8.5 Nonlinear oscillation solutions

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §8.5. Original wording. Continues 8.4b. Labs: 8.5b.

Same stepper, new force

Once $(x,v)$ is in dynamic form, swapping potentials is a one-line change in $\mathbf f$:

$$F=-k x^{p-1} \quad(p\text{ even; odd power of }x\text{ restores}),$$

or the soft model $F=-kx(1-\alpha x)$. Integrate with RK2/RK4 exactly as for the harmonic case.

What changes physically

• Period vs amplitude. For $p\neq 2$, larger energy ⇒ different period. Measure $T$ by detecting successive $x=0$ crossings with $v>0$ (or peaks of $x$).
• Energy still checks the ODE. For autonomous conservative $F(x)$, $E=\tfrac12 mv^{2}+V(x)$ should stay flat if the algorithm is accurate — even when $V$ is not $\tfrac12 kx^{2}$.
• Soft $\alpha$. Start with small amplitude; push past $x\sim 1/\alpha$ and watch escape.

Slide amplitude and softness below: the measured period stretches away from the harmonic $T_0=2\pi$ as $A\alpha$ grows.

Energy as a diagnostic

For the harmonic benchmark, compare Euler vs RK2 energy drift at fixed final time. Nonlinear runs reuse the same plot idea with the matching $V(x)$.

Harmonic place-markers

Keep the $p=2$ benchmark ($\omega=2\pi$, $T=1$) beside every nonlinear run: same IC, same $h$, compare waveforms and energy drift. Nonlinear labs lock one RK2 micro-step on $F=-x^{3}$ so the force wiring is unambiguous before long integrations.

Takeaway

Model	Expect
$F=-kx$	$T$ independent of $A$
Soft / hard $F(x)$	$T=T(A)$
Any conservative $F(x)$	flat $E$ if the stepper is good

Quiz

Nonlinear oscillations

1 / 3

For a soft spring F=−kx(1−αx), increasing amplitude usually…

Pick an answer to see if you are right.

Next: 8.5b Nonlinear oscillation labs.