Lagrangian mechanics. Determine the equations of motion for a particle of mass m constrained to move on the surface of a cone in the presence of gravity. The cone has half-angle 45° and is oriented vertically, so that (using cylindrical coordinates) r-z with θ unconstrained. The gravitational potential is mg. Thus there are two equations of motion, one for z (or, equivalently, for r) and one for θ. You do not need to solve the equations of motion. However, you should give an expression for one conserved quantity other than the total energy, and state its physical interpretation. (A conserved quantity is one that does not depend on time).

In order to obtain the Lagrangian of the system we must first write the Kinetic and Potential Energies. Lets orient our axes such that the axis of the cone coincide with the z axis. In cilindrical coordinates we have

[tex]v^{2} = \frac{dr}{dt}^{2} +r^{2} \frac{d\theta }{dt} ^{2} +\frac{dz}{dt} ^{2}[/tex] - (1)

But, since the particle is constrained to move on the surface of the cilinder, we have the following relation between r and z:

[tex]\frac{r}{z}=tan(45)[/tex]

or:

[tex]z = r cot(45)[/tex] - (2)

and:

[tex] \frac{dz}{dt} = \frac{dr}{dt} cot(45)[/tex]

replacing (2) in (1) we obtain:

[tex]v^{2} = \frac{dr}{dt}^{2} (1+cot(45))+r^{2}\frac{d\theta }{dt} ^{2}[/tex] - (3)

Now the kinetic energy is given as:

[tex]T = \frac{1}{2}m(\frac{dr}{dt}^{2} (1+cot(45))+r^{2}\frac{d\theta }{dt} ^{2})[/tex] - (4)

And the potential energy is given by:

[tex]V = -mgz = -mgr cot(45)[/tex]

So the Langrangian is given by:

[tex]L = T - V= \frac{1}{2}m(\frac{dr}{dt}^{2}(1+cot(45)+r^{2})\frac{d\theta }{dt} ^{2}) + mgr cot(45)[/tex]

And the equations of motion are:

For θ

[tex]\frac{d}{dt} (mr\frac{d\theta}{dt}) = 0-->mr{d\theta}{dt}=c[/tex]

For r

[tex]\frac{d}{dt}(m\frac{dr}{dt}(1+cot(45) )= mgcot(45)+mr\frac{d\theta}{dt} ^{2}\\m\frac{d^{2} r}{dt^{2} }(1+cot(45)= mgcot(45)+mr\frac{d\theta}{dt} ^{2}[/tex]

Obtained from the Euler-Langrange equations

Here the conserved quantity is given by the first equation of motion, namely:

[tex]mr\frac{d\theta}{dt}=c[/tex]

Which is the magnitude of the angular momentum