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import mmf_setup;mmf_setup.nbinit()
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt

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Assignment 4: Small Oscillations

Due: 11:59pm Friday 27 October 2023

Masses and Springs 1

Consider an equiliateral triangle of three equal masses \(m\) and springs with spring constant \(k\) and equilibrium length \(l\) as discussed in class. Consider this free to move in three dimensions (i.e. on the International Space Station).

1. How many normal modes do you expect for this system?

2. How many different frequencies do you expect?

3. What would be the degeneracy of these frequencies? Describe the modes with a sketch.

4. Compute at least one non-zero frequency.

Masses and Springs 2

Repeat the process for a tetrahedron with 4 equal masses and 6 equal springs. You may want to check your results with the code on CoCalc.

2 Masses

Provide a complete solution describing the motion of two coupled oscillators as shown below that move in 1D (left-right) without friction etc.

Be sure to check various limiting cases to make sure your answer is correct, such as equal masses and spring constants, setting some spring constants to zero, etc.

Hint: Your final answer will probably be simpler if your replace your masses and spring constants with the frequencies

\[\begin{align*} \omega_{j}^2 &= \frac{k_{j} + k_{12}}{m_{j}}, \\ \omega_{12}^2 &= \frac{k_{12}}{\sqrt{m_1m_2}}. \end{align*}\]

Use your answer to characterize how energy is transferred between two harmonic oscillators. I.e. consider the case where the middle spring has a very small spring constant \(k_{12} \ll k_{1, 2}\). This corresponds to two weakly coupled harmonic oscillators. Suppose you excite the left oscillator with angular frequency \(\omega_1 \approx \sqrt{k_1/m_1}\): what conditions must be true in order that a significant portion of the energy can be transferred to the second oscillator with angular frequency \(\omega_2 \approx \sqrt{k_2/m_2}\)?

Be sure to apply the checklist described at the start of the course when you consider the completeness of your solution:

1. Is the problem well defined and well formulated? (Does it have a definite answer? Could you tell a computer how to solve this problem?)

2. How will you describe the solution? (What variables will you use? What parameters are required?) Note: the solution will often appear much simpler if you describe the solution with appropriate variables.

3. What is your intuition for the problem? What do you think the system will do? Record your predictions before you start solving the equations.

4. Formulate the equations that will distinguish the physical solution from all other possible behaviors admitted by your description. What physics do you include? What effects do you ignore? Justify briefly. (If you were to setup an experiment to test your results, would you need to consider the effect of Jupiter’s gravitational field? Why or why not?)

5. Solve the equations to get the solution. Consider different strategies here as some approaches will be much cleaner and more efficient than others. I tend to start working quickly to see how things progress, then once I see my way through to a portion of the solution, I often start over again using a more streamlined approach that will ensure I make fewer mistakes with my algebra etc. Some suggestions:

   1. Identify all dimensionless parameters. The qualitative behavior of the system will depend (only) on these.

   2. Are there any simplifying limits where you can quickly solve the problem? If you get stuck, these might help you find the more general solution.

   3. If finding an analytic solution is challenging, maybe try quickly implementing a numerical solution so you can build a better intuition for what should happen. This introduces the added complication of making sure your numerical implementation is correct, however, I find I often learn most about a problem while trying to making the numerical implementation agree with my analytic calculations, so this extra effort is usually not wasted.

6. Check your solution:

   1. Does it agree with your intuition? If not, you need to either find your mistake, or possibly change your mental model of how this system works (once you verify that your solution is indeed correct).

   2. Is it dimensionally correct?

   3. Are there limiting case you can check easily?

   4. Can you compare with a colleague who independently solved the problem?

   5. Can you solve it a different way to compare? (Maybe with different variables or in a different coordinate system?)

   6. Can you do an experiment or relate this to a physical system where you know what happens?

7. Generalize: Sometimes once you have solved a problem, you can generalize your solution. In this case, do you see how you could solve a generalized problem with 3, 4, or more masses? Does your method generalize to the cased of different masses or different spring constants? Even if you do not actually solve the generalized problem, thinking about this can help you find more efficient techniques for solve similar problems in the future.

(Optional) Double Pendulum

Consider a double pendulum as shown below:

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import sympy
sympy.init_printing()


from phys_521 import double_pendulum
double_pendulum.draw_figure()

/home/docs/checkouts/readthedocs.org/user_builds/physics-521-classical-mechanics-i/checkouts/latest/src/phys_521/double_pendulum.py:79: SyntaxWarning: invalid escape sequence '\d'
  '''

The formal task here is to characterize and quantify the normal modes of this system, but I would like you to take this further and perform a thorough analysis of this system, investigating both analytic features (such as the normal modes) and features that are only apparent numerically. Upon request, I will provide code examples to help you analyze this system, but would like you to first try to solve it from scratch yourself. (Please use the notation given above so that we can compare answers.)

Here are some recommendations for analyzing this system. Note, I recommend first finding out as much information as possible using techniques that take very little of your time and effort. These quick solutions might be approximate, or very slow to run on a computer but get you something you can use to check your more refined solutions later. Once you have a set of quick and dirty results that you know are correct, you can start to implement a more elegant or efficient solution, checking it against the other solutions. As an example, consider performing a nasty integral - the integral might admit a nice analytic form after performing some clever substitutions, but you might first try simply approximating it numerically to see if you can get an idea of the answer. Then as you implement the more sophisticated transformations, you can keep checking to make sure that you have not make any mistakes.

Brief Exploration: Spend a few minutes thinking about the problem, maybe writing a Lagrangian, getting familiar with the coordinates and structure. The equations for this system are moderately complicated, so correctly deriving them and implementing them can be prone to errors, thus it is crucial that you think about how you can check your answers. I will suggest some general approaches here: not all will be relevant or useful for this problem.

Dimensionless Parameters: Identify the dimensionless parameters of the problem. What will set the scale for periods? When will the motion be simple, when will it be complicated? In your normal mode analysis, what will constitute “small”? These parameters will characterize the different qualitative behaviors of the system, allowing you to classify them.

Limiting Cases: Identify simple limiting cases. Can you reduce the system to a simpler system with appropriate limits? Write these limits down explicitly, and if possible, determine how the leading order corrections will scale as you move away from the ideal limits. This will provide a very useful way of quantitatively testing your code. (One issue is that some useful limits might be impossible to implement numerically because of divergence or singularities. Thus your numerical implementation will have to simulate a system close to, but not at, the limit. Since the system cannot be exactly situated at the limit in these cases, understanding how the errors will scale is crucial for testing.)

Conservation Laws and Symmetry: Identify conserved quantities, or relationships that follow from symmetry.

Brute Force Solution: Is there a simple way to implement brute force solution? Sometimes a problem can be formulated in a dead simple way that takes you almost no time, but perhaps is very slow for the computer, or can only give an approximate answer. If so, get a simple or limited version of your code working.

CAS: If your problem is messy, can you use a computer algebra system CAS like SymPy or Sage (both can be used on CoCalc) to make sure your results are correct? Sometimes these can even be used to generate the functions used in your numerical code. (Warning: Although I really like this approach in principle, I find that often I waste far more time trying to automatically generate code than I would spend if I just sat down and did the thing by hand once. However, if you need to generate many similar but different pieces of code with changing functional forms, then code generation becomes a very viable option. Keep in mind, however, that the final functional form generated by a CAS will likely not be optimal in terms of performance, or roundoff error. Often hand-manipulation will greatly improve the result, but this is where I end up wasting lots of time - trying to automate these simplifications.)

Numerical Implementation: Once you have a numerical implementation, test it carefully against the limiting case you found earlier. Also check that your units etc. are correct by running the code with different scalings of the parameters, but the same dimensionless combinations. The results should be identical, but with different units.

Testing: Check that your code obeys all the conservation laws and symmetry properties you derived earlier. If you find something that seems wrong, how do the errors behave? If they are smooth, there is probably some systematic error (missing factor, etc.). If they appear to fluctuate rapidly or randomly, they might be numerical errors due to roundoff, or an algorithm not converging. (With solve_ivp or similar integrators, you might have problems if you take to large steps. These can be fixed by taking smaller step sizes, or increasing the equivalent of the max_step parameter to allow the algorithm to do more work for each time step.)

Exploration: Armed with a validated numerical solution, start exploring various aspects of the physics. Check your intuition, and try to test analytic results and approximations. If you find something surprising, can you quantify it or understand what is happening qualitatively from the analytics?

Generalization: Consider generalizing the problem: are there any ways you can generalize your solution with a minimal amount of work to extend its range of validity? Can you test other aspects of physics or apply your solution to related problems? Once you have a working numerical solution to a problem, and have put some time into understanding the behavior of the system, it is a good idea to see if you can get some mileage out of it. (For example, the system under consideration here exhibits chaotic dynamics. It is thus provides a great opportunity for exploring and learning about chaos, Poincaré sections, Lyapunov exponents, etc.: