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- Homework Statement
- Let there be a pendulum which is free to move in space, the pendulum is connected to a wire with a constant length, which is connected to a fixed point. will use ##\theta, \phi## to point its location.
1. Write ##\vec{r}(\theta,\phi)##
2. Find the lagrangian
3. write equations of motion
4. Which coordinate is cyclic? and which size is invariant
5. plugin the invariant size and find a. equilibrium b. small fluctuations
- Relevant Equations
- ##E_k=\frac{m\dot{r}^2}{2}##
##E_p=mgh##
##L=E_K-e_P##
##\frac{\partial L}{\partial q_i }-\frac{d}{dt}\frac{\partial L}{\partial \dot{q_1}}##
##H(p,q)=p\dot{q}(p.q)-L##
So I answered 1 and 2, got:
1. ##\vec(r)(\theta,\phi)=l(sin \theta cos \phi, sin \theta sin \phi, -cos \theta)##
2. ##L=\frac{ml^2 (\dot{\theta}^2+\dot{\phi}^2 sin^2 \theta)}{2}+mglcos \theta##
3. a ##mlsin \theta -mgsin \theta =l^2 \ddot{\theta}## , b. ##ml^2 \ddot{\phi}=0##
4. I know that ##\phi## is a cyclic coordinate, because only its derivative is in the lagrangian. which mean that the momentum in the ##\phi## axis is constant.
Now it is seems the "invariant size" (sorry it is a direct translation) is related to the Hamiltonian (which I do not know why).
The Hamiltonian is defined as ##H(p,q)=\sum_{i=1}^{n}(p_i\dot{q_i})-L## where n is the number of coordinates.
So we first need to find ##p_\theta=\frac{\partial L}{\partial \dot{\theta}}=ml^2\dot{\theta}## and ##p_\theta=\frac{\partial L}{\partial \dot{\phi}}=ml^2\dot{\phi}##
No the Hamiltonian should not have ##\dot{\theta}## or ##\dot{\phi}## so ##\frac{p_\theta}{ml^2}=\dot{\theta}## and ##\frac{p_\phi}{ml^2}=\dot{\phi}##
So the Hamiltonian ##H=p_\theta*\frac{p_\theta}{ml^2}+p_\phi*\frac{p_\phi}{ml^2}+\frac{ml^2 (\dot{\theta}^2+\dot{\phi}^2 sin^2 \theta)}{2}+mglcos \theta## ?
I know that there are ##\dot{q_i}=\frac{\partial H}{\partial p_i}## and ##\dot{p_i}=-\frac{\partial H}{\partial q_i}## but how are they related to the "invariant size", equilibrium and "small fluctuations" (again direct translation)
Thanks
1. ##\vec(r)(\theta,\phi)=l(sin \theta cos \phi, sin \theta sin \phi, -cos \theta)##
2. ##L=\frac{ml^2 (\dot{\theta}^2+\dot{\phi}^2 sin^2 \theta)}{2}+mglcos \theta##
3. a ##mlsin \theta -mgsin \theta =l^2 \ddot{\theta}## , b. ##ml^2 \ddot{\phi}=0##
4. I know that ##\phi## is a cyclic coordinate, because only its derivative is in the lagrangian. which mean that the momentum in the ##\phi## axis is constant.
Now it is seems the "invariant size" (sorry it is a direct translation) is related to the Hamiltonian (which I do not know why).
The Hamiltonian is defined as ##H(p,q)=\sum_{i=1}^{n}(p_i\dot{q_i})-L## where n is the number of coordinates.
So we first need to find ##p_\theta=\frac{\partial L}{\partial \dot{\theta}}=ml^2\dot{\theta}## and ##p_\theta=\frac{\partial L}{\partial \dot{\phi}}=ml^2\dot{\phi}##
No the Hamiltonian should not have ##\dot{\theta}## or ##\dot{\phi}## so ##\frac{p_\theta}{ml^2}=\dot{\theta}## and ##\frac{p_\phi}{ml^2}=\dot{\phi}##
So the Hamiltonian ##H=p_\theta*\frac{p_\theta}{ml^2}+p_\phi*\frac{p_\phi}{ml^2}+\frac{ml^2 (\dot{\theta}^2+\dot{\phi}^2 sin^2 \theta)}{2}+mglcos \theta## ?
I know that there are ##\dot{q_i}=\frac{\partial H}{\partial p_i}## and ##\dot{p_i}=-\frac{\partial H}{\partial q_i}## but how are they related to the "invariant size", equilibrium and "small fluctuations" (again direct translation)
Thanks