- #31
Andreas C
- 196
- 20
Oh, I think I figured it out on my own. I got ##\ddot θ ## in terms of the angular velocity, acceleration, and θ.
How Do You Derive the Cosine of an Integral in Pendulum Motion?
- Context: Undergrad
- Thread starter Andreas C
- Start date
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- Tags
- Derivative Integral
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Discussion Overview
The discussion revolves around deriving the cosine of an integral in the context of pendulum motion using the Lagrangian approach. Participants explore the mathematical formulation and differentiation involved in the motion of a simple pendulum under gravitational influence, focusing on the derivatives of trigonometric functions and their implications in the Euler-Lagrange equation.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant expresses confusion about the derivative of a cosine function with respect to time and seeks clarification on the expression d(cosθ)/(dθ/dt).
- Another participant points out the need for using partial derivatives instead of total derivatives in the context of the Lagrangian mechanics.
- There is a discussion about the correct formulation of the Euler-Lagrange equation and the necessity of differentiating the Lagrangian with respect to both position and velocity variables.
- Participants explore the implications of signs in derivatives, particularly concerning stable and unstable equilibria in pendulum motion.
- One participant realizes a mistake in their calculations regarding the kinetic energy term and seeks confirmation on their revised understanding.
- There is a mention of the relationship between the derived equations of motion and the characteristics of harmonic oscillators, specifically under small amplitude approximations.
Areas of Agreement / Disagreement
Participants generally agree on the need for careful differentiation and the use of partial derivatives in the context of the Lagrangian. However, there are multiple competing views regarding the interpretation of signs in derivatives and the implications for stability in the pendulum's motion. The discussion remains unresolved on some aspects, particularly regarding the transition from the derived equations to their physical interpretations.
Contextual Notes
Limitations include potential misunderstandings about the application of derivatives in the context of Lagrangian mechanics, as well as unresolved mathematical steps in the derivation process. The discussion also highlights the dependency on the assumptions made regarding small angle approximations in pendulum motion.
Who May Find This Useful
This discussion may be useful for students and enthusiasts of classical mechanics, particularly those interested in Lagrangian dynamics and the mathematical treatment of pendulum motion.
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