First-Order Extrema in Classical Mechanics , Theoretical Minimum

In summary, in the lecture on First-Order Extrema in "Classical Mechanics" from the Theoretical Minimum series, Dr. Susskind discusses calculating extrema and saddle points to first order. This means expanding the equation in a Taylor series around the minimum and using a small value for x to achieve better accuracy. This concept may have been unfamiliar in the past, but it is now commonly used in math and physics.
  • #1
morangta
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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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  • #2
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
 
  • #3
What does that have to do with the stationary points?
 

What is a first-order extrema in classical mechanics?

A first-order extrema refers to a point where the derivative of a function is zero. In classical mechanics, this can represent a maximum, minimum, or saddle point in the system.

How is first-order extrema related to the theoretical minimum in classical mechanics?

The theoretical minimum in classical mechanics refers to the point where the system has the lowest possible energy. This point is often associated with a first-order extrema, where the system is at a minimum energy state.

What are some real-world examples of first-order extrema in classical mechanics?

Some examples include a pendulum at its lowest point, a ball rolling down a hill, or a satellite in stable orbit around a planet. In all of these cases, the system is at a minimum energy state, corresponding to a first-order extrema.

How are first-order extrema calculated in classical mechanics?

In classical mechanics, first-order extrema can be found by taking the derivative of the system's Lagrangian function with respect to the variables of the system. This derivative is then set equal to zero and solved to find the extrema points.

Why are first-order extrema important in classical mechanics?

First-order extrema are important in classical mechanics because they represent points of equilibrium or stability in a system. They can also provide insight into the behavior of a system and help predict future states of the system.

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