# Plotting a parameter Plane of a spring pendulum in Mathematica

• Mathematica
• MarkTheQuark
In summary: In the text, the authors mention a method for finding a parameter plane which minimizes the energy. However, they do not provide a step-by-step procedure for doing this in Mathematica, and the procedure they give is for an entirely different system.

#### MarkTheQuark

TL;DR Summary
I need help plotting a parameter plane of energy-ratio R and the frequency-ratio of a spring pendulum.
I'm reading an article about the order-chaos-order sequence of a spring pendulum [Ref 1], as I'm reading it I'm trying to reproduce the graphs and results through Mathematica.
However, I am new to this software.
I will list below some of the most important equations mentioned in the paper.
"In its equilibrium position the spring will be stretched, due to the weight rng, to a length: ## l_c = l_0 + \frac{mg}{k} ##
angular frequency of the spring: ## \omega_s = \sqrt{\frac{k}{m}} ##
frequency of the pendulum: ## \omega_p = \sqrt{\frac{g}{l_c}} = \sqrt{\frac{g}{l_0 + mg/k}} ##
Total Energy: ## E = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2) + mgy + \frac{1}{2} k (\sqrt{x^2 + y^2} - l_0)^2 ##
Minimum energy: ## E_{min} = -mg (l_0 + \frac{1}{2} \frac{mg}{k}) ##
With that, the author makes a contour plot of the potential energy [Fig 1], and a Parameter Plane of R and ## \mu ## [Fig 2], where R and ## \mu ## are given by:
## R \equiv - \frac{E}{E_{min}} ##
## \mu = 1 + \frac{k l_0}{mg} ##

So, how did he found this parameter plane? And how can I remake it in Mathematica?

The article in question:
Ref 1 - The order—chaos—order sequence in the spring pendulum
J.P. van der Weele and E. de Kleine
Physica A: Statistical Mechanics and its Applications, 1996, vol. 228, issue 1, 245-272

Figures:

#### Attachments

• Fig 1.png
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• Fig 2.png
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From the context, $\mu$ and $R$ must be non-dimensional constants which characterise the system, eiher because they appear in the equation of motion when appropriately scaled or they are conserved quantities fixed by the initial conditions, again when appropriately scaled; I assume the authors define these scalings somewhere in the text. (I think it's clear that position is scaled by $l_0$; the time scaling is probably by reference to one of the natural frequencies of the system.)

To reproduce the figure, you would have to conduct a large number of observations (ie. numerical simulations) of the system for a variety of values of the parameters and classify their behaviour as either chaotic or non-chaotic.

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## What is a spring pendulum?

A spring pendulum is a physical system that consists of a mass attached to a spring, which is then suspended from a fixed point. As the mass moves, it causes the spring to stretch or compress, resulting in a periodic motion.

## How can I plot a parameter plane of a spring pendulum in Mathematica?

To plot a parameter plane of a spring pendulum in Mathematica, you can use the ParametricPlot3D function. This allows you to specify the equations for the motion of the mass and spring in terms of the parameters, and then plot the resulting motion in a three-dimensional space.

## What parameters do I need to consider when plotting a parameter plane of a spring pendulum?

The main parameters to consider when plotting a parameter plane of a spring pendulum include the mass of the object, the spring constant, the initial position and velocity of the mass, and the length and angle of the spring. These parameters will affect the resulting motion and can be adjusted to create different plots.

## Can I customize the appearance of the parameter plane plot?

Yes, you can customize the appearance of the parameter plane plot in Mathematica by using various options for the ParametricPlot3D function. This includes changing the color, style, and thickness of the plotted lines, as well as adding labels and axes to the plot.

## Are there any tutorials or resources available for plotting a parameter plane of a spring pendulum in Mathematica?

Yes, there are several tutorials and resources available online that can guide you through the process of plotting a parameter plane of a spring pendulum in Mathematica. You can also refer to the official Mathematica documentation for more information and examples.