I am given the Lagrangian:
$\dot{x}^2+2t$
and need to show that:
the solution $S$ of the corresponding Hamilton-Jacobi equation is given by $S(t,x,\alpha,\beta) = -\alpha^2t + 2 \alpha x +\beta +t^2$ where $\alpha$ and $\beta$ are constants and hence show that the extremal for this problem are straight lines
I am also told that:
Assume that the solution $S$ is of the form $u(x) + v(x)$
I start off by solving for $H$ using the equation: $H = -L +p_x \dot{x}$ where $p_x = \frac{\partial L}{\partial \dot{x}}$ and therefore $\dot{x} = \frac{p_x}{2}$
This gives me $H = \frac{1}{4} {p_x}^2-2t$
To derive the Hamilton Jacobi equation, I use the fact that $H(t,x_i,\frac{\partial S}{\partial x_i})+\frac{\partial S}{\partial t} = 0$ to derive:
$2t - \dfrac{1}{4}\left(\dfrac{\partial S}{\partial x}\right)^2 = \dfrac{\partial S}{\partial t}$
At this point, I am pretty sure the above is correct and "along the right path".
The problem I have now, is how do I solve for $S$ and prove that it equals to the equation given? Any guidance would be appreciated.