- #1
TimJ
- 18
- 0
1. The problem statement
I'm trying to solve the brachistochrone problem between two points on the surface of a sphere.
2. The attempt at a solution
The Lagrangian for this problem in spherical coordinates is
[tex]
L=\frac{1}{2}m \left(r^2 \left (\frac{d \theta}{dt}\right)^2+r^2 \sin^2(\theta) \left(\frac{d \phi}{dt}\right)^2\right)-mgr\cos(\theta)
[/tex]
After applying the Euler-Lagrange equation we get:
For [tex] \theta [/tex]:
[tex] \frac{\partial L}{\partial (\frac{d \theta}{dt})} = m r^2\frac{d \theta}{dt}[/tex]
[tex] \frac{\partial L}{\partial \theta} = m r^2 \left(\frac{d \phi}{dt}\right)^2 \sin(\theta) \cos(\theta) + mg r \sin(\theta) [/tex]
[tex]
\frac{\partial L}{\partial \theta} - \frac{d}{dt} \left ( \frac{\partial L}{\partial (\frac{d \theta}{dt})} \right ) = 0 \;\; \Rightarrow \;\;
m r^2 \left(\frac{d \phi}{dt}\right)^2 \sin(\theta) \cos(\theta) + mg r \sin(\theta) -
m r^2 \frac{d^2 \theta}{dt^2}=0
[/tex]
For [tex] \phi[/tex]:
[tex] \frac{\partial L}{\partial \phi} = 0 \;\; \Rightarrow \;\; \frac{\partial L}{\partial (\frac{d \phi}{dt})}=m r^2 \sin^2(\theta) \frac{d \phi}{dt}= const.[/tex]
Now we have two equations:
[tex]
\frac{d^2 \theta}{dt^2}-\frac{g}{r}\sin(\theta)-\sin(\theta) \cos(\theta) \left(\frac{d \phi}{dt}\right)^2=0
[/tex]
and
[tex]
m r^2 \sin^2(\theta) \frac{d \phi}{dt}= const. = A
[/tex]
And here is where it stopped. I don't know how to solve this two equations.
I'm trying to solve the brachistochrone problem between two points on the surface of a sphere.
2. The attempt at a solution
The Lagrangian for this problem in spherical coordinates is
[tex]
L=\frac{1}{2}m \left(r^2 \left (\frac{d \theta}{dt}\right)^2+r^2 \sin^2(\theta) \left(\frac{d \phi}{dt}\right)^2\right)-mgr\cos(\theta)
[/tex]
After applying the Euler-Lagrange equation we get:
For [tex] \theta [/tex]:
[tex] \frac{\partial L}{\partial (\frac{d \theta}{dt})} = m r^2\frac{d \theta}{dt}[/tex]
[tex] \frac{\partial L}{\partial \theta} = m r^2 \left(\frac{d \phi}{dt}\right)^2 \sin(\theta) \cos(\theta) + mg r \sin(\theta) [/tex]
[tex]
\frac{\partial L}{\partial \theta} - \frac{d}{dt} \left ( \frac{\partial L}{\partial (\frac{d \theta}{dt})} \right ) = 0 \;\; \Rightarrow \;\;
m r^2 \left(\frac{d \phi}{dt}\right)^2 \sin(\theta) \cos(\theta) + mg r \sin(\theta) -
m r^2 \frac{d^2 \theta}{dt^2}=0
[/tex]
For [tex] \phi[/tex]:
[tex] \frac{\partial L}{\partial \phi} = 0 \;\; \Rightarrow \;\; \frac{\partial L}{\partial (\frac{d \phi}{dt})}=m r^2 \sin^2(\theta) \frac{d \phi}{dt}= const.[/tex]
Now we have two equations:
[tex]
\frac{d^2 \theta}{dt^2}-\frac{g}{r}\sin(\theta)-\sin(\theta) \cos(\theta) \left(\frac{d \phi}{dt}\right)^2=0
[/tex]
and
[tex]
m r^2 \sin^2(\theta) \frac{d \phi}{dt}= const. = A
[/tex]
And here is where it stopped. I don't know how to solve this two equations.