Day 31 - Euler-Lagrange Equation

Seminars this Week

WEDNESDAY, April 2, 2025

  • Astronomy Seminar, 1:30 pm, 1400 BPS, Andy Tzandikas, Univ. of Washington, Title: Searching for the Rarest Stellar Occultations

  • PER Seminar, 3:00 pm., BPS 1400, Abigail Daane, Professor of Physics, South Seattle College, Title: The obstacles, stumbles, and growth in examining the “decolonization” of physics education

THURSDAY, April 3, 2025

  • Colloquium, 3:30 pm, 1415 BPS, Alex Sushkov, Boston University, Title: Nuclear magnetic resonance at the quantum sensitivity limit

Reminders

We proposed a solution to the line problem that involved an error term , which is a small perturbation to the true path . This leads to a perturbed function:

where is a small parameter.

We proposed that there's a functional that depends on a function , its derivative , and the independent variable such that:

Reminders

By taking the derivative of the functional with respect to , we can find the condition for which the functional is stationary (i.e., a minimum or maximum).

This (with a lot of math) led us to the following expression:

Clicker Question 31-1

We completed this derivation with the following mathematical statement:

where is an arbitrary function. What does this imply about the term in square brackets?

  1. The term in square brackets must be a pure function of .
  2. The term in square brackets must be a pure function of .
  3. The term in square brackets must be a pure function of .
  4. The term in square brackets must be zero.
  5. The term in square brackets must be a non-zero constant.

Clicker Question 31-2

Returning to the line problem,

here, , where .

Apply the Euler-Lagrange equation to find the expression for the function in this case. Write your result to find the expression for the term in square brackets:

Click when you have an answer!

Clicker Question 31-3

With,

where is a constant, the solution expresses a straight line.

  1. True and I can prove it!
  2. True, but I'm not sure how to prove it.
  3. False, I think this is incorrect.
  4. I don't know.

Clicker Question 31-4

We derived the time that it takes to run from a point on the shore to a point in the water, :

To find the minimal time, what derivative should we take?

  4. Something else?

Clicker Question 31-5

For the brachistochrone problem, the ball moves purely under the influence of gravity. Consider that the ball has moved a vertical distance from rest. What is the speed of the ball at this point?

  4. I'm not sure, but
  5. Something else?