In Hamilton-Jacobi theory Hamilton's principal function $S$ is a function of $n+1$ constants. But we take one of the $n+1$ constants as an additive constant. I don't get this step?
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The HJ equation is a non-linear first-order PDE for $S$ in $(n+1)$ variables $(q^1, \ldots, q^n, t)$, but the PDE does not depend on $S$ directly, only its derivatives. Therefore one additive integration constant $S\to S+\alpha_{n+1}$ is trivial.
For more information, see also this related Phys.SE post.
answered Oct 27, 2018 at 12:28
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$\begingroup$ PDE in (n+1) variables implies we need ( n+1) constants to find complete solution . if we set one constant as additive and solve PDE in n constants , How this will bring a complete solution? $\endgroup$– user208263Commented Oct 27, 2018 at 17:16
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$\begingroup$ Why don't $S= S+\alpha_n+\alpha_{n+1}$ @Qmechanic $\endgroup$ Commented Jan 30, 2022 at 12:10