Skip to main content

Theory and Modern Applications

TableĀ 1 Commutator table

From: Lie symmetry analysis and invariant solutions of 3D Euler equations for axisymmetric, incompressible, and inviscid flow in the cylindrical coordinates

\([ \boldsymbol{V}_{\boldsymbol{1}}, \boldsymbol{V}_{\boldsymbol{2}}]\)

\(\boldsymbol{X}_{\boldsymbol{1}}\)

\(\boldsymbol{X}_{\boldsymbol{2}}\)

\(\boldsymbol{X}_{\boldsymbol{3}}\)

\(\boldsymbol{X}_{\boldsymbol{4}}\)

\(\boldsymbol{X}_{\boldsymbol{1}}\)

0

\(f_{3} ' \frac{\partial }{\partial z} + f_{3}^{\prime \prime } \frac{\partial }{\partial u} + a_{1} \frac{\partial }{\partial p}\)

\(\frac{\partial }{\partial t} + a_{2} \frac{\partial }{\partial z} + a_{3} \frac{\partial }{\partial u} + a_{4} \frac{\partial }{\partial p}\)

\(a_{5} \frac{\partial }{\partial z} + a_{6} \frac{\partial }{\partial u} + a_{7} \frac{\partial }{\partial p}\)

\(\boldsymbol{X}_{\boldsymbol{2}}\)

\(-( f_{3} ' \frac{\partial }{\partial z} + f_{3}^{\prime \prime } \frac{\partial }{\partial u} + a_{1} \frac{\partial }{\partial p} )\)

0

\(-t f_{3} ' \frac{\partial }{\partial z} + a_{8} \frac{\partial }{\partial u} - \frac{2}{r^{2} v} \frac{\partial }{\partial v} + a_{9} \frac{\partial }{\partial p}\)

\(f_{3} \frac{\partial }{\partial z} + f_{3} ' \frac{\partial }{\partial u} + \frac{4}{r^{2} v} \frac{\partial }{\partial v} + a_{10} \frac{\partial }{\partial p}\)

\(\boldsymbol{X}_{\boldsymbol{3}}\)

\(-( \frac{\partial }{\partial t} + a_{2} \frac{\partial }{\partial z} + a_{3} \frac{\partial }{\partial u} + a_{4} \frac{\partial }{\partial p} )\)

\(-(-t f_{3} ' \frac{\partial }{\partial z} + a_{8} \frac{\partial }{\partial u} - \frac{2}{r^{2} v} \frac{\partial }{\partial v} + a_{9} \frac{\partial }{\partial p} )\)

0

\(a_{11} \frac{\partial }{\partial z} + a_{12} \frac{\partial }{\partial u} + a_{13} \frac{\partial }{\partial p}\)

\(\boldsymbol{X}_{\boldsymbol{4}}\)

\(-( a_{5} \frac{\partial }{\partial z} + a_{6} \frac{\partial }{\partial u} + a_{7} \frac{\partial }{\partial p} )\)

\(-( f_{3} \frac{\partial }{\partial z} + f_{3} ' \frac{\partial }{\partial u} + \frac{4}{r^{2} v} \frac{\partial }{\partial v} + a_{10} \frac{\partial }{\partial p} )\)

\(-( a_{11} \frac{\partial }{\partial z} + a_{12} \frac{\partial }{\partial u} + a_{13} \frac{\partial }{\partial p} )\)

0