Quantum Mechanics etc can be expressed in terms of Lagrangian Mechanics and Hamiltonian Principles. But sometimes I just can't help wonder whether is it ever possible (in the future, maybe) to discover a physical law that can't be expressed in terms of Lagrangian Equations?

Or to put it in other words, can we list down for all the physical laws that can be expressed in terms of Lagrangian equations, what are the mathematical characteristics of them( such as it must not contain derivatives higher than 2, all the solutions must be linear etc)?

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All solutions must be linear? Why? – QGR Feb 7 '11 at 14:12
Related: – user566 Feb 7 '11 at 14:19
@QGR, I'm not saying that all solutions to physics law must obey linearity!! It's just an example I use to illustrate what I mean by "mathematical characteristics". – Graviton Feb 7 '11 at 14:22
This is pretty much a duplicate of the question cited by Moshe, vote to close as a duplicate. – pho Feb 7 '11 at 15:26
@Morshe, not too similar! I'm also asking about what are the mathematical characteristics physical laws must obey if they are expressible in terms of Lagrangian mechanics. – Graviton Feb 8 '11 at 1:52

4 Answers 4

up vote 3 down vote accepted

Hamilton's dynamics occurs on a phase space with an equal number of configuration and momentum variables $\{q_i,~p_i\}$, for $i~=~1,\dots n.$ The dynamics according to the symplectic two form ${\underline{\Omega}}~=~\Omega_{ab}dq^a\wedge dp^b$ is a Hamiltonian vector field $$ \frac{d\chi_a}{dt}~=~\Omega_{ab}\partial_b H, $$ with in the configuration and momentum variables $\chi_a~=~\{q_a,~p_a\}$ gives $$ {\dot q}_a~=~\frac{\partial H}{\partial p_a},~{\dot p}_a~=~-\frac{\partial H}{\partial q_a} $$ and the vector $\chi_a$ follows a unique trajectory in phase space, where that trajectory is often called a Hamiltonian flow.

For a system the bare action is $pdq$ ignoring sums. The Hamiltonian is found with imposition of Lagrangians as functions over configuration variables. This is defined then on half of the phase space, called configuration space. It is also a constraint, essentially a Lagrange multiplier. The cotangent bundle $T^*M$ on the configuration space $M$ defines the phase space. Once this is constructed a symplectic manifold is defined. Therefore Lagrangian dynamics on configuration space, or equivalently the cotangent bundle defines a symplectic manifold. This does not mean a symplectic manifold defines a cotangent bundle. The reason is that symplectic or canonical transformations mix the distinction between configuration and momentum variables.

As a result there are people who study bracket structures which have non-Lagrangian content. The RR sector on type IIB string is non-Lagrangian. The differential structure is tied to the Calabi-Yau three-fold, which defines a different dynamics.

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If you have a classical theory specified by some partial differential equations, you can automatically come up with a Lagrangian by introducing a Lagrange multiplier for each PDE.

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Not too sure what you mean here. Classical theory, as opposed to, quantum theory? – Graviton Feb 7 '11 at 14:23
@Graviton: doesn't matter. You can use the classical Lagrangian to quantize your theory. Whether that works in a particular case is a different question though. – Marek Feb 7 '11 at 16:33

You cannot model friction very well with Lagrangian Mechanics.

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And why would that be? – Marek Feb 7 '11 at 16:32
It is true that in introductory classes to classical mechanics usually only systems with energy conservation are considered. But it is possible to extend the framework to systems with dissipation, see e.g. Brogliato et. alt., “Dissipative Systems, Analysis and Control”, Springer, 2nd edition. – Tim van Beek Feb 7 '11 at 16:39
Looking at how you would describe coulomb friction in a lagrangian? – ja72 Feb 7 '11 at 17:10
@ja72 you could read this paper, which solves coulomb friction using a lagrangian: – spencer nelson Feb 7 '11 at 17:51
Why is this downvoted? This is exactly correct--- odd order differential equations, disspiative ones, cannot come out of an explicitly time independent Lagrangian, because they cannot conserve a time independent symplectic volume--- their time evolution takes any motion in phase space to a single point generically! – Ron Maimon Oct 19 '11 at 17:21

This question is rather generic in several different respects. For example it is not clear that even Schrodinger's equation is an equation that can be "expressed in Hamiltonian principles". Yes we have $\delta/\delta t \Psi=H \Psi$ but is this an expression in "Hamiltonian Principles"? How different must the equation become to not meet this requirement - assuming it does now, perhaps adding a non-linear term?

Furthermore "Hamilton's Principle" does not itself apply in the Quantum context, although the action paths that it introduces are used in Feynman Path Integrals. Hamilton's Principle being a classical concept.

Another generality is in the range of Physics. The whole area of Thermodynamics comes to mind. Now there are phase space explanations, but is that "Hamiltonian Principles?"

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