How Do You Quantize the Hamiltonian for a Particle on a Unit Circle?

[the1ceman]

Homework Statement

The Lagrangian of a non-relativistic particle propagating on a unit circle is
$$L=\frac{1}{2}\dot{\phi}^{2}$$

where ϕ is an angle 0 ≤ ϕ < 2π.
(i) Give the Hamiltonian of the theory, and the Poisson brackets of the ca-
nonical variables. Quantize the theory by promoting the Poisson brackets into
commutators, and write the angular momentum operator, L, which is the con-
jugate (momentum) variable of ϕ, in the position representation. Note that in
the position representation
$$\hat{\phi}|\phi\rangle=\phi|\phi\rangle\;,\;\langle\phi'|\phi\rangle=\delta(\phi'-\phi)$$

Homework Equations

3. The attempt
i am stuck on the part where i have to write down L, how do i find its form in the $\phi$ representation? Please help

 

[the1ceman]

Anyone?

 

[the1ceman]

Please some1 help!