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Following partial differential equation

$\frac{1}{2 m}\left[\frac{\partial S}{\partial q}\right]^{2}+\frac{1}{2} k q^{2}+\frac{\partial S}{\partial t}=0$

is solved by substituting

$S=S_{1}(q)+S_{2}(t)$

I couldn't understand motivation behind this substitution, can somebody point out?

ConfusedHuman
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    The motivation is the usual for partial differential equations in general: by plugging that ansatz in the equation, you find yourself in a situation in which one side of the equation depends only on $q$ and the other only on $t$. In particular, this is the Hamilton-Jacobi equation for a harmonic oscillator. Since the Hamiltonian does not depend on time, the separability is directly given by your anzats, with your $S_2(t)=-Et$, see link. $E$ is then both the energy and the separation constant of the PDE. – secavara Mar 07 '21 at 10:07

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