I'm trying to post a question but keep getting the error
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Reducing my question to just the opening sentence, I still get the same message.
Consider the anisotropic harmonic potential in two dimensions $(q_1,q_2)$ given by $$ V(q_1,q_2) = \frac{m}{2} \, q_1^2 + \frac{k}{2} \, q_2^2. $$
Could anybody explain what the problem is? Here is the full question:
Consider the anisotropic harmonic potential in two dimensions $(q_1,q_2)$ given by
$$ V(q_1,q_2) = \frac{m}{2} \, q_1^2 + \frac{k}{2} \, q_2^2, $$
or `V = m/2 q1^2 + k/2 q2^2;` in Mathematica.
The Newtonian e.o.m.s of a particle moving through this potential are
$$ \ddot{q}_1 = -q_1, \qquad \ddot{q}_2 = -\omega^2 \, q_2, $$
where $\omega = \sqrt{k/m}$ is the angular frequency of oscillations in the $q_2$-direction. Given the initial conditions $q_i(0) = q_{i,0}$ and $p_i(0) = p_{i,0}$, the e.o.m.s are solved by
$$ \begin{aligned} q_1(t) &= q_{1,i} \cos(t) + \frac{p_{1,i}}{m} \, \sin(t),\\ q_2(t) &= q_{2,i} \cos(\omega t) + \frac{p_{2,i}}{m \omega} \, \sin(\omega t), \end{aligned} \qquad \text{with $p_i = m \dot{q}_i$.} $$
In Mathematica:
DSolve[{q1''[t] == -q1[t], q2''[t] == -\[Omega]^2 q2[t], q1[0] == q10,
q1'[0] == p10/m, q2[0] == q20, q2'[0] == p20/m}, {q1[t], q2[t]}, t]
//FullSimplify
{{q1[t] -> q10 Cos[t] + (p10 Sin[t])/m, q2[t] -> q20 Cos[t \[Omega]] + (p20 Sin[t \[Omega]])/(m \[Omega])}}
I generated a 3d plot of the potential.
Plot3D[V /. {m -> 1, k -> 3}, {q1, -5, 5}, {q2, -5, 5},RegionFunction -> Function[{q1, q2}, m/2 q1^2 + k/2 q2^2 <= 12 /. {m -> 1, k -> 3}]]
[![enter image description here][2]][2]
What I would like to do now is draw the particle as a ball moving through this potential with a fixed energy, i.e. it always reaches the same height before rolling back down again and up the other side, thereby creating oscillatory motion. The frequency $\omega(\phi)$ is dependent on which direction the particle is moving. In theory all that would need to be done is to somehow add a ball to the plot whose position is given by the solution to the e.o.m.s above. I tried using the `Manipulate[]` command but couldn't get it to work. Any help would be much appreciated.
[2]: https://i.stack.imgur.com/8NtyI.png
