# Equation of motion for a double pendulum

firstly I am a Grade 12 Physics student currently doing my final experimental investigation which I decided to do on double pendulums. Before I got into this, I was under the impression that there was some equation I could use which I could plug in my starting conditions such as starting angle, weight of pendulum arms, their lengths, and gravity to pull out a function of x and y against time. Now I am writing the report, it seems it isnt that simple and there is a lot of terminology used on sites that I am unfamiliar with and I really don't know what I am looking for.

My maths/physics knowledge extends to know about integral/differentials, but nothing as complex as these Lagrangians, Hamiltonians or other university-level mathematical principles.

I have researched many sites, such as Wikipedia, Wolfram ScienceWorld, and other physics stack exchange questions. However, I still have only a vague idea of what I am looking for.

On the stack exchange question I linked, I became familiar with the dot notation. I guess my question is, what is the equation/function which I can use to find the path of a double pendulum given starting conditions. Looking at the wikipedia page shows me many equations of motion. Do I need the hamiltonian equations? The one linked on that stack exchange? The one on that stack exchange question uses $p_{\theta1}$ and its $p_{\theta2}$ equivalent which I don't know what they are either.

Sorry if my question is not clear enough. Also, my teacher told me that I do not need to know the proof for the equation of motion, only that I include the final equation in my report to show how accurate my data is (which I know is off due to the chaotic motion).

• In my opinion, for a project, it's interesting that there's no analytic solution. This could be the basis for learning/explaining numerical integration. – BMS Aug 7 '14 at 13:39

From the Lagrangian one can obtain the equations of motion, called the Euler-Lagrange Equations. These equations are, in general and also in this case, differential equations. As far as I understand your level of knowledge, you don't know anything about this subject?

In differential equations you are looking for a whole function as a solution, not only a variable ( in this case the solution is the function you are looking for: plug in the time and get out the position of each mass). The functions not only occur in an algebraic equation, there are also derivatives of this function in this equations. So in your case the two functions you are looking for are $\theta_1(t)$ and $\theta_2(t)$. Unfortunately you can not analytically solve these equations stated in the wiki article: " It is not possible to go further and integrate these equations analytically, to get formulae for θ1 and θ2 as functions of time. It is however possible to perform this integration numerically using the Runge Kutta method or similar techniques."

This means: without numerics you have no chance, at least as far as I know.

• I am currently learning about Differential Equations in maths. So i know a fair bit, but we haven't gone through the whole unit yet. – VikeStep Aug 7 '14 at 11:50
• I feel like Runge-Kutta may be a little too advanced for OP. However, I do believe a simple rprogram could be written to simulate the situation numerically – Danu Aug 7 '14 at 12:14
• The problem is that these equations can not be solved analytically, you have to use numerics as wiki quotes. So even you learn solving pdes you will not be able to solve non linear, coupled pdes. – Noldig Aug 7 '14 at 12:14
• @Danu The Runge Kutta method is from Wikipedia, I only wanted to point out that you will not be able to find an analytic solution – Noldig Aug 7 '14 at 12:15
• demonstrations.wolfram.com/DoublePendulum there is even a demonstration nb on the mathematica page, so yes of course – Noldig Aug 7 '14 at 12:17

For understanding the equations of motion of a double pendulum you need to understand some lagrangian. It is not necessary but highly recommended as the system of double pendulum is chaotic. You will have two degrees of freedom ∅1 and ∅2. Four components of displacement, 2 for each degree of freedom, hence 4 components of velocity. Now you need to calculate the total k.e. and p.e. of the system. Define L as L=k.e. - p.e. Take the partial of L wrt ∅1 and ∅2 as follows d/dt(dL/d∅') - dL/d∅. You will get the 2 equations of motion.