The EulerLagrange(f, t, x(t)) command computes the Euler-Lagrange equations of a functional $J=\underset{a}{\overset{b}{\int }}f\left(t\,x\left(t\right)\,x\'\left(t\right)\right)\ⅆt$ subject to $x\left(a\right)=A$ and $x\left(b\right)=B$.

In general, the Euler-Lagrange equations are not independent.

The Euler-Lagrange equations are returned as expressions.

If they can be calculated, the trivial first integrals are also returned.

The first integrals are set equal to generated global indexed variables $K_i$ that denote arbitrary constants.

For higher-order functionals, for example, f(t, y(t), y'(t), y''(t)), use variables to represent derivatives. For example, set x1(t) = y(t) and x2(t)=y'(t), and then determine the Euler-Lagrange equations of the functional f + L*( x1'(t) - x2(t) )^2. To find the equations for the higher-order problem, substitute x2(t) = x1'(t) into the result.

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

$\left[\mathrm{ConjugateEquation}\,\mathrm{Convex}\,\mathrm{EulerLagrange}\,\mathrm{Jacobi}\,\mathrm{Weierstrass}\right]$

$f≔{\left({\mathrm{diff}\left(x\left(t\right)\,t\right)}^2+{\mathrm{diff}\left(y\left(t\right)\,t\right)}^2\right)}^{\frac{1}{2}}$

$f≔\sqrt{{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}$

$\mathrm{EulerLagrange}\left(f\,t\,\left[x\left(t\right)\,y\left(t\right)\right]\right)$

$\left\{\frac{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)\left(2\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)\right)+2\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right)\right)}{2{\left({\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-\frac{\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)}{\sqrt{{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}}\,\frac{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)\left(2\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}x\left(t\right)\right)+2\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right)\right)}{2{\left({\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-\frac{\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)}{\sqrt{{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}}\,\frac{\frac{\ⅆ}{\ⅆt}x\left(t\right)}{\sqrt{{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}}=K_1\,\frac{\frac{\ⅆ}{\ⅆt}y\left(t\right)}{\sqrt{{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}}=K_2\,\sqrt{{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}-\frac{{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2}{\sqrt{{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}}-\frac{{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}{\sqrt{{\left(\frac{\ⅆ}{\ⅆt}x\left(t\right)\right)}^2+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}}=K_3\right\}$

$g≔\frac{{\left(1+{\mathrm{diff}\left(y\left(t\right)\,t\right)}^2\right)}^{\frac{1}{2}}}{{y\left(t\right)}^{\frac{1}{2}}}$

$g≔\frac{\sqrt{1+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}}{\sqrt{y\left(t\right)}}$

$\mathrm{EulerLagrange}\left(g\,t\,y\left(t\right)\right)$

$\left\{-\frac{\sqrt{1+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}}{2{y\left(t\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}+\frac{{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2\left(\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\right)}{{\left(1+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{y\left(t\right)}}+\frac{{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}{2\sqrt{1+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}{y\left(t\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-\frac{\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)}{\sqrt{1+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}\sqrt{y\left(t\right)}}\,\frac{\sqrt{1+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}}{\sqrt{y\left(t\right)}}-\frac{{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}{\sqrt{1+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2}\sqrt{y\left(t\right)}}=K_1\right\}$