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Potential of a simple harmonic oscillator: $$U=\frac{1}{2}k x^2$$

I'm asked to calculate the trajectory of a particle moving in this potential, with initial conditions $x(t=0) = 0$ and $v(t=0)=v_0$.

What exactly do they mean by "calculate the trajectory?" What sort of thing am I supposed to end up with?

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  • $\begingroup$ What tools do you have? That is, have you learned Lagrangian or Hamiltonian dynamics, or is this for a Newtonian mechanics course? $\endgroup$
    – Kyle Kanos
    Commented Oct 1, 2014 at 14:01
  • $\begingroup$ This question leads on to calculating this trajectory in phase space, I'm just starting Hamiltonian dynamics $\endgroup$
    – Gray
    Commented Oct 1, 2014 at 14:05
  • $\begingroup$ So basically the question is asking you to write out the Hamiltonian to find $dp/dt$ and $dx/dt$, then get $x(t)$ from that. $\endgroup$
    – Kyle Kanos
    Commented Oct 1, 2014 at 14:18

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"Calculate the trajectory" just means calculate $x(t)$, given the potential energy and the initial conditions.

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To calculate the trajectory means to find the relation between $v$, $\omega$, $A$, and $x$ in the $v$-$x$ coordinate system which leads to the ellipsoid curve.

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