Linear and Nonlinear Oscillators in Classical Mechanics
Explore the concepts of linear and nonlinear oscillators, derivatives, differential equations, and harmonic oscillators in classical mechanics and electromagnetism. Learn about the linearization of differential equations, Taylor expansion of forces, and methods of solving harmonic oscillators. Dive into homogeneous and inhomogeneous second-order linear differential equations and damped harmonic oscillators with practical examples and visual aids.
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Presentation Transcript
1 1 Linear and Nonlinear Oscillators Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU
2 Primes and Dots x-prime means derivative of x with respect to s, where s is some spatial coordinate. x-dot mean derivative of x with respect to t, where t is some temporal coordinate. 2 3/18/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
3 3 Linearization of a second order differential equation 3 3/18/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
4 Taylor Expansion of a general 1D force on a particle Consider an arbitrary force on a particle: Usually, it can be linearized as a damped harmonic oscillator: 4 3/18/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
5 Linear Differential Equation A differential equation is linear if the following is true Linear Example: Nonlinear Example: 5 3/18/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
6 6 Method of Undetermined Coefficients (Solving the Harmonic Oscillator) 6 3/18/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
7 Linear Differential Equation A differential equation is linear if the following is true Linear Example: Nonlinear Example: 7 3/18/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
8 Homogeneous Second-Order Linear Diff. Eq. Damped Harmonic Oscillator: Start with an ansatz for the form of the solution: The general solution is a linear combination of the independent solutions: 8 3/18/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
9 Inhomogeneous Second-Order Linear Diff. Eq. Damped Harmonic Oscillator with a constantly progressing force: Find one solution that matches the differential form of the force: 9 3/18/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
10 Constantly Progressing Force (cont.) The general solution is the sum of complimentary solution (the general solution to the homogeneous equation) and the particular solution (a solution to the inhomogeneous equation): 10 3/18/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
11 Constantly Progressing Force (cont.) This is just the oscillator, but with it s pivot point moving at a constant velocity: 11 3/18/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
12 More Oscillator Examples See Lecture 2. 12 3/18/2025 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU
