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Consider a harmonic oscillator defined by the coupled differential equations \begin{align} \begin{split} \dot{X} &= \omega Y \\ \dot{Y} &= - \omega X \, . \end{split} \tag{1} \end{align} Defining new variables $a = X + i Y$ and $a^* = X - i Y$, produces a new uncoupled system of equations \begin{align} \begin{split} \dot{a} &= - i \omega \, a \\ \dot{a}^* &= i \omega \, a^* \, . \end{split} \tag{2} \end{align}

In classical physics [1] (or just in the mathematical context of this transformation used to solve a pair of coupled differential equations) what are the variables a and $a^*$ called?

[1]: In the context of quantum mechanics, the variables $a$ an $a^*$ would in fact be operators and would be called the "raising" and "lowering" operators.

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  • $\begingroup$ Aren't they canonical variables; at least this is the term I was familiar with when decoupling differential equations. Those, though, were generally not complex, so I wonder if the usage of $i$ here changes that? $\endgroup$
    – Kyle Kanos
    Dec 30, 2019 at 17:41
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    $\begingroup$ I think your equations 3 & 4 are incorrect. Shouldn't they have an $i$ out front with the $\omega$? $\endgroup$
    – Geoffrey
    Dec 30, 2019 at 18:08
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    $\begingroup$ @Geoffrey In the course of getting to the starting form Daniel suggests you multiply the position and momentum by constants so that $X$ and $Y$ have the same units (otherwise the initial expressions would also be dimensionally inconsistent). Alas that confuses the issue of interpretation. $\endgroup$ Dec 30, 2019 at 21:13
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    $\begingroup$ Daniel, it might help the discussion to exhibit the transformation for some specific system (mass on a spring?), though I admit that risks contaminating the resulting interpreation with the specifics of that particular system. $\endgroup$ Dec 30, 2019 at 21:15
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    $\begingroup$ A few months ago I learned how to number groups of equations, and I thought it might be useful here. $\endgroup$ Dec 30, 2019 at 21:30

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I would call $a$ and $a^*$ the complex amplitude of the oscillator. Or I guess $a$ is the complex amplitude itself and $a^*$ is the complex conjugate of the amplitude but the distinction is unimportant as they carry the same information (just like in the quantum case).

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  • $\begingroup$ There is a simple way to figure out what $a$ and $a^*$ are: express the observable things for the oscillator via these "complex amplitudes". To be rigorous - this is the only right way of "seeing" their true meaning. $\endgroup$ Dec 30, 2019 at 18:07
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    $\begingroup$ In phase space quantization, where they are c-numbers instead of operators, they are called complex phase space amps, indeed. $\endgroup$ Dec 30, 2019 at 21:36
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I would call them normal modes, which are by definition the degrees of freedom of a system that oscillate at a single frequency.

Beyond terminology there is a whole body of classical theory behind this term making it useful, for example for more complex oscillators or continuous oscillating fields.

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