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Take simple harmonic motion as an example. Start from the solution.

We know that the solution is $x=A\cos(\omega t+\phi)$ and we have $\dot x=-A\omega \sin(\omega t+\phi)$, where $A$ and $\phi$ are constants. Then we can guess the "energy" as $\omega^2x^2+\dot x^2=\omega^2 A^2$, which is a first integral of the system.

What is the standard procedure to "guess" all these first integrals out of $x$ and $\dot x$ for any system?

rioiong
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  • Some of it is experience and having seen appropriate functions before. In the SHM example, solving with $f''(t)= -n^2f(t)$ we know that the derivative of sin is cos, and derivative of cos is -sin, so a variation of sin and cos will be a solution of the DE. It remains to find the exact form and to show that these solutions are the only ones. Plenty of textbooks do this. – Peter Feb 12 '21 at 13:18

1 Answers1

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This is baked in Hamilton-Jacobi theory but the ability to “see” first integrals depends on the coordinate system. In your specific case the Lagrangian for this system does not depend explicitly on $t$ so $H$ is conserved and corresponds to the total energy.

ZeroTheHero
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