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The Lagrangian and Hamiltonian in Classical mechanics are given by $\mathcal{L} = T - V$ and $\mathcal{H}=T+V$ respectively. Usual notation for kinetic and potential energy is used.

But, in GR they are defined as $$\mathcal{L} = \frac{1}{2}g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu, \\ \mathcal{H}=\frac{1}{2}g^{\mu\nu}\dot{x}_\mu\dot{x}_\nu.$$

The Hamiltonian above is defined to be a "Super-Hamiltonian" according to MTW.

So, the Hamiltonian associated with a test particle in Schwarzschild geometry is then given by $$ \frac{1}{2} \left(A(r)^{-1} p_t^2 - B(r)^{-1}p_r^2-\frac{p_{\theta}^2}{r^2} - \frac{p_\phi^2}{r^2 \sin^2 \theta} \right). $$ This is only a kinetic energy term - - Where is the potential energy?

Rumplestillskin
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2 Answers2

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Writing a Hamiltonian down like that looks strange to me because if you look at those two expressions above they are identical: both scalars and in the end $\dot x_\mu \dot x^\mu/2$. In my edition of MTWs Gravitation they do not refer to a Hamiltonian in that form (at least I did not find one).

The typical way of deriving the Hamiltonian from a given Lagrangian is using canonical momenta/Legendre transformation: The canonical momentum conjugate to $x^\mu$ equals the momentum one-form of the particle \begin{equation}p_\mu=\frac{\partial\mathcal{L}}{\partial \dot x^\mu}=g_{\mu\nu}\dot x^\nu.\tag{1}\end{equation} The Hamiltonian is given by the Legendre transformation of the Lagrangian: $$\mathcal{H}=p_\mu \dot x^\mu-\mathcal{L}=\frac{1}{2}g^{\mu\nu}p_\mu p_\nu.\tag{2}$$

The Hamiltonian in form of eq. (2) is used by MTW in eq. (33.27c) and is called "super-Hamiltonian" in their book.

That being said using eq. (1) we may write (2) as $\mathcal{H}=\frac{1}{2}g^{\mu\nu}\dot x_\mu \dot x_\nu$ but this kind of makes our Legendre transformation pointless.

MTW use the prefix "super" and the related terms superspace and "super-Hamiltonian" to refer to different points and concepts of the related to the ADM formalism:

[Wikipedia, "Superspace", https://en.wikipedia.org/wiki/Superspace, 2017.03.07]: The word "superspace" is also used in a completely different and unrelated sense, in the book Gravitation by Misner, Thorne and Wheeler. There, it refers to the configuration space of general relativity, and, in particular, the view of gravitation as geometrodynamics, an interpretation of general relativity as a form of dynamical geometry. In modern terms, this particular idea of "superspace" is captured in one of several different formalisms used in solving the Einstein equations in a variety of settings, both theoretical and practical, such as in numerical simulations. This includes primarily the ADM formalism, as well as ideas surrounding the Hamilton–Jacobi–Einstein equation and the Wheeler–DeWitt equation.

The first time they use the term "super-Hamiltonian" is for eq. (21.12): $$\mathcal{H}(\pi_{ij},g_{mn})=0,$$ and they reference B. S. DeWitt, Phys. Rev. 160, 1113 (1967) and he presents the ADM Hamiltonian, related momenta and quantities in the 3+1 formalism.

N0va
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OP is considering a massive relativistic point particle with mass $m=1$ in curved spacetime. It seems relevant to mention that MTW's "super-Lagrangian" and "super-Hamiltonian" can be seen as the $e=1$ gauge of the Lagrangian$^1$

$$\begin{align} L~:=~&\frac{\dot{x}^2}{2e}-\frac{e (mc)^2}{2} -V , \cr \dot{x}^2~:=~&\sum_{\mu,\nu=0}^3 g_{\mu\nu}~\dot{x}^{\mu}\dot{x}^{\nu}, \cr \dot{x}^{\mu}~:=~&\frac{dx^{\mu}}{d\lambda}, \end{align}\tag{1} $$

and the corresponding Hamiltonian Lagrangian

$$\begin{align} L_H~:=~& \sum_{\mu=0}^3p_{\mu} \dot{X}^{\mu} - H, \cr H~:=~& eT+V,\cr T~:=~&\frac{1}{2}(p^2+(mc)^2), \cr p^2~:=~& \sum_{\mu,\nu=0}^3g^{\mu\nu}(X)~p_{\mu} p_{\nu},\end{align} \tag{2} $$

see e.g. this, this & this Phys.SE post. Here $\lambda$ is the worldline parameter, $e$ is an auxiliary einbein field, and $H$ is the Hamiltonian.

The perhaps more familiar square root form appears by integrating out the einbein field $e>0$. Various possible gauge choices are given in my Phys.SE answer here for a free particle in Minkowski spacetime.

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$^1$ In this answer we choose the Minkowski signature $(-,+,+,+)$.

Qmechanic
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