How do I find constraints on the Nambu-Goto Action? Let $X^\mu (t,\sigma ^1,\ldots ,\sigma ^p)$ be a $p$-brane in space-time and let $g$ be the metric on $X^\mu$ induced from the ambient space-time metric.  Then, the Nambu-Goto action on $X^\mu$ is defined to be
$$
S:=-T\int dt\, d\sigma \sqrt{-\det (g)}.
$$
(We use the convention of space-time signature $(-,+,+,+)$.)
Let us try to compute the Hamiltonian for this theory.  The first step is to calculate the conjugate momenta:
$$
P_\mu :=\frac{\partial L}{\partial (\partial _tX^\mu)},
$$
where of course $L:=-T\sqrt{-\det (g)}$.  It turns out that you are unable to invert this map to write $\partial _tX^\mu$ as a function of $X^\mu$, $P^\mu$, and $\partial _{\sigma ^i}X^\mu$, which means that the image of the Legendre transformation $(X^\mu ,\partial _{\sigma ^i}X^\mu ,\partial _tX^\mu )\mapsto \left( X^\mu ,\partial _{\sigma ^i}X^\mu ,\frac{\partial L}{\partial (\partial _tX^\mu )}\right)$ is not surjective, but instead the image of this map is a proper sub-manifold (of course, in general the image of a manifold under a smooth map will not in general be a manifold, but in our particular case, that should not be a problem) of $T^pT^*M$, and so (locally anyways) will be specified by $N$ constraints $\phi _j(X,\partial _{\sigma ^i}X,P)=0$, where $N$ is the dimension of the kernel of the derivative of the Ledgendre transformation.
In the case of the Nambu-Goto Lagrangian, I have found that $N=1+p$.  I found this by calculating the multiplicity of $0$ as an eigenvalue of $\frac{\partial ^2L}{\partial (\partial _tX^\mu )\partial (\partial _tX^\nu )}$.  The question is:  now that I know how many constraints there should be, how do I systematically find what those constraints actually are?
For what it's worth, I know that one constraint is
$$
P^2+T^2\det (k)=0,
$$
where $k$ (I have suppressed the time-dependence of $k(t)$ in the notation) is the metric on the space-like sub-manifold $X_t^\mu (\sigma ^1,\ldots ,\sigma ^p):=X^\mu (t,\sigma ^1,\ldots ,\sigma ^p)$ induced from the metric $g$ on $X^\mu$, and that the other $p$ constraints are
$$
\partial _{\sigma ^i}X\cdot P=0
$$
for $1\leq i\leq p$.  I even know how to verify that these are in fact constraints.  What I don't know, however, is how to come up with these constraints without simply pulling them out of my ass.
Is there a systematic, yet computationally feasible way of determining what these $1+p$ constraints should be?
Furthermore, I have a hunch that each of these constraints arises from a corresponding re-parameterization invariance, so if that is indeed the case, it would be wonderful if someone could elucidate this connection for me.  (Is there a re-parameterization invaraince-$\Rightarrow$-constraint theorem analogous to Noether's Theorem for symmetries and conserved quantities?)
 A: I managed to find a quasi-systematic way to do this.  The idea that allowed me to do this was inspired by Noether's Theorem.
Re-parameterization invariance is a symmetry of the system, a symmetry much stronger than an ordinary global symmetry.  Similarly, however, a constraint is also a conserved quantity, but it is something much stronger than that.  Knowing that there was some relation between the two, I suspected there might be a way to derive constraints given re-parameterization invariance in a similar way that Noether's Theorem allows you to derive conserved quantities from a known symmetry.  I thus managed to hack together a modification of the'proof' of Noether's Theorem that allowed me to calculate the constraints.  Unfortunately, however, putting the constraints entirely in terms of $X$, $\partial _{\sigma ^i}X$, and $P$ was not completely systematic, but it was still much more straightforward than simply coming up with the constraints out of thin air.  Anyways, here's what I did.  For simplicity, I only addressed the case of the string ($p=1$).
For a mapping of the string $X\mapsto X'$ which depends on a parameter $\varepsilon$, I abbreviate $\frac{d}{d\varepsilon}\big| _{\varepsilon =0}$ by $\delta$.  This notation is common amongst physicists, but they often do mention exaclty what they mean by it.  Under such a transformation of the string alone, I have
\begin{align*}
\delta L & =\frac{\partial L}{\partial X}\cdot \delta X+\frac{\partial L}{\partial (\partial _tX)}\cdot \delta (\partial _tX)+\frac{\partial L}{\partial (\partial _\sigma X)}\cdot \delta (\partial _\sigma X) \\
& =\frac{\partial}{\partial t}\left[ \frac{\partial L}{\partial (\partial _tX)}\right] \cdot \delta X+\frac{\partial L}{\partial (\partial _tX)}\cdot \delta (\partial _tX)+\frac{\partial}{\partial \sigma}\left[ \frac{\partial L}{\partial (\partial _\sigma X)}\right] \cdot \delta X \\
& +\frac{\partial L}{\partial (\partial _\sigma X)}\cdot \delta (\partial _\sigma X) \\
& =\frac{\partial}{\partial t}\left[ \frac{\partial L}{\partial (\partial _tX)}\cdot \delta X\right] +\frac{\partial}{\partial \sigma}\left[ \frac{\partial L}{\partial (\partial _\sigma X)}\cdot \delta X\right] .
\end{align*}
Note that I have assumed that the derivatives commute with the transformation (i.e. $\delta (\partial _tX)=\partial _t(\delta X)$ and $\delta (\partial _\sigma X)=\partial _\sigma (\delta X)$).  In the case of our re-parametrization invariance, this turns out to be the case, though I did have to check this and I don't see any reason why this should be true in general (though do point it out if you are aware of a reason)).
Assuming that $\delta L$ is of the form $\delta L=\partial _tf+\partial _\sigma g$, we can re-arrange this equation to obtain
$$
\frac{\partial}{\partial t}\left[ \frac{\partial L}{\partial (\partial _tX)}\cdot \delta X-f\right]=\frac{\partial}{\partial \sigma}\left[ g-\frac{\partial L}{\partial (\partial _\sigma X)}\cdot \delta X\right] .
$$
Thus, under the assumption that the appropriate functions vanish at infinity, the integral over $\sigma$ of the quantity in the time derivative will be conserved.  We suspect this might be a constraint.  This is enough motivation to write down that quantity to see if it is in fact a constraint (note that the above derivation is only supposed to be motivation to check, not an actual proof).
In the case of time re-parametrization $X(t)\mapsto X(t+\varepsilon \xi )$, $\delta X=\xi \dot{X}$ and $\delta L=\partial _t(\xi L)$ (the first you can see right away, the second I had to actually sit down and calculate).  Thus, we suspect that
$$
\xi \dot{X}\cdot P-\xi L=\text{const}
$$
might be a constraint.  In fact, if you calculate $P$ and plug it in, we indeed see that this expression vanishes identically.  So indeed it is true that
$$
\partial _tX\cdot P=-L.
$$
If you do the same thing with $\sigma$ re-parameterization, you find
$$
\partial _\sigma X\cdot P=0.
$$
Note that in this case $\delta L$ is a $\sigma$ derivative, as opposed to a time derivative as before, and so doesn't show up.  Fantastic!  The only thing that remains to be down is to eliminate the pesky $\dot{X}$.  To do this, we have to actually compute $P$.
It turns out that
$$
P_\mu =\frac{T^2}{L}\left( (\partial _tX\cdot \partial _\sigma X)\partial _\sigma X_\mu-(\partial _\sigma X)^2\partial _tX_\mu \right) .
$$
The idea is that we can use the time re-parameterization constraint to eliminate $\partial _tX$ from the expression for $P$ by contracting $P$ with itself:
$$
P^2=\frac{T^2}{L}\left( (\partial _tX\cdot \partial _\sigma X)\partial _\sigma X\cdot P-(\partial _\sigma X)^2\partial _tX\cdot P\right) =-T^2(\partial _\sigma X)^2.
$$
Et voila!  There be the sought after constraints!
And now I move on with my life . . .
