Day 35 - Lagrangian Examples

Announcements

• Last "Class" Week
• Homework 8 due Friday, April 18
• Next Week: Project Work and Discussion
• Final Project Due April 28th (no later than 11:59 pm)
• No Final Exam

Seminars This Week

TUESDAY, April 15, 2025

• Theory Seminar, 11:00am., FRIB 1200, Speaker: Fnu Aaina Thapa, LLNL, Title: “Deducing neutron capture on short-lived nuclei”
• High Energy Physics Seminar, 1:00 pm, 1400 BPS, Speaker: Fei Yao, Brookhaven National Lab, Title: Extracting Meson Distribution Amplitudes from Nonlocal Euclidean Correlations at Next-to-Next-to-Leading Order

WEDNESDAY, April 16, 2025

• Astronomy Seminar, 1:30 pm, 1400 BPS, Undergraduate Thesis Talks will be given the next two weeks
• PER Seminar, 3:00 pm., BPS 1400, Speaker: Rebeckah Fussell, Cornell University, Title: Comparing approaches to using large language models in science education research
• FRIB Nuclear Science Seminar, 3:30pm., FRIB 1300 Auditorium, Speaker: Professor Alex Brown (FRIB), Title: Nuclear Science Advances at the MSU Cyclotron, NSCL, FRIB, ...

THURSDAY, April 17, 2025

• Colloquium, 3:30 pm, 1415 BPS, Speaker: Jessie Christiansen, Caltech/IPAC, Title: From Kepler to the Habitable Worlds Observatory: The Emerging Picture of Planet Populations
• Astronomical Horizons Public Lecture Series, 7:30 pm, Skye Theater, Abrams Planetarium, Speaker: Marcel Yanez Reyes, Title: From Event Horizons to Particle Collisions: The Geometry of the Extreme.

Stand Up for Higher Education

• Graduate Employee Union
• Union of Nontenure Track Faculty
• Union of Tenure System Faculty

Thursday, April 17th at 3pm

Please make time to show up!

www.dayofactionforhighered.org

Reminders

We used the Lagrangian formalism to derive the equations of motion for a plane pendulum. We chose the and coordinates.

This gave us the following derivatives for the Lagrangian:

We made a mistake by not including the constraint

We made a mistake by not including the constraint in our Lagrangian.

We can change variables to and .

Now we include the constraint , so that .

Clicker Question 35-1

For the plane pendulum, we changed the Lagrangian from Cartesian coordinates to plane polar coordinates. In Cartesian, we found the Lagrangian depended on . In polar, it only depended on and .

What does that tell you about the dimensions of the system? The system is:

1. in 3D space, so it's 3D.
2. described by two spatial dimensions (), so it's 2D.
3. described by one spatial dimension (), so it's 1D.

Clicker Question 35-2

With , we can find the equations of motion.

Which of the following equations of motion is correct?

5. None of these

Clicker Question 35-3

For the Atwood's machine, is connected to by a string of length . Each mass has a length of string extended as measured from the center of the pulley () of and , respectively. The string wraps around half the pulley.

Which of the following represents the equation of constraint for the system?

5. None of these

Take the time derivative of the constraint equation. What do you notice?

Clicker Question 35-4

With a Lagrangian of the form , we can find the generalized forces and generalized momenta.

What are and for the Atwood's machine?

1. and
2. and
3. and
4. and
5. None of these

Clicker Question 35-5

Now, we allow the pulley (mass, ) to rotate. The Lagrangian is given by:

Where is the moment of inertia of the pulley. What is the moment of inertia of the pulley?

5. None of these

Clicker Question 35-6

The rope moves without slipping on the pulley. A rotation of corresponds to a displacement of for the first mass, . What is the new equation of constraint for the system?

5. More than one of these