Day 31 - Euler-Lagrange Equation

Day 31 - Euler-Lagrange Equation#

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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 \(\eta(x)\), which is a small perturbation to the true path \(y(x)\). This leads to a perturbed function:

\[Y(x) = y(x) + \alpha \eta(x)\]

where \(\alpha\) is a small parameter.

We proposed that there’s a functional \(f(Y,Y',x)\) that depends on a function \(Y(x)\), its derivative \(Y'(x)\), and the independent variable \(x\) such that:

\[\int_{s_1}^{s_2} f(Y,Y',x) \, dx > \int_{s_1}^{s_2} f(y,y',x) \, dx\]

Reminders#

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

\[\frac{d}{d\alpha} \int_{s_1}^{s_2} f(Y,Y',x) \, dx \bigg|_{\alpha=0} = 0\]

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

\[\int_{s_1}^{s_2} \eta(x) \left[\dfrac{\partial f}{\partial y} - \dfrac{d}{dx}\left(\dfrac{\partial f}{\partial y'}\right)\right] \, dx = 0\]

Clicker Question 31-1#

We completed this derivation with the following mathematical statement:

\[\int_{s_1}^{s_2} \eta(x) \left[\dfrac{\partial f}{\partial y} - \dfrac{d}{dx}\left(\dfrac{\partial f}{\partial y'}\right)\right] \, dx = 0\]

where \(\eta(x)\) 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 \(x\).

  2. The term in square brackets must be a pure function of \(y\).

  3. The term in square brackets must be a pure function of \(y'\).

  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,

\[l = \int_{x_1}^{x_2} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]

here, \(f(y,y',x) = \sqrt{1 + (y')^2}\), where \(y' = \frac{dy}{dx}\).

Apply the Euler-Lagrange equation to find the expression for the function \(f(y,y',x)\) in this case. Write your result to find the expression for the term in square brackets:

\[\dfrac{d}{dx}\left[?\right] = 0\]

Click when you have an answer!

Clicker Question 31-3#

With,

\[y' = \pm\sqrt{\dfrac{c^2}{1 + c^2}}\]

where \(c\) 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, \(T\):

\[T = \dfrac{1}{v_1} \left(x_1^2 + (y-y_1)^2\right)^{1/2}+\dfrac{1}{v_2} \left(x_2^2 + (y_2-y)^2\right)^{1/2}\]

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

  1. \(\dfrac{dT}{dx}\)

  2. \(\dfrac{dT}{dy}\)

  3. \(\dfrac{dT}{dt}\)

  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 \(\Delta y\) from rest. What is the speed of the ball at this point?

  1. \(v = gt\)

  2. \(v = 2 g\Delta y\)

  3. \(v = \sqrt{2g\Delta y}\)

  4. I’m not sure, but \(<\sqrt{2g\Delta y}\)

  5. Something else?