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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?

Qmechanic
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1 Answers1

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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.

Qmechanic
  • 201,751
  • 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? –  Oct 27 '18 at 17:16
  • Why don't $S= S+\alpha_n+\alpha_{n+1}$ @Qmechanic – QFT addict. Jan 30 '22 at 12:10