<p>In orbital dynamics there are often two 'classes'- stable and non-stable, or periodic and non-periodic. Is this just a specific class of linear and non-linear harmonics? Would Klemperer's rosette orbit be an example of a non-linear orbit because it is neither closed nor stable?</p>

CC BY-SA 4.0
2022-02-24T18:40:59.243Z
The only linear (non-curved) orbital paths are directly towards or away from the more massive body, but the motion along these paths is non-linear due to acceleration from gravity.  **Thus, all orbital motion is nonlinear in $\mathbb{R}^3$.**  One might argue that a circular orbit is linear in a spherical coordinate system aligned perpendicular to the orbital plane, since then $r$, $\phi$, and $\dot{\rho}$ are constant, which I suppose is true, but not very useful.
Connor Garcia
CC BY-SA 4.0
2022-02-28T12:13:22.227Z
@ConnorGarcia I interpreted the OP as asking about "linear harmonic oscillation", not linear paths.
Carl Witthoft
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2022-02-28T20:15:18.953Z
@CarlWitthoft Under that interpretation, there is no simple (linear) harmonic motion since gravitational forces scale with $r^2$.  The obvious only exception is when $r$ is fixed, for a circular orbit, in which the motion could be considered harmonic along each dimension in the orbital plain: https://en.wikipedia.org/wiki/Simple_harmonic_motion#Uniform_circular_motion.
Connor Garcia
<orbital-mechanics>
Non-stable vs non-linear orbits
NotSoSN
# Non-stable vs non-linear orbits

(Asked by: username_0 on 2022-02-24)

In orbital dynamics there are often two 'classes'- stable and non-stable, or periodic and non-periodic. Is this just a specific class of linear and non-linear harmonics? Would Klemperer's rosette orbit be an example of a non-linear orbit because it is neither closed nor stable?

**Tags:** orbital-mechanics
> **Comments:**
> **username_1**: The only linear (non-curved) orbital paths are directly towards or away from the more massive body, but the motion along these paths is non-linear due to acceleration from gravity.  **Thus, all orbital motion is nonlinear in $\mathbb{R}^3$.**  One might argue that a circular orbit is linear in a spherical coordinate system aligned perpendicular to the orbital plane, since then $r$, $\phi$, and $\dot{\rho}$ are constant, which I suppose is true, but not very useful.  
> **username_2**: @ConnorGarcia I interpreted the OP as asking about "linear harmonic oscillation", not linear paths.  
> **username_1**: @CarlWitthoft Under that interpretation, there is no simple (linear) harmonic motion since gravitational forces scale with $r^2$.  The obvious only exception is when $r$ is fixed, for a circular orbit, in which the motion could be considered harmonic along each dimension in the orbital plain: https://en.wikipedia.org/wiki/Simple_harmonic_motion#Uniform_circular_motion.  

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