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Jan
28
comment About Poincare section for the double pendulum
Thanks for the clarification. I understand it a bit more now. I read a book today about finding the fixed point mathematically. There the author just try to solve the differential equation when all time derivative of each generalized coordinates to zero with linearization. So in our case, I linearize all angle but from the second equation $l_\alpha'=0$, I obtain that $\alpha=\beta=0$ and then plug that back to the last equation but I will get null solution for $l_\beta$. I understand that we could solve it from normal mode but what's wrong with this math?
Jan
27
comment About Poincare section for the double pendulum
So can I say fixed point always corresponding to regular motion and is that the reason why I have to linearize the map? So based on your description, it seems that fixed point depends on energy also or it always gives when energy is the lowest energy?
Jan
27
comment About Poincare section for the double pendulum
I seen that the fixed point is plotted in the Poincare section, so to find the fixed point, should I simply replace $\alpha$ with zero and make $\dot{l_\alpha}=0$ and solve for $\beta$ and $l_\beta$? I did that and solve it numerically, but it gives me a solution with $l_\beta=0$ and $\beta$ is non zero. However, if you read the plot the fixed point should locate around $(\beta=0, l_\beta=-0.01)$ and $(\beta=0, l_\beta=0.024)$
Jan
27
comment About Poincare section for the double pendulum
Hi there, I have another question regarding the double pendulum. In the plot you made, there is two fixed points shown in positive and negative region of $l_\beta$. I am wondering in what way I can calculate those fixed points? Should I solve the equations of $\dot{\alpha}=0, \dot{\beta}=0, \dot{l_\alpha}=0, \dot{l_\beta}=0$? The equations seem too complicated and I don't find a way to solve it.
Jan
22
comment How does one measure the frequency of a laser?
so how to measure wavelength?
Jan
21
asked Do acousto-optic modulators shift the frequency of the diffracted beams?
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21
asked How does one measure the frequency of a laser?
Jan
16
comment About Poincare section for the double pendulum
I think I confuse the change of angle over time with the angular momentum. Actually, it should be angular speed instead of angular momentum. So $\dot{\alpha}$ is angular speed and $l_\alpha$ is angular momentum, right? So can we calculate the Poincare section by restrict that $l_\alpha>0$? If so, does it make any sense to do so physically?
Jan
15
asked Will magnetic field lines change when a magnet gets closer to other magnets?
Jan
15
asked about field gradient
Jan
15
comment About Poincare section for the double pendulum
I think I understand most of that now but one of your comment there you said $\dot{\alpha}$ is not same as $l_\alpha$? So the time derivative of angle is not angular momentum?
Jan
15
comment About Poincare section for the double pendulum
Thanks a lot. That looks amazing.
Jan
15
accepted About Poincare section for the double pendulum
Jan
13
comment About Poincare section for the double pendulum
I tried many different energy, finally find that when the constant energy chosen as -2.9996992384564987, it gave the plot pretty close to the one shown in the book. So does it mean the fix point really depends on the energy chosen? My last question is in the book, there should the big circle as the boundary, how to find and plot that boundary?
Jan
13
revised About Poincare section for the double pendulum
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Jan
13
comment About Poincare section for the double pendulum
First, the bottom fix point in the book is about -0.01 but in my plot, it shifted up to -0.02. Secondly, in my result, the biggest closed curve in the second half cross the $l_\alpha$ axis at about 0.018 but in the book, it crosses at about 0.07.