Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum
In this lesson, we learn about non-linear autonomous systems and how to sketch their trajectories. Specifically, we focus on a non-linear pendulum and how to find its critical points. Critical points are points where the right-hand side of the equations is zero, and the velocity vector is also zero. By finding these critical points, we can better understand the motion of the system. While non-linear systems can be difficult to solve, physical intuition can help us locate critical points and gain qualitative information about the system's behavior.
Non-linear Autonomous Systems: Finding the Critical Points and Sketching Trajectories; the Non-linear Pendulum -- Lecture 31. Some very deep non-linear ODE discussion.
Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 29, 2008). License: Creative Commons BY-NC-SA.
More info at: http://ocw.mit.edu/terms
- Lecture Notes - Lecture Notes
- Supplementary Notes - Chapter 25 Supplementary Notes written by Prof. Miller
- What is a non-linear system of differential equations?
- How do you solve non-linear ODEs?
- How do you find critical points?
- How do you sketch trajectories?
- What is the non-linear pendulum?
- What is a spiral in ODEs?
- What is a sink in ODEs?
- What is a Jacobian matrix?
- How do you use a Jacobian matrix?
- What are some examples of using a Jacobian matrix?
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Staff Review
This video is a very new idea to this course. This lecture deals with non-linear autonomous systems of ODEs. To solve them, you must find critical points and sketch trajectories for the systems. A lot of sketching graphs and writing systems of differential equations from the pictures is done. Some actual concrete problems are also discussed. Jacobian matrices and actual problems using them are shown as well.
