A point particle, mass m, is trapped in the minimum of a potential: $$ V(x) = \frac{cx}{x^2 + a^2} $$
It is given an initial velocity v0 in the positive x-direction. What will the particle do? If v0 is small enough, the particle will oscillate like an harmonic oscillator. Find the oscillation frequency. How does it depend on a, c, m?
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In the animation, the harmonic approximation of the particle motion is given by the green time-trace and the green dot in the V(x)-graph. In red is the true particle motion given (numerically solved).
The left graph shows the potential V(x). The right one gives the position as a function of time.
The green particle moves as if it is trapped in a square potential, found by trunckating a Taylor expansion of the potential V(x), after the second order term (see left graph).
For what velocity range will the red particle espace to -∞ ? For what velocity to +∞ ? How does this depend on a, c, m?
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