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An inverted hollow cone (cut in half) is set up with its vertex at the origin $O$ and an angle $\alpha$ between the horizontal ($x$ and $y$-axes) and the cone (so $\alpha$ close to $0$ would be a flat circle while close to $\pi/2$ would be a cylinder).

A particle is then launched and moves around the inside of the semicone (slipping), we could assume friction is negligible. Of course, there is a specific initial speed $v$, angle $\alpha$, and starting location of particle up along the incline of the cone $s$ which would produce circular motion if specific conditions are met. However, at differing initial conditions the motion will oscillate around some equilibrium (never reaching such). I have looked into the Lagrangian equations one could form for the particles based on a cylindrical coordinate system, and while this is all good I still cannot understand how to clearly model some aspects of the particle's motion

I'd like to be able to quantitatively model the displacement of the particle up the incline $s$ as a function of its angular displacement around the semicone $\phi$ ($0\leq\phi\leq\pi$), so that I could say, for example, at $\phi=\pi/4$, the ball will be $s$ cm up the incline (depending on $\alpha, v,$ and $s_{\textrm{initial}}$).

I know that this would have to do with formulating the Lagrangian of the particle, but then I have gotten stuck on having a bunch of differential equations with derivatives whose values are unknown to me.

How can this be represented in a calculable way, preferably not involving simulations? (I'm very new to DEs and Lagrangians, sorry...)

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  • $\begingroup$ You should really provide your working on the problem so we have a better handle on where you are going wrong. What's specifically about the differential equations was difficult to solve? $\endgroup$
    – Triatticus
    May 13 at 21:57
  • $\begingroup$ Would you be interested in building some sort of computational model? There doesn't seem to be an explicit closed form solution in terms of compositions of simple functions, so somewhere one probably needs to draw the line and move on to numerical implementations. I assume, for small oscillations around a circular orbit, one could carry out some reasonable series approximations and calculate a more explicit pair of equations linking the variables $s, \,\phi$ and time $t$. But are you interested in that? $\endgroup$ May 17 at 20:51
  • $\begingroup$ Well, I very much underestimated how hard modeling motion like this would be. But, as this is not pure math, approximating a solution or a function with respect to a variable is my next goal - and hoping that this can be applied to real life! I'm not sure where exactly to start. $\endgroup$
    – thekerbal
    May 19 at 0:27

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In polar coordinates, the constraint fixing the particle on the cylinder reads : $$z = r\tan(\alpha)$$

Therefore, the kynetic energy is (setting $m = 1$ for simplicity) : $$T = \frac{1}{2}(\dot r^2 + \dot z^2+r^2\dot \theta^2)=\frac{1}{2}((1+\tan(\alpha)^2)\dot r^2 + r^2\dot\theta^2)$$ The potential energy is : $$V = gz = gr\tan(\alpha)$$

Therefore, the Lagrangian is : $$L = \frac{1}{2}((1+\tan(\alpha)^2)\dot r^2 + r^2\dot\theta^2)-mgr\tan(\alpha)$$ Using this, we can derive the Euler-Lagrange equations : \begin{align} (1+\tan^2(\alpha))\ddot r &= g\tan(\alpha)-r\dot \theta^2 \\ \frac{d}{dt}(r^2\dot\theta)&=0 \end{align}

The second equation is the conservation of angular momentum. It implies that there is a cosntant $C$ such that $r^2 \dot \theta = C$. Then, plugging this into the other equation, we get a differential equation for $r$ : $$(1+\tan^2(\alpha))\ddot r -g\tan(\alpha) + \frac{C}{r^3} = 0 $$

This is equivalent to a particle with mass $1$ and moving in an effective potential : $$V_{\text{eff}}(r) = \frac{1}{1+\tan^2(\alpha)}\left(g\tan(\alpha)r + \frac{C}{2r^2}\right)$$

I don't think that there is an analytical solution to this equation. However, we can derive some qualitative properties of this potential. For example, for a given value of $C$, the radial coordinate will oscillate around the equilibrium $r_* = \sqrt[R]{\frac{C}{g\tan(\alpha)}}$, which is the radius of the corresponding circular orbit.

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  • $\begingroup$ Sorry, just wanted to notify you that you have a typo for the effective potential. i think, the last term should be something like $+ \frac{C}{2 r^2}$.. $\endgroup$ May 14 at 0:31
  • $\begingroup$ Thanks ! I edited the mistake (that's what I get from doing physics past midnight :) $\endgroup$ May 14 at 7:50
  • $\begingroup$ Thank you so much for this! To be honest, I'm not sure how the Euler-Lagrange equation is actually derived, but that is not as relevant. But more importantly, is there a non-analytical approach to solving the differential equation? I don't want the work done for me, but is there a direction I can go in - perhaps using Newtonian mechanics instead - such that I can obtain a computable relationship between, say, $r$ or $s$ with respect to $t$ or the angular displacement $\phi$? $\endgroup$
    – thekerbal
    May 14 at 16:25
  • $\begingroup$ For a non-analytical approach use Runge-Kutta numerical integration. $\endgroup$
    – F. Wright
    May 14 at 21:59

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