On inverse variational principles for a measurement of side-trough spillways

摘要

The hydraulic analysis of side-trough spillways based on the law of conservation of linear momentum is considered. From mathematical point of view the necessary condition that the differential equation provides an extremum of the functional leads to the Euler-Lagrange equation. The inverse variational problem consists of a construction of a functional, using the given differential equation, so that this equation would be an Euler-Lagrange equation for the constructed functional. Two basic approaches have been used for that purpose:a/some special cases when the functional is constructed by means of symmetrical and positive operator and b/ numerical methods for solution of Euler-Lagrange equation. The case b/ leads to "direct" variational principles as Reyleigh-Riesz method, etc. In the case of one side spillway the solution for water depth is used in order to assess the global volume of the trough.