Sn(u), Jacobi elliptic function, for simple pendulum of any amplitude

• mishima
In summary, the problem is asking for a solution to the equation ##\sqrt{\frac{2g}{l}} \int_0^t dt'## where ##\theta=0## at ##t=0##. The equation uses the same substitutions that were used in problem 17, so the solution is the same.
mishima
Homework Statement
Boas Chapter 11, Section 12, Problem 22. Show that the exact solution when ##\alpha## is not small is
$$sin \frac \theta 2=sin\frac \alpha 2 sn \sqrt \frac g l t$$
where k = ##sin\frac \alpha 2## is the modulus of the elliptic function.
Relevant Equations
$$sn(u) = \int_0^\phi \frac{d\theta}{\sqrt{1-k^{2}sin^{2}\theta}}=sin\phi$$
I understand how to reach

$$\int_0^\phi \frac{d\theta}{\sqrt{1-k^{2}sin^{2}\theta}}=\sqrt \frac g l t$$

from physics but from there I don't get how to turn that into this new (for me) sn(u) form.

Anything I can do to make this more answerable?

I've read the sn, cn, and dn functions were widely used in the past but not so much these days...hard to find some worked examples.

mishima said:
Problem Statement: Boas Chapter 11, Section 12, Problem 22.

Have you done problem 17 from the same section. If not, then first do problem 17.

mishima and dextercioby
Thanks, I did that one earlier, and could recognize the same ##\frac {sin\frac \theta 2} { sin \frac \alpha 2}## substitution in the exact solution, but I don't understand how this makes the sn(u) function.

mishima said:
Thanks, I did that one earlier, and could recognize the same ##\frac {sin\frac \theta 2} { sin \frac \alpha 2}## substitution in the exact solution, but I don't understand how this makes the sn(u) function.

Can you show your solution to question 17?

mishima
Alright.

The section narrative provided the starting point of
$$\int_0^\alpha \frac {d\theta} {\sqrt {cos\theta-cos\alpha}}=\sqrt {\frac {2g} l} \frac {T_\alpha} 4$$
where ##\alpha## is the amplitude of the pendulum's swing, and ##T_\alpha## is the period for a swing from ##\alpha## to ##-\alpha##. ##~T_\alpha/4## then represents a swing through ##\pi/2## and is the t in problem 22.

I can change the cosines into sines using the double angle trig identity:

$$\int_0^\alpha \frac {d\theta} {\sqrt {2sin^2\frac {\alpha} 2-2sin^2\frac \theta 2}}=\sqrt {\frac {2g} l} \frac {T_\alpha} 4$$

Now I can make the denominator more like an elliptic integral by pulling out the following factor:

$$\int_0^\alpha \frac {d\theta} {\sqrt 2 sin\frac {\alpha} 2\sqrt {1-\frac {sin^2\frac {\theta} 2 }{sin^2\frac {\alpha} 2}}}=\sqrt {\frac {2g} l} \frac {T_\alpha} 4$$

Now the substitution ##x=\frac {sin^2\frac {\theta} 2 }{sin^2\frac {\alpha} 2}##, its differential, and corresponding limits of integration changes it from Legendre to Jacobi form. This step is of course making my spider senses tingle with problem 22, but again I don't quite understand the sn(u) notation:

$$\sqrt 2 \int_0^1 \frac {dx} {cos\frac \theta 2 \sqrt {1-x^2}}$$

Then its just a matter of using the pythagorean formula for sines and cosines to turn the cosine in the denominator into the Jacobi form, then one last substitution to get the k in the right place:

$$\sqrt 2 \int_0^1 \frac {dx} {\sqrt {1-x^2sin^2\frac \alpha 2} \sqrt {1-x^2}}$$

But that's just

$$\sqrt 2 K(sin\frac \alpha 2)$$

Good.

mishima said:
This step is of course making my spider senses tingle with problem 22, but again I don't quite understand the sn(u) notation

To link up with sn, let' s redo problem 17, but with different limits of integration. Suppose that ##\theta=0## at ##t=0##, and that the angular position at general ##t## is ##\theta##. Then, equation (12.6) of the 3rd edition gives

$$\sqrt{\frac{2g}{l}} \int_0^t dt' = \int_0^\theta \frac{d \theta'}{\sqrt{\cos \theta' - \cos \alpha}} .$$

Now, use the same substitutions that are used in problem 17. What happens?

What is the Sn(u) function?

The Sn(u) function is a special function known as the Jacobi elliptic function. It is used to describe the motion of a simple pendulum of any amplitude. It is a periodic function with a period of 4K, where K is the complete elliptic integral of the first kind.

How is the Sn(u) function related to the simple pendulum?

The Sn(u) function is used to describe the displacement of a simple pendulum from its equilibrium position as a function of time. It takes into account the amplitude, length, and gravitational acceleration of the pendulum to accurately describe its motion.

What is the significance of the Jacobi elliptic function in studying simple pendulum motion?

The Jacobi elliptic function is significant in studying simple pendulum motion because it provides a more accurate and complete description of the pendulum's motion compared to the simple harmonic motion model. It takes into account the nonlinearity of the pendulum's motion and allows for the analysis of larger amplitudes that cannot be accurately described by the simple harmonic motion model.

How can the Sn(u) function be used to determine the period of a simple pendulum?

The period of a simple pendulum can be calculated using the Sn(u) function by finding the value of u at which the function reaches its maximum value. This value of u corresponds to half of the period of the pendulum's motion. The full period can then be calculated by multiplying this value by 4K, where K is the complete elliptic integral of the first kind.

Are there any limitations to using the Sn(u) function for simple pendulum motion?

While the Sn(u) function provides a more accurate description of simple pendulum motion, it is still an approximation and may not fully capture all the complexities of a real pendulum. It also assumes a perfect pendulum with no friction or air resistance, which may not be the case in real-world situations.

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