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I have the following exercise, but I have no solutions, so I'd need a check, since I'm pretty new to Lagrangian mechanics.

Let $P$ be a point particle of mass $m$ constrained to move on the upper lap of a circular orthogonal cone, parametrized by $z=\sqrt{x^2+y^2}$ and gravity acts on the system. As Lagrangian coordinates, use polar coordinates $(r,\theta)$ on the $(x,y)$-plane.

Write the Lagrangian and Lagrange equations

My solution

My main problem is that I don't know how to consider the $z$-coordinate. I mean, the text only says to use polar on $(x,y)$ and I'm a bit confused.

Anyway, since I have that $z=\sqrt{x^2+y^2}$, then $z=r$, where $r$ is the radius.

I start writing the Kinetic energy: $T=\frac{1}{2} m ||\vec{v_p}||^2$,

where $v_p=[\dot{r} \cos(\theta) - r \dot \theta \sin(\theta),\dot r\sin(\theta)+r \dot \theta\cos(\theta),\dot r]$

Then, $||v_p||^2= 2\dot r^2+ r^2 \dot \theta^2$

So, \begin{align} T=\frac{1}{2}m[2\dot r^2+ r^2 \dot \theta^2] \end{align}

The potengial energy is simply

\begin{align} V=mgz=mgr \end{align}

and then the Lagrangian is

\begin{align} \mathcal{L}=\frac{1}{2} m [2 \dot r^2+r^2 \dot \theta^2] - mg r \end{align}

and find Lagrange equations is just a matter of computations now


Could it be okay or am I missing something?

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    $\begingroup$ What happened to the mass term from kinetic energy in the Lagrangian? $\endgroup$ – exp ikx Apr 24 at 8:05
  • $\begingroup$ ups, let me fix it, it's a typo, thanks $\endgroup$ – VoB Apr 24 at 8:06
  • $\begingroup$ Anyway, I need to know if my procedure is correct or not $\endgroup$ – VoB Apr 24 at 8:08
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    $\begingroup$ I think your procedure is OK. However, something of crucial relevance is actually missed in the text of the exercise: the surface must be frictionless, otherwise Lagrange equations are not the correct equations of motion. $\endgroup$ – Valter Moretti Apr 24 at 8:45
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    $\begingroup$ It is okay as long as you are following the same convention for the sign of $g$ throughout. Usually, we put $V = -mgz$, but of course, it depends on what is your reference point. $\endgroup$ – exp ikx Apr 24 at 11:51

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