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I tried to apply Hamilton-Jacobi equation to uniform circular motion and it doesn't come out right.

Lets say a particle of mass $m$ undergoes uniform circular motion around another particle of mass $M$.

$$H=T+V = (1/2) m v^2 - GmM/r$$

$$S = \int{Ldt} = \int(T-V)dt = \int((1/2) m v^2 + GmM/r)dt$$

Applying the equation:

$$H+\partial{S}/\partial{t} = 0$$

=>

$$((1/2) m v^2 - GmM/r) +\partial({\int((1/2) m v^2 + GmM/r)dt})/\partial{t} = 0$$

For uniform circular motion $v$ is constant and $r$ is constant.

=> $((1/2) m v^2 - GmM/r)$ = - $\partial({(1/2) m v^2 (t-t0) + GmM/r(t-t0)})/\partial{t}$

$(1/2) m v^2 - GmM/r$ = - $(1/2) m v^2 - GmM/r$

=>

$v = 0$

what is going wrong with this equation?

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Fakrudeen
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1 Answers1

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Hint: For starter, it seems OP is conflating the action functional $S[q;t_f,t_i]$ and Hamilton's principal function $S(q,\alpha,t)$. For their definitions and how they are interrelated, see e.g. my Phys.SE answer here.

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