Ω∈Rm is a closed region. The variable f:Ω→Rp is an n differentiable function with fixed boundary conditions on ∂Ω. The function L is real-valued and has continuous first partial derivatives, and the 0th to nth partial derivatives of f and the independent variable x∈Ω will be arguments of L. Define a functional
I:=f↦∫ΩL(⋯)dV,
where dV is the volume element in Ω. Then the extremal of I satisfies a set of PDEs with respect to f. The set of PDEs consists of p equations, the ith of which is
j=0∑nμ∈Pj,m∑(−1)j∂μ∂(∂μfi)∂L=0, |
(1) |
where Pj,m is the set of all (not necessarily strictly) ascending j-tuples in {1,…,m}j, and
∂μ:=∏k∂xμk∂lenμ.
Equation 1 is the Generalization of Euler–Lagrange equation.