How Does Goldstein Derive the Kinetic Energy Equation in Classical Mechanics?

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SUMMARY

The discussion focuses on the derivation of the kinetic energy equation in classical mechanics as presented by Goldstein. Participants analyze the expression md²r/dt² · dr/dt = d/dt (1/2 m (dr/dt)²), emphasizing the application of the chain rule in calculus. The left-hand side (LHS) represents the derivative of kinetic energy, while the right-hand side (RHS) simplifies to the kinetic energy formula. The conversation highlights the importance of correctly applying the chain rule to understand the relationship between these two expressions.

PREREQUISITES
  • Understanding of classical mechanics principles
  • Familiarity with calculus, specifically the chain rule
  • Knowledge of derivatives and their applications in physics
  • Basic understanding of kinetic energy concepts
NEXT STEPS
  • Study the chain rule in calculus with a focus on its application in physics
  • Explore the derivation of kinetic energy in classical mechanics using Goldstein's "Classical Mechanics" textbook
  • Learn about the relationship between force, mass, and acceleration in classical mechanics
  • Investigate advanced topics in derivatives, such as higher-order derivatives and their physical interpretations
USEFUL FOR

Students of physics, educators teaching classical mechanics, and anyone interested in the mathematical foundations of kinetic energy derivation.

bengeof
how is md^2r/dt^2 . dr/dt = d/dt (1/2 m (dr/dt)^2 )

Thank You
 
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bengeof said:
how is md^2r/dt^2 . dr/dt = d/dt (1/2 m (dr/dt)^2 )

Thank You

Chain rule.
 
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Likes   Reactions: Ibix
Can you work it out for me explicitly ?
 
Can you state the chain rule? Can you apply it to the time derivative of ##(dr/dt)^2##?
 
Ibix said:
Can you state the chain rule? Can you apply it to the time derivative of ##(dr/dt)^2##?

2dr/dt . . is that right ?
 
bengeof said:
I was able to work out the right hand side. But the left hand side is my problem. .

That makes no sense. The LHS is the answer to the derivative on the RHS. There's nothing to work out on the left.

bengeof said:
2dr/dt . . is that right ?

The stackexchange link I gave you is all about why that is not the answer!
 
okay Got it. Thank you so much !
 
  • #10
The rest of Goldstein may present difficulties.
 

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