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Given a harmonic oscillator $\ddot{x}+\omega_0^2x=0$ if we calculate the Wronskian using Abel's Identity we get $W[x_1,x_2](t)=W_0e^{\int_{t_0}^t 0 ds}=W_0$

which means (for our two solutions $x_1$ and $x_2$) that $$ det\begin{pmatrix} \dot{x_1} & \dot{x_2} \\ x_1 & x_2 \end{pmatrix}=\dot{x_1}x_2-\dot{x_2}x_1=W_0$$

is some conserved quantity. To me it smells like it should be equal to something of the form $\frac{E}{m}$ (because if we multiply $\dot{x_1}x_2-\dot{x_2}x_1$ by $m$ we get something in units of energy(we don't, I made a mistake, but anyway the question makes sense). Is the conservation of the Wronskian actually the conservation of something energy-ish (to the factor of $m$) for our system or is it something else, if not what is it (physically, what is its physical meaning)?

Ps: I think it probably isn't energy (it definitely isn't energy) because in the case of parametric resonance (aka. $\ddot{x}+\omega_0^2(t)x=0$) the Wronskian is also conserved but the Energy isn't. I might be wrong :)

pss: This answer states that the conservation of the Wronskian means the conservation od angular moment, however I find it unclear what angular momentum would mean for a 1D system

Alexandar Ruño
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  • You have two solutions. Energy of what? It's better to think in terms of a two-dimensional system, where $W_0$ is just the angular momentum. – AccidentalFourierTransform Apr 05 '18 at 17:50
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    possible duplicate: https://physics.stackexchange.com/q/240049/84967 – AccidentalFourierTransform Apr 05 '18 at 17:52
  • I have two linearly independent solutions whose linear combination $Ax_1+Bx_2$ gives us some solutions for some initial conditions, the initial conditions will also determine the Energy $E$

    So Energy of what, the Energy of the system (for some particular choice of initial conditions)

     – Alexandar Ruño Apr 05 '18 at 17:55

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