Comparing the derivation of equation of motion for double pendulum by method of F=ma and by energy (Lagrangian) method
by Nasser M. Abbasi
May 11, 2011
The goal of this note is to show how to use a symbolic program to help solve a typical basic engineering problem that requires large amount of algebraic manipulation. By applying a CAS program such as Mathematica on such a problem, one can then concentrate more on the problem formulation itself by having the program do the long algebraic computation, which can be very time consuming if done by hand, and also can be error prone.
This also allows one to be able to experiment with different formulation, as one is now relieved from the time consuming part and is able to try different solutions and different idea, which can lead to better insight of the problem.
In this note, we solve the double pendulum problem both ways. One way is by using the energy method, and another way is by using F=ma approach. Both would require long time to do by hand due to the large amount of algebra involved. And at the end, both solutions are verified to be the same.
system diagram
This diagram shows the velocity and acceleration components of the 2 masses in the double pendulum.
Lagrangian solution
Write down the kinetic energy T
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Write down the potential energy
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Write down the Lagrangian L
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Find equation of motion for θ
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Find equation of motion for φ
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decouple the above 2 equations
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Numerically solve the 2 equations found above for θ(t) and φ(t), use some arbitrary parameters
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Make plots of the solutions. Assume 
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F=ma solution
Write the free body diagram
We have 4 unknowns 
Write F=ma for forces radially along θ along for 
Resolve forces perpendicular for 
Write F=ma for forces radially along φ along for 
Resolve forces perpendicular for 
Solve the above 4 equestions
We see that the solution from F=ma is the same as the solution obtained from Lagrange method. To verify
We see from the above that the solutions are identical.
