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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.

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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? $\endgroup$ – robin raj Oct 27 '18 at 17:16

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