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Can someone help me explain this paradox please.
A simple harmonic oscillator $ma=-kx$ is a system that oscillates in one dimension. But the text book says one-dimensional system can't oscillate. Why is that?? Thank you in advance.

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  • $\begingroup$ I suspect you are quoting the text out of context. Perhaps you could supply more detail, or a link. $\endgroup$ – Gerry Myerson Jun 28 '12 at 7:12
  • $\begingroup$ There's no detail. The really short section of the textbook just says one dimensional system can't oscillate. And the question use the simple harmonic oscillator ma= -kx to state that it oscillate in one dimension and just asks to explain the paradox. Thanks $\endgroup$ – Johnny Jun 28 '12 at 7:25
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    $\begingroup$ If all the book does is say, "a one-dimensional system can't oscillate" then there is no paradox - the book is just wrong. But I'd really like to see what it says myself before I make any judgement on it. $\endgroup$ – Gerry Myerson Jun 28 '12 at 7:34
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    $\begingroup$ Perhaps "one-dimensional" is used in two different senses here. The harmonic oscillator is two-dimensional in the sense that $a$ is the second derivative of $x$, giving you two constants of integration to work with. $\endgroup$ – Per Manne Jun 28 '12 at 7:35
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Either you are incompletely quoting the textbook (surely even a "really short section" would be longer than one sentence) or the textbook is poorly written.

Mathematically, a first-order ordinary differential equation in one dimension cannot oscillate. But $mx'' = -kx$ is a second-order ODE.

Physically, a system with a one-dimensional phase space cannot oscillate. But the phase space of a simple harmonic oscillator has two dimensions: one for position, $x$, and one for velocity, $x'$.

These are two sides of the same coin, because you can reduce any one-dimensional ODE of order $n$ to an $n$-dimensional first-order ODE.

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The textbook you are talking about is (almost certainly) the Strogatz: Nonlinear dynamics and Chaos, exercise 2.6.1. It's considered to be a great textbook about the subject and it may hardly be defined poorly written.

Anyway I agree with Rahul Narain, I think the trick here is the use of the word "one-dimensional". In the short section 2.6 Strogatz looks at $\dot{x}=f(x)$ as a flow on a line and use this to briefly explain the impossibility of oscillations under a topological point of view. Anyway he is always talking about first order systems.

The fact that $m\ddot{x}=-kx$ is one dimensional doesn't change the fact that that is a second order ODE. In fact, I think that the author appositely omits to talk about "orders" in the text of the exercise just to confuse the reader a little bit. As it has been already pointed out, the phase space of the above ODE is not the line, but it consists of two dimensions (position and velocity) and you could also break the second order 1-dim ODE into a 2-dim first order ODE. Hence the possibility oscillations.

Notice also that always in section 2.6 the example in which he considers a mechanical analog regarding $\dot{x}=f(x)$ as a limiting case of Newton's law in which $m\ddot{x}$ is negligible (e.g. a mass attached to a spring placed in a vat of extremely viscous fluid), works just because he is just cutting $m\ddot{x}$ off the equation to reduce it to the form of $\dot{x}=f(x)$. That is, he is just cutting off the term that would make the equation a second order one!

Well, I guess my answer is basically the same as Rahul's after all. I just wanted to give more details about the book that I happened to have right on my desk.

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