I had a course in General Relativity this semester and wrote a paper on a Hamiltonian formulation of General Relativity, the ADM formalism. It is written at a level appropriate for a final year undergraduate student of physics, with some familiarity of Classical Mechanics and General Relativity assumed. Basically it’s a review of Lagrangian and Hamiltonian mechanics followed by a walkthrough of Wald Appendix E and Misner, Thorne & Wheeler chapter 21, so hopefully it’ll be useful for those of you who are trying to grasp the ADM formalism but are missing the background. Here’s the abstract:

Einstein’s equations can be derived from a variation of the Einstein-Hilbert action. It is also possible to use the Einstein-Hilbert action to find a constrained Hamiltonian formulation of General Relativity. Using Dirac’s canonical quantization method, one can attempt a quantization of gravitation by way of the Hamiltonian. In this paper, we review one Hamiltonian formulation of General Relativity, the ADM formalism. We explain how an initial value formulation brings new insights into General Relativity. We briefly discuss how the canonical quantization of gravity leads to the Wheeler-deWitt equation and the application of the ADM formalism to numerical relativity.

Here’s the paper.