### 31 Hamilton-Jacobi PDE

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#### 31.1 Hamilton-Jacobi type PDE

problem number 169

Taken from Maple pdsolve help pages, which is taken from Landau, L.D. and Lifshitz, E.M. Translated by Sykes, J.B. and Bell, J.S. Mechanics. Oxford: Pergamon Press, 1969

Solve for $$S \left ( t,\xi ,\eta ,\phi \right )$$ \begin{align*} -{\frac{\partial }{\partial t}}S \left ( t,\xi ,\eta ,\phi \right ) &=1/2 \,{\frac{\left ({\frac{\partial }{\partial \xi }}S \left ( t,\xi ,\eta ,\phi \right ) \right ) ^{2} \left ({\xi }^{2}-1 \right ) }{{\sigma }^{2}m \left ( -{\eta }^{2}+{\xi }^{2} \right ) }}+1/2\,{\frac{ \left ({\frac{ \partial }{\partial \eta }}S \left ( t,\xi ,\eta ,\phi \right ) \right ) ^{ 2} \left ( -{\eta }^{2}+1 \right ) }{{\sigma }^{2}m \left ( -{\eta }^{2}+{ \xi }^{2} \right ) }}+1/2\,{\frac{ \left ({\frac{\partial }{\partial \phi }}S \left ( t,\xi ,\eta ,\phi \right ) \right ) ^{2}}{{\sigma }^{2}m \left ({\xi }^{2}-1 \right ) \left ( -{\eta }^{2}+1 \right ) }}+{\frac{a \left ( \xi \right ) +b \left ( \eta \right ) }{-{\eta }^{2}+{\xi }^{2}}} \end{align*}

Mathematica

$\text{DSolve}\left [-s^{(1,0,0,0)}(t,\zeta ,\eta ,\phi )=\frac{s^{(0,0,0,1)}(t,\zeta ,\eta ,\phi )^2}{2 \left (\zeta ^2-1\right ) \left (-\eta ^2-1\right ) m \sigma ^2}+\frac{\left (-\eta ^2-1\right ) s^{(0,0,1,0)}(t,\zeta ,\eta ,\phi )^2}{2 m \sigma ^2 \left (\zeta ^2-\eta ^2\right )}+\frac{\left (\zeta ^2-1\right ) s^{(0,1,0,0)}(t,\zeta ,\eta ,\phi )^2}{2 m \sigma ^2 \left (\zeta ^2-\eta ^2\right )}+\frac{a(\zeta )+b(\zeta )}{\zeta ^2-\eta ^2},s(t,\zeta ,\eta ,\phi ),\{t,\zeta ,\eta ,\phi \}\right ]$

Maple

$S \left ( t,\xi ,\eta ,\phi \right ) ={\it \_c}_{{4}}\phi +{\it \_c}_{{1}}t+{\it \_C1}+{\it \_C2}+{\it \_C3}+{\it \_C4}-\int \!{\frac{\sqrt{-2\,{\eta }^{4}m{\sigma }^{2}{\it \_c}_{{1}}+2\,b \left ( \eta \right ){\eta }^{2}m{\sigma }^{2}+2\,{\eta }^{2}{\it \_c}_{{1}}{\sigma }^{2}m-2\,{\eta }^{2}{\it \_c}_{{3}}{\sigma }^{2}m-2\,b \left ( \eta \right ){\sigma }^{2}m+2\,{\it \_c}_{{3}}{\sigma }^{2}m-{{\it \_c}_{{4}}}^{2}}}{{\eta }^{2}-1}}\,{\rm d}\eta -\int \!{\frac{\sqrt{-2\,m{\sigma }^{2}{\xi }^{4}{\it \_c}_{{1}}-2\,a \left ( \xi \right ) m{\sigma }^{2}{\xi }^{2}+2\,{\xi }^{2}{\it \_c}_{{1}}{\sigma }^{2}m-2\,m{\sigma }^{2}{\xi }^{2}{\it \_c}_{{3}}+2\,a \left ( \xi \right ){\sigma }^{2}m+2\,{\it \_c}_{{3}}{\sigma }^{2}m-{{\it \_c}_{{4}}}^{2}}}{{\xi }^{2}-1}}\,{\rm d}\xi$