I found out the conserved current for harmonic oscillator with angular frequency 1, particle falling in gravity close to ground ($g=1$) using maple. I'm unable to understand the result:
$$J[t](t, x(t), \frac{dx(t)}{dt}) = F1(x(t)^2 + (\frac{dx(t)}{dt})^2, -arctan(x(t)/\frac{dx(t)}{dt}) + t)$$
For particle falling under gravity:
$$J[t](t, x(t), \frac{dx(t)}{dt}) = F1(-t + \frac{dx(t)}{dt}, t^2/2 - t*\frac{dx(t)}{dt}+ x(t))$$
What do these terms mean? Also, how does one calculate conserved current for these systems? I thought that conserved current is a 4-vector that satisfies continuity equation. I'm unable to make that connection here.
