Given the functional U = integral_(t_0)^(t_1) f(y_1, ..., y_n ;y_1', ..., y_n') d t + G(y_10, ..., y_(n r) ;y_11, ..., y_(n1)), find in a class of arcs satisfying p differential and q finite equations ϕ_α(y_1, ..., y_n ;y_1', ..., y_n') = 0 for α = 1, ..., p ψ_β(y_1, ..., y_n) = 0 for β = 1, ..., q as well as the r equations on the endpoints χ_γ(y_10, ..., y_(n r) ;y_11, ..., y_(n1)) = 0 for γ = 1, ..., r, which renders U a minimum.

calculus of variations