# Write a Lagrangian for a constrained particle

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?

• What happened to the mass term from kinetic energy in the Lagrangian? – exp ikx Apr 24 at 8:05
• ups, let me fix it, it's a typo, thanks – VoB Apr 24 at 8:06
• Anyway, I need to know if my procedure is correct or not – VoB Apr 24 at 8:08
• 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. – Valter Moretti Apr 24 at 8:45
• 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. – exp ikx Apr 24 at 11:51