Day 10 - Integrating EOMs Numerically

Day 10 - Integrating EOMs Numerically#

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Announcements#

  • HW 4 is posted; starting project work

  • Midterm 1 is coming up (Assigned 29 Sep; Due 10 Oct)

  • REMINDER: Office hours, for now (Mihir-MN; Danny-DC):

    • Tuesday 6-8pm (MN, Zoom)

    • Thursday 6-8pm (MN, Zoom)

    • Friday 2-4pm (DC, 1248 BPS)

  • Zoom Link: https://msu.zoom.us/j/96882248075

    • password: phy321msu

  • Friday’s Class: Homework 4 Exercise 3 + Q&A

Seminars this week#

WEDNESDAY, September 17, 2025#

Astronomy Seminar, 1:30 pm, 1400 BPS, In Person and Zoom, Host~ Speaker: Matthew Murphy, Michigan State University Title: Zoom Link: https://msu.zoom.us/j/887295421?pwd=N1NFb0tVU29JL2FFSkk0cStpanR3UT09 Meeting ID: 887-295-421 Passcode: 002454

Seminars this week#

WEDNESDAY, September 17, 2025#

FRIB Nuclear Science Seminar, 3:30pm., FRIB 1300 Auditorium and online via Zoom Speaker: Katie Yurkewicz of Argonne National Laboratory (MSU ALUM) Title: From Nuclei to Narratives: Lessons Learned on the Journey from NSCL Student to Lab Communications Leader Please see website for full abstract. Please click the link below to join the webinar: Join Zoom: https://msu.zoom.us/j/96657965451?pwd=Isaf23sK5agzaao0Kwaei7AaWHkc4W.1 Meeting ID: 966 5796 5451 Passcode: 479842

Seminars this week#

FRIDAY, September 19, 2025#

QuIC Seminar, 12:30pm, -1:30pm, 1300 BPS, In Person
Speaker: Jeremiah Rowland, Michigan State University Title: Measurement Reduction Techniques for Measuring Hamiltonians Full Scheule is at: https://sites.google.com/msu.edu/quic-seminar/ For more information, reach out to Ryan LaRose

Reminder: email me your extra credit seminar write-ups

Goals for this week#

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  • Establish a model for drag forces

  • Develop an understanding of the process for modeling forces

  • Produce equations of motion that can be investigated

  • Start probing the behavior of these systems with math and computing

Model-to-EOM Pipeline for Classical Mechanics#

  1. ✅ Develop conceptual description of the system; make justifiable assumptions

  2. ✅ Using a framework of physics (i.e., Newton’s Laws, Lagrangian Dynamics), develop a mathematical model of the system

  3. ✅ Produce the equations of motion (EOM) by following the framework (i.e., ordinary differential equations)

  4. 😵 Solve for trajectories of the system (e.g., \(x(t)\), \(v(t)\), \(v(x)\))

Most EOMs are nonlinear#

We need approximate methods to produce trajectories#

Clicker Question 10-1#

We found that the equation of motion for the spring-mass system was:

\[\ddot{x} = -\dfrac{k}{m}x = -\omega^2 x\]

Your friends have proposed the following general solutions:

\[1.\;x(t) = A\cos(\omega t) \qquad 2.\;x(t) = B\sin(\omega t) \qquad 3.\;x(t) = A\cos(\omega t) + B\sin(\omega t) \]
\[ 4.\;x(t) = A\cos(\omega t + \phi) \qquad 5.\;x(t) = B\sin(\omega t + \phi) \qquad 6.\;x(t) = A\cos(\omega t + \phi) + B\sin(\omega t + \phi) \]

How many of them are correct? (1) Only one (2) Two (3) Three (4) Four (5) All of them

Clicker Question 10-2#

To convert the second order ODE for a spring mass system,

\[\ddot{x} = -\dfrac{k}{m}x = -\omega^2 x\]

to two first order ODEs, we first introduce an equation for the velocity:

\[v = \dot{x}.\]

This is a first order ODE, and the first one we need. What is the other first order ODE?

Clicker Question 10-2#

For spring-mass system (\(\ddot{x} = -\omega^2 x\)), and using the first order ODE the defines velocity \(v = \dot{x}\), which other first order ODE can be used with the velocity equation to represent the spring mass system?

  1. \(\dot{x} = -\omega^2 x\)

  2. \(\dot{v} = -\omega^2 x\)

  3. \(\ddot{x} = a\)

  4. \(\dot{v} = a\)

  5. Something else

Clicker Question 10-3#

We derived the Euler step for constant acceleration:

\[v[i+1]=v[i]+a*\Delta t\]
\[x[i+1]=x[i]+v[i]*\Delta t\]

We can implement in Python for arrays x and v with:

v[i+1]=v[i]+a*deltat        # Constant Acceleration
x[i+1]=x[i]+v[i]*deltat     # Euler Step

How can we implement a non-constant acceleration?

Clicker Question 10-3#

How can we implement a non-constant acceleration?

v[i+1]=v[i]+a*deltat        # Constant Acceleration
x[i+1]=x[i]+v[i]*deltat     # Euler Step

  1. Add in a[i] to the first equation

  2. Pre-compute the array a outside the loop

  3. Compute a[i] in the loop

  4. Change v[i] to v[i+1]

  5. More than one of these