I can't answer the following question about a (simple) physical system I have studied using Lagrangian mechanic techniques.
So, we have a straight rigid rod in a horizontal plane, symmetrically fixed in the origin of the cartesian axes. On the rod is a material point P, which has a mass equal to $m$. The lagrangian coordinates are:
$s=P-O$, distance from the origin to the point
$\theta$, the angle between the positive $x$ axis and the point position on the rod. We have an elastic force acting on the system, so $k$ from now on is the elastic coefficient.
The request is to "qualitatively describe the motion" by reducing the problem to a unidimensional one and finding the value of the parameters for which we have oscillations around the origin (so $s$ assumes both positive and negative values) and for which we have only negative or positive values of $s$.
I've already reduced the problem to a unidimensional one using the fact that $H$ (hamiltonian function) and the component of the angular moment projected on the $z$ axis are conserved, thus obtaining
$H=E_0= \frac{m}{2} \dot s^2 + \frac{L_{0_z}^2}{2ms^2} + \frac{ks^2}{2} \\ $,
where $L_{0_z}$ is the value assigned to the conserved component of the angular moment. We clearly have an equation term related to the kinetic energy, and the other terms are a sort of "effective potential".
I can't proceed from here, though I have some concepts clear in my mind, in particular:
- in the extreme point of the oscillations, kinetic energy is null and we only have potential energy, which there assumes its maximum value;
- kinetic energy in the origin has a maximum if oscillations are happening;
I'm sure that by putting these ideas together I'll get to the result, but I'm not there yet. Any help or input is appreciated, thanks in advance.
