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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}),
\end{displaymath} (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.


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

© 1996-9 Eric W. Weisstein