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Obtain solution of the free particle problem in two dimensions using Hamilton Jacobi equation and obtain expression for the Hamilton's principal function.

Consider \(N\) molecules of a gas obeying van der Waals equation of state given by
\[\left(P+ \frac{a N^2}{V^2}\right)\big(V-Nb\big) = Nk_B T\]
where \(a\) is a measure of the attractive forces between the molecules and \(b\) is another constant proportional to the size of a molecule. The other symbols have their usual meanings. Show that during an isothermal expansion (Temperature is kept onstant) from volume \(V_1\) to volume \(V_2\) quasi-statically and reversibly, the work done is
\[ W =-Nk_B T \log\left(\frac{V_2-Nb}{V_1-Nb}\right) + a^2 \Big(\frac{1}{V_1}-\frac{1}{V_2}\Big)\]

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