Coupled 2nd-Order Non-linear ODEs

  • Thread starter cepheid
  • Start date


Staff Emeritus
Science Advisor
Gold Member
1. The problem statement, all variables and given/known data

I'm trying to solve the equations:

[tex] \ddot{\phi} + 2\left(\frac{\cos \theta}{\sin \theta}\right) \dot{\theta}\dot{\phi} =0 [/tex]​


[tex] \ddot{\theta} - \sin \theta \cos \theta \dot{\phi^2} =0 [/tex]​

for [itex] \theta(\lambda), \phi(\lambda) [/itex] where the dots represent differentiation w.r.t the parameter [itex] \lambda [/itex].

2. Relevant equations

Chain rule, other differentiation tricks

3. The attempt at a solution

I've tried to tackle the first equation by writing:

[tex] \frac{\cos \theta}{\sin \theta} \dot{\theta} = \frac{1}{\sin \theta} \frac{d}{d\theta}(\sin \theta) \dot{\theta} = \frac{d}{d\lambda}[\ln (\sin \theta) ] [/tex]​

I'm wondering if it's correct to do that. It seems to follow from the chain rule, but I'm not sure. If so, then the equation becomes:

[tex]\frac{\ddot{\phi}}{\dot{\phi}} = -2\frac{d}{d\lambda}[\ln (\sin \theta) ] [/tex]

[tex]\frac{1}{\dot{\phi}}\frac{d \dot{\phi}}{d\lambda} = -2\frac{d}{d\lambda}[\ln (\sin \theta) ] [/tex]

[tex]\frac{d \ln ( \dot{\phi})}{d\lambda} = -2\frac{d}{d\lambda}[\ln (\sin \theta) ] [/tex]​

Then I integrated both sides w.r.t. lambda:

[tex]\ln ( \dot{\phi}) = - 2\ln (\sin \theta) + \mathrm{~const.} [/tex]

[tex] \dot{\phi} \propto \frac{1}{\sin^2 \theta} [/tex]​

Then I tried substituting this result into the second ODE, but it gave me:

[tex] \ddot{\theta} \propto \frac{\cos \theta}{\sin^3 \theta} [/tex]​

which I don't know what to do with.
Last edited:


Staff Emeritus
Science Advisor
Gold Member
Just bumping the thread. Any insight you might have, esp. regarding the last line, would be most helpful.

Want to reply to this thread?

"Coupled 2nd-Order Non-linear ODEs" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Top Threads