It is necessary to find a generating line of the rotation-surface with
minimum area. It is known that the line to be found is a catenary. Depending on
the concrete boundary conditions, there can exist either two solutions of the
Euler equation or alone or none. These solutions are catenaries. In
traditional courses, one does not pay attention enough to the case of absence
of the classic solution. In the numerical solution by the Euler method, the
broken-wise solution is easily discovered. These are two disks for which the
generating line consists of two vertical segments and a horizontal one lying on
the axis of rotation. Moreover, the broken-wise solution can be a global
minimum in the zone of existing the solution for the Euler equation (Ahiezer,
1955; Young, 1969; Cesari, 1983). If the left-end point is fixed at *(0, 1)*,
then one can construct a partition of the quadrant of feasible locations for
the right-end point. If the right-end point is located in the left area (see
Fig. 1),
then the global minimum is a catenary. If this point is in the central area,
then the global minimum is a broken-wise solution (disks) while the catenary
gives only a local minimum. If the point is in the right area, then the only
(global) minimum is the broken-wise solution, no solutions of the Euler
equation exist. These facts can be found in (Anisimov, 1904;
Ahiezer, 1955;
Young, 1969; Cesari, 1983), however, they are not widely known.

Fig. 1 Press mouse button on black field |

More accessible site

Ahiezer, N.I. (1955). *Lectures on the calculus of variations.*
TehGIZ, Moscow [in Russian].

Anisimov, V. (1904). *Course on the calculus of variations.* V. I,
Warshaw [in Russian].

Cesari, L. (1983) *Optimization Theory and Applications Problems with
Ordinary Differential Equations.* Springer-Verlag, N.Y., Heidelberg,
Berlin.

Young, L.C. (1969). *Lectures on the calculus of variations and optimal
control theory.* W.B. Saunders Co., Philadelphia, London, Toronto.

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