Canonical formalism in light-cone coordinates Consider scalar field theory $$ \mathcal L = \frac{1}{2} (\partial \phi)^2 -V(\phi).$$ I want to understand Hamiltonian formalism in light-cone coordinates. I choose convention $$x^{\pm}=\frac{1}{\sqrt{2}}(x^0 \pm x^3)$$ and obtain $$\pi \equiv \frac{\partial \mathcal L}{\partial (\partial_+ \phi)}=\partial_- \phi$$ and the following Hamiltonian: $$ \mathcal H= \frac{1}{2}(\nabla_{\perp}\phi)^2+V(\phi).$$ It doesn't depend on $\pi$ and I get Hamilton's equation $\partial_+ \phi=0$, which is incorrect. This leads to following question: what did I do wrong and how to proceed correctly in this case. Of course my final goal is quantization of the theory.
Remark: I suspect that my problem is related to the fact that there are characteristic curves of this theory tangent to hypersurface $x^+$. Nevertheless, I hope there is a way around.
 A: There are at least two approaches, which yield the same result: 


*

*Dirac-Bergmann analysis: There is a second-class constraint
$$\chi~:=~\pi-\partial_-\phi~\approx~0, \tag{1}$$
which leads to the Dirac-bracket (5).

*Jackiw-Faddeev method: Recall that $x^+$ is light-cone time, so that the 
Lagrangian density 
$${\cal L}~=~\partial_-\phi ~\partial_+\phi -{\cal H}, \qquad {\cal H}~:=~\frac{1}{2}(\partial_{\perp}\phi)^2+{\cal V}(\phi), \tag{2}$$
is already on first-order form. The symplectic one-form potential can be transcribed from the kinetic term in eq. (2):
$$  \vartheta(x^+) ~=~\int\! dx^- d^2x^{\perp}
~\partial_-\phi(x)~\mathrm{d}\phi(x), \tag{3} $$
where $\mathrm{d}$ denotes the exterior derivative in infinitely many dimensions. The symplectic two-form is then 
$$  \omega(x^+)~=~\mathrm{d}\vartheta(x^+)$$ 
$$~=~\frac{1}{2}\int\! dx^- d^2x^{\perp}\int\!dy^- d^2y^{\perp}~(-2)\delta^{\prime}(x^-\!-\!y^-)~\delta^2(x^{\perp}\!-\!y^{\perp})~\mathrm{d}\phi(x)\wedge\mathrm{d}\phi(y).\qquad \tag{4}$$
The equal-time Dirac bracket on fundamental fields is the inverse matrix of the matrix for the symplectic two-form (4):
$$ \{\phi(x^+,x^-,x^{\perp}),\phi(x^+,y^-,y^{\perp})\}_{DB}
~=~-\frac{1}{4} {\rm sgn}(x^-\!-\!y^-) \delta^2(x^{\perp}\!-\!y^{\perp}) .\tag{5} $$
One may check that Hamilton's equation 
$$ \partial_+\phi(x)~\approx~\{\phi(x),H(x^+)\}_{DB}, \qquad 
H(x^+)~:=~\int\! dx^- d^2x^{\perp}~{\cal H}(x), \tag{6}$$
reproduces Euler-Lagrange (EL) equation.
