Bolza Problem

Given the functional

$$U=\int_{t_0}^{t_1} f(y_1,\ldots,y_n;{y_1}',\ldots,{y_n}')\,dt+G(y_{10},\ldots,y_{nr}; y_{11},\ldots,y_{n1}),$$ (1)

find in a class of arcs satisfying $p$ differential and $q$ finite equations

$\phi_\alpha(y_1,\ldots,y_n;{y_1}',\ldots,{y_n}') = 0 \qquad \hbox{for }\alpha=1,\ldots,p\quad$	(2)
$\psi_\beta(y_1,\ldots,y_n) = 0 \qquad \hbox{for }\beta=1,\ldots,q\quad$	(3)

as well as the $r$ equations on the endpoints

$\chi_\gamma(y_{10},\ldots,y_{nr}; y_{11},\ldots,y_{n1}) = 0 \qquad\hbox{for }\gamma=1,\ldots,r,\quad$

(4)

one which renders $U$ a minimum.

References

Goldstine, H. H. A History of the Calculus of Variations from the 17th through the 19th Century. New York: Springer-Verlag, p. 374, 1980.