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Hi I'm currently learning Hamiltonian and Lagrangian Mechanics (which I think also encompasses the calculus of variations) and I've also grown interested in functional analysis. I'm wondering if there is any connection between functional analysis and Hamiltonian/ Lagrangian mechanics? Is there a connection between functional analysis and calculus of variations? What is the relationship between functional analysis and quantum mechanics; I hear that that functional analysis is developed in part by the need for better understanding of quantum mechanics?

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The answer for your questions is Yes. In particular, for Quantum Mechanics, see von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Anyway you can find also more information and reference, about this relations in wikipedia. –  Leandro Jul 1 '10 at 0:07
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Also see Reed and Simon, Methods of Modern Mathematical Physics, vols 1 - 4. One might argue that the entire tome (well, maybe less so the first half of volume 2 and parts of volume 3) is about application of functional analysis as inspired by the study of Schrodinger equation. –  Willie Wong Jul 1 '10 at 0:13
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@Willie: I'm very much a non-applications kind of analyst, but doesn't very basic linear ODE theory have a tinge of functional analysis -- at least in early attempts to get somewhere? –  Yemon Choi Jul 1 '10 at 3:21
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@Yemon: The proof of Picard-Lindeloef (and cousins) is a functional analysis proof, since it's a fixed point theorem in Banach spaces. It still doesn't give the theory a functional analytic flavour. The key problem is that the functions, one considers do not live in nice spaces. (Exceptions are known. e.g. Sturm--Liouville Theory, but that is more quantum mechanics). –  Helge Jul 1 '10 at 9:39
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@Yemon: I am going to channel a physicist acquaintance of mine to illustrate why I don't really consider the sort of stuff in basic ODE theory functional analysis (though you are absolutely right that there is a an application of functional analysis). He said, during a (physics) seminar, to the nodding approval of the (physics) big wigs in the room: "... and as we all know, ODEs good; PDEs bad." –  Willie Wong Jul 1 '10 at 10:25
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(1) Depends on what you mean by Hamiltonian and Lagrangian mechanics.

If you mean the classical mechanics aspect as in, say, Vladimir Arnold's "Mathematical Methods in ..." book, then the answer is no. Hamiltonian and Lagrangian mechanics in that sense has a lot more to do with ordinary differential equations and symplectic geometry than with functional analysis. In fact, if you consider Lagrangian mechanics in that sense as an "example" of calculus of variations, I'd tell you that you are missing out on the full power of the variational principle.

Now, if you consider instead classical field theory (as in physics, not as in algebraic number theory) derived from an action principle, otherwise known as Lagrangian field theory, then yes, calculus of variations is what it's all about, and functional analysis is King in the Hamiltonian formulation of Lagrangian field theory.

Now, you may also consider quantum mechanics as "Hamiltonian mechanics", either through first quantization or through considering the evolution as an ordinary differential equation in a Hilbert space. Then through this (somewhat stretched) definition, you can argue that there is a connection between Hamiltonian mechanics and functional analysis, just because to understand ODEs on a Hilbert space it is necessary to understand operators on the space.

(2) Mechanics aside, functional analysis is deeply connected to the calculus of variations. In the past forty years or so, most of the development in this direction (that I know of) are within the community of nonlinear elasticity, in which objects of study are regularity properties, and existence of solutions, to stationary points of certain "energy functionals". The methods involved found most applications in elliptic type operators. For evolutionary equations, functional analysis plays less well with the calculus of variations for two reasons: (i) the action is often not bounded from below and (ii) reasonable spaces of functions often have poor integrability, so it is rather difficult to define appropriate function spaces to study. (Which is not to say that they are not done, just less developed.)

(3) See Eric's answer and my comment about Reed and Simon about connection of functional analysis and quantum mechanics.

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Well,I'm not sure about classical mechanics,but functional analysis certainly has many applications in quantum mechanics via the modeling of wavefunctions by PDEs and operators defined on Hilbert and Banach spaces.

A great book for beginning the study of these properties is the classic text by S.B.Sobolev,Some Applications of Functional Analysis in Mathematical Physics,now I believe in it's 4th edition and avaliable through the AMS. A more comprehensive text is the 4-volume work by Barry Simon and Louis Reed, which covers not only basic functional analysis,but all the basic applications to modern physics,such as spectral analysis and scattering theory.

Lastly,some less well known applications can be found in Elliott Lieb and Micheal Loss' Analysis.

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While Lou Reed has surely enriched the lives of many mathematicians, it is primarily through his musical work with the Velvet Underground rather than any collaboration with Barry Simon. You must be thinking of the mathematician Michael Reed. –  Tom LaGatta Jul 1 '10 at 21:22
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One of the biggest problems in mathematical physics is actually to understand the link between Hamiltonian/Lagrangian mechanics and functional analysis. This is because classical mechanics is formulated in the former setting while quantum mechanics is formulated in the functional analysis setting. The act of going from classical mechanics to quantum mechanics is called quantization and basically consists of assigning functional analytic operators to classical observables, in a way that respects the Poisson and Lie brackets. For example in classical quantization we assign position to the operator of multiplication by x and we assign to momentum the operator $-i\frac{d}{dx}$. Both of these act on (a dense subset of) the space $L^2(\mathbb R)$, which is taken to be the space of wave functions in one dimension. You may want to take a look at the orbit method, which is the mathematics involved in a quantization scheme called geometric quantization.

Some relevant MO discussion about this are:

What is Quantization ?

What does "quantization is not a functor" really mean?

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Hamilton-Jacobi PDE is a formulation of classical mechanics (as far as I understand; I am no expert in physics) and the unique weak solution is found by a certain calculus of variations problem inspired by optimal control theory.

Hamilton-Jacobi is also, I think, somewhat related to the Schrödinger equation.

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Very good point. HJE skipped my mind (maybe because the OP mentioned explicitly Hamitonian and Lagrangian mechanics). So it does brings in a tie to calculus of variations. And as a PDE, the general existence of the solution does have a bit of a flavour of functional analysis. –  Willie Wong Jul 1 '10 at 10:11
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One instance, where classical mechanics has to be treated with 'functional analysis' are infinite dimensional systems. The prototypical example is the Korteweg-de Vries equation $$ u_t + u_{xxx} + 6 u u_x = 0 $$ which a priori looks like a non-linear PDE. The key now is that it is completely integrable, which means that one can associate to an equivalent evolution for operators on Hilbert spaces. Define $$ L(t) = - \frac{d^2}{dx^2} + u(x,t) $$ as an operator on $L^2(\mathbb{R})$. Then this operator obeys $$ L_t = [P, L], $$ where $P$ is another operator, one can construct from $u$. (The specific form doesn't matter). The operators $P$ and $L$ are known as Lax Pair. (The $P$ stands for Peter not for Pair ☺ ). This is just the Heisenberg picture of quantum mechanics, so one can use the tools developed there, i.e. functional analysis, to investigate this equation. Of special importance is something known as scattering theory.

Just on a final point: KdV is a limit of Navier--Stokes, which is a classical system.

P.S.: In shameless self-promotion for some details on another system, the Toda Lattice, where it is easier to see that it is classical mechanics (one can write down the Hamiltonian easily), see here. I just made the post about KdV, since it is well-known.

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I think you may have copied the KDV equation wrong. (Check the last term on the LHS.) And if you are going to mention scattering theory, you might as well spell out that $(L,P)$ are what is known as a Lax pair to aid people in literature searching. :) –  Willie Wong Jul 1 '10 at 10:18
    
Fixed these things. Unfortunately, this forum does not support smileys. There should be an ;-) somewhere instead of ☺ –  Helge Jul 1 '10 at 11:33
    
i see the smiley just fine. –  Willie Wong Jul 1 '10 at 11:54
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There is a very good discussion of this issue in L. Takhtajan's excellent text Quantum Mechanics for Mathematicians; see especially section 2.1. Chapter 1 also treats classical mechanics in a way that naturally extends to the quantum picture.

The idea as I read it is this: both classical and quantum mechanics consider some underlying phase space, and a collection of observables, physical values you can measure. These naturally form an algebra.

In classical mechanics you assume that you can measure different observables simultaneously without the measurements affecting one another; this turns out to correspond to the condition that the algebra of observables is commutative. A good example is thinking of observables as continuous functions on the phase space, and the Gelfand representation says that this is essentially the only example. So a functional analysis result says that you don't need to do too much functional analysis here (or rather, it's of a fairly trivial kind).

In quantum mechanics, the algebra of observables might not be commutative. A good example of such a thing is operators on a Hilbert space (again, in some sense the only example). If you could use a finite-dimensional Hilbert space, you'd just be doing linear algebra. But it turns out the commutation relations that the physics requires can only be satisfied by unbounded operators. This forces you to use infinite-dimensional Hilbert spaces, and puts you into the realm of functional analysis.

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