Double Pendulum Potential Energy

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Discussion Overview

The discussion revolves around the potential energy equations for a double pendulum, exploring different formulations and their implications. Participants examine the definitions and origins of potential energy in this context, considering both theoretical and mathematical aspects.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a potential energy equation for a double pendulum and questions its validity compared to a commonly accepted formula.
  • Another participant suggests that having two different origins for potential energy is incorrect.
  • A later reply acknowledges the concern and indicates a willingness to recalculate based on the feedback received.
  • It is noted that the partial derivatives of potential energy with respect to the angles are identical for both proposed equations, suggesting that the differences may not affect the dynamics.
  • Another participant points out that the extra constant terms in the proposed equation are typically disregarded in such problems, even if they arise from a correct calculation.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of the potential energy formulations, with some suggesting that the differences may not significantly impact the analysis. The discussion remains unresolved regarding which equation is preferable.

Contextual Notes

There are unresolved assumptions regarding the choice of reference points for potential energy and the implications of constant terms in the equations.

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Hello, everybody.

This website and many others define the potential energy of a double pendulum as:

V=-(m_1+m_2) g l_1 cos\theta_1-m_2 g l_2 cos\theta_2

However, I came up with the following equation:

V= (m_1+m_2) g l_1 (1-cos\theta_1)+m_2 g l_2 (1-cos\theta_2)

I started from the position of what looks like equilibrium (when the pendulum is fully stretched and hanging freely). They seem to start at the point where the pendulum is "pinned."

Which equation should I use? Am I missing something here?

Thanks in advance.
 
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It seems to me you have two origins for potential energy which is wrong!
 
Thank you. I think I see what you mean. Let me recalculate it.
 
Okay. I guess it doesn't really matter in the end, since the partial derivatives \frac{\partial V}{\partial \theta_1} and \frac{\partial V}{\partial \theta_2} are identical for both equations, namely:

\frac{\partial V}{\partial \theta_1}=(m_1+m_2) \sin\theta_1 g l_1 \frac{\partial V}{\partial \theta_2}=m_2 g l_2 \sin\theta_2
 
Yes, your PE has just some constant terms extra. The constant terms are usually dropped anyway in this kind of problems, even if they result from a correct calculation.
 

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