- #1
badluckmath
- 9
- 3
- Homework Statement
- Show that ## \omega = \frac{\sqrt{a^{2}+g^{2} }}{l} ## for small angles
- Relevant Equations
- After solving the Hamilton equations, we find that ## \frac{d^{2}\theta}{dt^{2}} = -gsin(\theta)/l- acos(\theta)/l ##
Here is an image of the problem:
The problem consist in finding the moviment equation for the pendulum using Lagrangian and Hamiltonian equations.
I managed to get the equations , which are shown insed the blue box:
Using the hamilton equations, i finally got that the equilibrium angle ##\theta_{e}## : $$\theta_{e} = \tan^{-1}(\frac{-a}{g})$$m which is the angle where ## \frac{d^{2}\theta}{dt^{2}} =0 ##.
Now, i got stuck. I tried to solve an EDO using the small angles aproximation, but it doesn't seems to lead me anywhere, because i don't really have information for the initial values.
I managed to get the equations , which are shown insed the blue box:
Using the hamilton equations, i finally got that the equilibrium angle ##\theta_{e}## : $$\theta_{e} = \tan^{-1}(\frac{-a}{g})$$m which is the angle where ## \frac{d^{2}\theta}{dt^{2}} =0 ##.
Now, i got stuck. I tried to solve an EDO using the small angles aproximation, but it doesn't seems to lead me anywhere, because i don't really have information for the initial values.