Day 35 - Lagrangian Examples

Day 35 - Lagrangian Examples#

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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

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Reminders#

We used the Lagrangian formalism to derive the equations of motion for a plane pendulum. We chose the \(x\) and \(y\) coordinates.

\[T(\dot{x}, \dot{y}) = \dfrac{1}{2} m (\dot{x}^2 + \dot{y}^2) \quad V(y) = mgy\]
\[\mathcal{L} = T - V = \dfrac{1}{2} m (\dot{x}^2 + \dot{y}^2) - mgy\]

This gave us the following derivatives for the Lagrangian:

\[\frac{\partial \mathcal{L}}{\partial x} = 0 \quad \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}} \right) = \frac{d}{dt} \left( m\dot{x} \right) = 0\]
\[\frac{\partial \mathcal{L}}{\partial y} = -mg \quad \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{y}} \right) = -m\ddot{y}\]

We made a mistake by not including the constraint#

We made a mistake by not including the constraint \(x^2 + y^2 = L^2\) in our Lagrangian.

We can change variables to \(r\) and \(\phi\). $\(x = r \cos(\phi) \quad y = r \sin(\phi)\)$

\[T(\dot{x}, \dot{y}) = \dfrac{1}{2} m (\dot{x}^2 + \dot{y}^2) = \dfrac{1}{2} m \left( r^2 \dot{\phi}^2 + 2r\dot{r}\dot{\phi} + \dot{r}^2 \right) = T(r, \dot{r}, \phi, \dot{\phi})\]
\[V(y) = mgy = mg r \sin(\phi) = V(r, \phi)\]

Now we include the constraint \(r = L\), so that \(\dot{r} = 0\).

\[T(\phi, \dot{\phi}) = \dfrac{1}{2} m L^2 \dot{\phi}^2 \quad V(\phi) = mgL \cos(\phi)\]
\[\mathcal{L} = \dfrac{1}{2} m L^2 \dot{\phi}^2 - mgL \cos(\phi)\]

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 \(y,\dot{x},\dot{y}\). In polar, it only depended on \(\phi\) and \(\dot{\phi}\).

\[\mathcal{L}(x,y,\dot{y}) \longrightarrow \mathcal{L}(\phi, \dot{\phi})\]

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 (\(x,y\)), so it’s 2D.

  3. described by one spatial dimension (\(\phi\)), so it’s 1D.

Clicker Question 35-2#

With \(\mathcal{L} = \dfrac{1}{2} m L^2 \dot{\phi}^2 - mgL \cos(\phi)\), we can find the equations of motion.

\[\dfrac{\partial \mathcal{L}}{\partial \phi} - \dfrac{d}{dt} \left( \dfrac{\partial \mathcal{L}}{\partial \dot{\phi}} \right) = 0\]

Which of the following equations of motion is correct?

  1. \(\ddot{\phi} = -\frac{g}{L} \sin(\phi)\)

  2. \(\ddot{\phi} = -\frac{g}{L} \cos(\phi)\)

  3. \(\ddot{\phi} = -\sqrt{\frac{g}{L} \sin(\phi)}\)

  4. \(\ddot{\phi} = -\sqrt{\frac{g}{L} \cos(\phi)}\)

  5. None of these

Clicker Question 35-3#

For the Atwood’s machine, \(M\) is connected to \(m\) by a string of length \(l\). Each mass has a length of string extended as measured from the center of the pulley (\(R\)) of \(y_1\) and \(y_2\), respectively. The string wraps around half the pulley.

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

  1. \(y_1 + y_2 = l - R \phi\)

  2. \(y_1 - y_2 = l + R \phi\)

  3. \(y_1 + y_2 = l - \pi R\)

  4. \(y_1 - y_2 = l + \pi R\)

  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 \(\mathcal{L} = \frac{1}{2}(M+m)\dot{y}^2_1 - (M-m)gy_1\), we can find the generalized forces and generalized momenta.

\[F_{y_1} = \frac{\partial \mathcal{L}}{\partial y_1} = -\frac{\partial V}{\partial y_1} \quad p_{y_1} = \frac{\partial \mathcal{L}}{\partial \dot{y}_1} = \frac{\partial T}{\partial \dot{y}_1}\]

What are \(F_{y_1}\) and \(p_{y_1}\) for the Atwood’s machine?

  1. \(F_{y_1} = -mg\) and \(p_{y_1} = m\dot{y}_1\)

  2. \(F_{y_1} = -Mgy_1\) and \(p_{y_1} = M\dot{y}_1\)

  3. \(F_{y_1} = -(M-m)g\) and \(p_{y_1} = (M+m)\dot{y}_1\)

  4. \(F_{y_1} = -(M+m)g\) and \(p_{y_1} = (M-m)\dot{y}_1\)

  5. None of these

Clicker Question 35-5#

Now, we allow the pulley (mass, \(M_p\)) to rotate. The Lagrangian is given by: $\(\mathcal{L} = \frac{1}{2}(M+m)\dot{y}_1^2 + \frac{1}{2}I\dot{\phi}^2 - (M-m)gy_1\)$

Where \(I\) is the moment of inertia of the pulley. What is the moment of inertia of the pulley?

  1. \(I = \frac{1}{2}M_pR^2\)

  2. \(I = \frac{1}{3}M_pR^2\)

  3. \(I = M_pR^2\)

  4. \(I = \frac{1}{4}M_pR^2\)

  5. None of these

Clicker Question 35-6#

The rope moves without slipping on the pulley. A rotation of \(R d\phi\) corresponds to a displacement of \(dy_1\) for the first mass, \(M\). What is the new equation of constraint for the system?

  1. \(y_1 + y_2 = l - R \phi\)

  2. \(dy_1 = R d\phi\)

  3. \(y_1 = R\phi\)

  4. \(\dot{y}_1 = R \dot{\phi}\)

  5. More than one of these