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?

(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. 


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 4volume 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. 


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: 


HamiltonJacobi 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. HamiltonJacobi is also, I think, somewhat related to the Schrödinger equation. 


One instance, where classical mechanics has to be treated with 'functional analysis' are infinite dimensional systems. The prototypical example is the Kortewegde Vries equation $$ u_t + u_{xxx} + 6 u u_x = 0 $$ which a priori looks like a nonlinear 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 NavierStokes, which is a classical system. P.S.: In shameless selfpromotion 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 wellknown. 


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 finitedimensional 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 infinitedimensional Hilbert spaces, and puts you into the realm of functional analysis. 

