LagrangeEquations receives an expression representing a Lagrangian and returns a sequence of Lagrange equations, of the form $\mathrm{expression}=0$, with as many equations as coordinates are indicated in the list or set F. In the case of only one degree of freedom (one coordinate), F can also be the coordinate itself, and the output consists of a single Lagrange equation.

The formula in the traditional case where the Lagrangian depends on 1st order derivatives of the coordinates and there is only one parameter, $t$, is

where $\frac{\partial }{\partial {\overrightarrow{r}}_i}L$ formally represents the derivative with respect to the coordinates of the $i^{\mathrm{th}}$ particle, equal to the Gradient when working in Cartesian coordinates; $\frac{\partial }{\partial {\overrightarrow{v}}_i}L$ represents the equivalent operation, replacing each coordinate by the corresponding velocity, i.e. its derivative with respect to $t$, and $\frac{\ⅆ}{\ⅆt}$ represents the total derivative with respect to $t$, the parameter parametrizing the coordinates. Note that in more general cases the number of parameters can be many. For example, in electrodynamics, the "coordinate" is a tensor field $A_{\mathrm{\mu }}\left(x\,y\,z\,t\right)$, there are then four coordinates, one for each of the values of the index $\mathrm{\mu }$, and there are four parameters $\left(x\,y\,z\,t\right)$.

The second argument F indicates the coordinates without their dependency, passed as names. For example, in the case of one single parameter $t$ and a coordinate $q\left(t\right)$, pass $q$. It is expected that these names appear in the Lagrangian consistently, always with the same functionality.

LagrangeEquations can handle tensors and vectors of the Physics package as well as derivatives using vectorial differential operators (see d_ and Nabla), works by performing functional differentiation (see Fundiff), and handles 1st, and higher order derivatives of the coordinates in the Lagrangian automatically. Unlike the similar command VariationalCalculus:-EulerLagrange, LagrangeEquations does not return first integrals.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\,\mathrm{coordinates}=\mathrm{cartesian}\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{coordinatesystems}=\left\{X\right\}\,\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$L≔\frac{1}{2}{\mathrm{diff}\left(x\left(t\right)\,t\right)}^2-\frac{1}{2}k{x\left(t\right)}^2$

$L≔\frac{{\stackrel{\mathbf{.}}{x}\left(t\right)}^2}{2}-\frac{k{x\left(t\right)}^2}{2}$

$\mathrm{LagrangeEquations}\left(L\,x\right)$

$x\left(t\right)k+\stackrel{\mathbf{..}}{x}\left(t\right)=0$

$\mathrm{CompactDisplay}\left(\mathrm{\phi }\left(t\right)\right)$

$\mathrm{\phi }\left(t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\phi }$

$L≔\frac{1}{2}m\left(-2\mathrm{diff}\left(\mathrm{\phi }\left(t\right)\,t\right)al\mathrm{\omega }\sin \left(\mathrm{\omega }t-\mathrm{\phi }\left(t\right)\right)+{\mathrm{diff}\left(\mathrm{\phi }\left(t\right)\,t\right)}^2{l}^2+2\cos \left(\mathrm{\phi }\left(t\right)\right)gl\right)$

$L≔\frac{m\left(-2\stackrel{\mathbf{.}}{\mathrm{\phi }}al\mathrm{\omega }\sin \left(\mathrm{\omega }t-\mathrm{\phi }\right)+{{\stackrel{\mathbf{.}}{\mathrm{\phi }}}^{}}^2{l}^2+2\cos \left(\mathrm{\phi }\right)gl\right)}{2}$

$\mathrm{LagrangeEquations}\left(L\,\mathrm{\phi }\right)$

$ml\left(a{\mathrm{\omega }}^2\cos \left(t\mathrm{\omega }-\mathrm{\phi }\right)-\sin \left(\mathrm{\phi }\right)g-\stackrel{\mathbf{..}}{\mathrm{\phi }}l\right)=0$

$\mathrm{Define}\left(A\left[\mathrm{\mu }\right]\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

$\mathrm{CompactDisplay}\left(A\left(X\right)\right)$

$A\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}A$

$F\left[\mathrm{\mu },\mathrm{\nu }\right]≔\mathrm{d\_}\left[\mathrm{\mu }\right]\left(A\left[\mathrm{\nu }\right]\left(X\right)\right)-\mathrm{d\_}\left[\mathrm{\nu }\right]\left(A\left[\mathrm{\mu }\right]\left(X\right)\right)$

$F_{\mathrm{\mu },\mathrm{\nu }}≔{\partial }_{\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)-{\partial }_{\mathrm{\nu }}\left(A_{\mathrm{\mu }}\right)$

$L≔{F\left[\mathrm{\mu },\mathrm{\nu }\right]}^2$

$L≔\left({\partial }_{\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)-{\partial }_{\mathrm{\nu }}\left(A_{\mathrm{\mu }}\right)\right)\left({\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)-{\partial }_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)\right)$

$\mathrm{LagrangeEquations}\left(L\,A\right)$

$-4\mathrm{\square }\left(A_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)+4{\partial }_{\mathrm{\mu }}\left({\partial }_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)\right)=0$

$\mathrm{with}\left(\mathrm{Vectors}\right)$

$\left[\mathrm{\&x}\,\mathrm{`+`}\,\mathrm{`.`}\,\mathrm{Assume}\,\mathrm{ChangeBasis}\,\mathrm{ChangeCoordinates}\,\mathrm{CompactDisplay}\,\mathrm{Component}\,\mathrm{Curl}\,\mathrm{DirectionalDiff}\,\mathrm{Divergence}\,\mathrm{Gradient}\,\mathrm{Identify}\,\mathrm{Laplacian}\,\nabla \,\mathrm{Norm}\,\mathrm{ParametrizeCurve}\,\mathrm{ParametrizeSurface}\,\mathrm{ParametrizeVolume}\,\mathrm{Setup}\,\mathrm{Simplify}\,\mathrm{`^`}\,\mathrm{diff}\,\mathrm{int}\right]$

$\mathrm{interface}\left(\mathrm{imaginaryunit}=i\right)$

$I$

$\mathrm{macro}\left(h=\mathrm{\hslash }\right)\:$

$\mathrm{Setup}\left(\mathrm{realobjects}=\left\{G\,h\,m\,t\,V\left(x\,y\,z\,t\right)\right\}\right)$

$\left[\mathrm{realobjects}=\left\{\mathrm{\hslash }\,G\,m\,\mathrm{\phi }\,r\,\mathrm{\rho }\,t\,\mathrm{\theta }\,x\,y\,z\,V\left(X\right)\right\}\right]$

$\mathrm{CompactDisplay}\left(\mathrm{\psi }\left(X\right)\,V\left(X\right)\right)$

$\mathrm{\psi }\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\psi }$

$V\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}V$

$L≔\frac{\frac{1}{2}\left(-h\left(i\mathrm{diff}\left(\mathrm{conjugate}\left(\mathrm{\psi }\left(X\right)\right)\,t\right)\mathrm{\psi }\left(X\right)m+h{\mathrm{Norm}\left(\mathrm{\%Gradient}\left(\mathrm{\psi }\left(X\right)\right)\right)}^2\right)+\left(-G{\mathrm{abs}\left(\mathrm{\psi }\left(X\right)\right)}^4+\mathrm{diff}\left(\mathrm{\psi }\left(X\right)\,t\right)i\mathrm{conjugate}\left(\mathrm{\psi }\left(X\right)\right)h-2V\left(x\,y\,z\,t\right){\mathrm{abs}\left(\mathrm{\psi }\left(X\right)\right)}^2\right)m\right)\cdot 1}{m}$

$L≔\frac{-\mathrm{\hslash }\left(\mathrm{i}{\stackrel{\&conjugate0;}{\mathrm{\psi }}}_t\mathrm{\psi }m+\mathrm{\hslash }{‖\nabla \mathrm{\psi }‖}^2\right)+\left(-G{\left|\mathrm{\psi }\right|}^4+\mathrm{i}\left(\stackrel{\mathbf{.}}{\mathrm{\psi }}\right)\stackrel{\&conjugate0;}{\mathrm{\psi }}\mathrm{\hslash }-2V{\left|\mathrm{\psi }\right|}^2\right)m}{2m}$

$\mathrm{LagrangeEquations}\left(L\,\mathrm{\psi }\right)$

$\frac{{\stackrel{\&conjugate0;}{\mathrm{\psi }\left(X\right)}}_{x,x}{\mathrm{\hslash }}^2+{\stackrel{\&conjugate0;}{\mathrm{\psi }\left(X\right)}}_{y,y}{\mathrm{\hslash }}^2+{\mathrm{\hslash }}^2{\stackrel{\&conjugate0;}{\mathrm{\psi }\left(X\right)}}_{z,z}-2m\left(G{{\stackrel{\&conjugate0;}{\mathrm{\psi }}}^{}}^2\mathrm{\psi }+\mathrm{i}{\stackrel{\&conjugate0;}{\mathrm{\psi }}}_t\mathrm{\hslash }+\stackrel{\&conjugate0;}{\mathrm{\psi }}V\right)}{2m}=0$

$\left(\mathrm{Laplacian}=\mathrm{\%Laplacian}\right)\left(\mathrm{\psi }\left(X\right)\right)$

${\mathrm{\psi }}_{x,x}+{\mathrm{\psi }}_{y,y}+{\mathrm{\psi }}_{z,z}={\nabla }^2\mathrm{\psi }$

$\mathrm{simplify}\left(\mathrm{conjugate}\left(\right)\,\left\{\right\}\right)$

$\frac{2\mathrm{i}\mathrm{\hslash }\left(\stackrel{\mathbf{.}}{\mathrm{\psi }}\right)m+{\mathrm{\hslash }}^2\left({\nabla }^2\mathrm{\psi }\right)-2m\mathrm{\psi }\left(G\stackrel{\&conjugate0;}{\mathrm{\psi }}\mathrm{\psi }+V\right)}{2m}=0$

$ih\mathrm{isolate}\left(\,\mathrm{diff}\left(\mathrm{\psi }\left(X\right)\,t\right)\right)$

$\mathrm{i}\left(\stackrel{\mathbf{.}}{\mathrm{\psi }}\right)\mathrm{\hslash }=\frac{-{\mathrm{\hslash }}^2\left({\nabla }^2\mathrm{\psi }\right)+2m\mathrm{\psi }\left(G\stackrel{\&conjugate0;}{\mathrm{\psi }}\mathrm{\psi }+V\right)}{2m}$

$\mathrm{CompactDisplay}\left(\mathrm{\Phi }\left(X\right)\right)$

$\mathrm{\Phi }\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\Phi }$

$L≔\frac{1}{2}\mathrm{d\_}\left[\mathrm{\mu }\right]\left(\mathrm{\Phi }\left(X\right)\right)\mathrm{d\_}\left[\mathrm{\mu }\right]\left(\mathrm{\Phi }\left(X\right)\right)-\frac{m^2}{2}{\mathrm{\Phi }\left(X\right)}^2+\frac{\mathrm{\lambda }}{4}{\mathrm{\Phi }\left(X\right)}^4$

$L≔\frac{{\partial }_{\mathrm{\mu }}\left(\mathrm{\Phi }\right){\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(\mathrm{\Phi }\right)}{2}-\frac{m^2{\mathrm{\Phi }}^2}{2}+\frac{\mathrm{\lambda }{\mathrm{\Phi }}^4}{4}$

$\mathrm{LagrangeEquations}\left(L\,\mathrm{\Phi }\right)$

${\mathrm{\Phi }}^3\mathrm{\lambda }-\mathrm{\Phi }{m}^2-\mathrm{\square }\left(\mathrm{\Phi }\right)=0$

The Physics[LagrangeEquations] command was introduced in Maple 2023.

For more information on Maple 2023 changes, see Updates in Maple 2023.