First-Order Extrema in Classical Mechanics , Theoretical Minimum

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SUMMARY

In the 3rd lecture of the Classical Mechanics series by Dr. Susskind, the concept of first-order extrema is introduced, emphasizing the use of Taylor series expansions to analyze potential energy near stationary points. The discussion highlights that first-order approximations provide insights into the behavior of systems when small perturbations occur. The example provided illustrates how to derive first-order approximations using the function y(x) = sqrt(a^2 - x^2), demonstrating the relevance of these mathematical techniques in classical mechanics.

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  • Understanding of Taylor series expansions
  • Familiarity with classical mechanics concepts
  • Basic knowledge of calculus, particularly derivatives
  • Experience with potential energy functions
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  • Study Taylor series applications in physics
  • Explore stationary points in potential energy landscapes
  • Learn about perturbation theory in classical mechanics
  • Investigate higher-order approximations and their significance
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This discussion is beneficial for physics students, educators, and anyone interested in deepening their understanding of classical mechanics and mathematical techniques used in theoretical physics.

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First-Order Extrema in "Classical Mechanics", Theoretical Minimum

In the 3rd lecture of Classical Mechanics, 2011, by Dr. Susskind in his Theoretical Minimum series, he talks about calculating extrema, saddle points, etc. to "first order".

"if you move a little bit, the potential is zero, to first order"

What does he mean, first order? When I was in college in the 60's, if we wanted better accuracy, we just made Δx smaller, and eventually got the accuracy we wanted.

No one talked about "first order" in the 60's. I think I would have remembered. Maybe not. I've noticed he keeps saying that. Is there something new in math and physics that I am not aware of?

Thanks.
 
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He is talking about expanding the equation in a taylor series around the minimum

example

y(x) = sqrt(a^2-x^2) ; for x<<1 y(x) = sqrt(1 -(x/a)^2) ~ 1- (1/2)*(x/a)^2 and this would be to first order
 
What does that have to do with the stationary points?
 

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