VariationalCalculus[EulerLagrange] - construct the Euler-Lagrange equations
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Calling Sequence
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EulerLagrange(f, t, x(t))
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Parameters
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f
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expression in t, x(t), and x'(t)
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t
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independent variable
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x(t)
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unknown function (or list of functions)
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Description
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In general, the Euler-Lagrange equations are not independent.
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The Euler-Lagrange equations are returned as expressions.
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If they can be calculated, the trivial first integrals are also returned.
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The first integrals are set equal to generated global indexed variables that denote arbitrary constants.
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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.
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Examples
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>
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Geodesics in the plane
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Brachistochrone
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![{-(1/2)*(1+(diff(y(t), t))^2)^(1/2)/y(t)^(3/2)+(diff(y(t), t))^2*(diff(y(t), `$`(t, 2)))/((1+(diff(y(t), t))^2)^(3/2)*y(t)^(1/2))+(1/2)*(diff(y(t), t))^2/((1+(diff(y(t), t))^2)^(1/2)*y(t)^(3/2))-(diff(y(t), `$`(t, 2)))/((1+(diff(y(t), t))^2)^(1/2)*y(t)^(1/2)), (1+(diff(y(t), t))^2)^(1/2)/y(t)^(1/2)-(diff(y(t), t))^2/((1+(diff(y(t), t))^2)^(1/2)*y(t)^(1/2)) = K[1]}](/support/helpjp/helpview.aspx?si=9080/file03355/math139.png)
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