Question

Equations of Motion:

If I want to solve differential equation:\[\frac{ d }{ dt }(ml ^{2}\theta ^{'}\sin ^{2}\phi)\]

How do I know which elements to differentiate with respect to time?

Thanks

Answer 1

theta, since it is only thing that could depend on time

Answer 2

\(\theta\) seems to be a function, but we don't see the variable t at all. there are no other variables? (it is not \(\theta(x)\)...)
the other symbols are often parameters.. it's difficult from our point of view.

what you wrote cannot be called an equation.. is there a '= ...' sign that you did not write?

Answer 3

@sarahusher You need to know which variables connect to what. You need some form of explicit definition of each variable.
For example, is m mass or momentum or moment of inertia?
Stuff like that.

Answer 4

Sorry I'll explain
I'm solving a Lagrangian System, which I have worked out to be\[L=\frac{ 1 }{ 2 }ml ^{2}\phi'^{2}+\frac{ 1 }{ 2 }ml ^{2}\theta'^{2}\sin ^{2}\phi-mglcos \phi \]
and I need to find Lagrange's Equations of Motion such that:\[\frac{ d }{ dt }(\frac{ \delta L }{ \delta q' })-\frac{ \delta L }{ \delta q }=0\]

Where
q is the coordinate (ie theta or phi)
q' is the derivative

Answer 5

both \(\phi\),\(\theta\),\(\theta'\) and \(\phi'\) are all functions of \(t\).
then you must use the rule \((fg)'=f'g+fg'\) with \(f=\theta'\) and \(g=C\sin^2(\phi)\). (with the correct C).

Answer 6

both = all here..

Answer 7

it'll give \(f'g+fg'=\theta''\sin^2\phi = \theta'(2\sin(\phi)\cos(\phi)\phi')\).
(times \(ml^2\))

Answer 8

sorry, the second '=' sign is a '+'.

Answer 9

Oh okay, I see! Thankyou so much!

Answer 10

np... a pleasure.