8.1 Nonlinear oscillators

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §8.1. Rewritten in original wording — not a paste of textbook figures or listings. Continues 7.8 Code-listing sketches. Next: 8.2–8.3 ODE review.

Why leave the harmonic world?

Every first course solves $m x''=-kx$. Real springs, pendula at large angles, and molecular bonds leave that paradise. The computational story of this chapter is: write Newton’s law, cast it as a first-order system, and integrate with a controlled ODE algorithm while watching energy and period.

Newton’s second law for one coordinate, then two potentials with different restoring forces. Pick each equation among lookalikes.

For small $|x|$ the soft model recovers $-kx$. Past $x>1/\alpha$ the “spring” no longer restores. Only $p=2$ in the power-law family is harmonic (isochronous); other even $p$ stay periodic in a bound well but with amplitude-dependent period.

Physics takeaways

• Bound motion in a 1D well is periodic (return to the same $(x,v)$).
• Isochronous (amplitude-independent period) only for the harmonic case $p=2$.
• Soft $\alpha$ models can cease to be bound if the particle climbs past $x=1/\alpha$.
• External drives and friction come later (§8.6); first we need ODE machinery.

Quiz

Nonlinear oscillators

1 / 4

From V≈½kx²(1−⅔αx), the force is…

Pick an answer to see if you are right.

Next: 8.2–8.3 ODE review and dynamic form.