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?



Functional Analysis and its relation to 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. 


