The minimum plane path for movable end-points and the nonsimultaneous variations

Abstract

The paper deals with the problem of existence of the minimum path for movable end-points in the one-of-degree-of-freedom mechanical system. The criteria for obtaining of extremum path for movable end-points is extended with new criteria for minimum. The nonsimultaneous variational calculus is applied. It is assumed that the actual path belongs to sub-set C2 of admissible curves. The series expansion up to the second order small values is applied and the first and the second variation of functional are calculated. It is proved that the necessary and sufficient conditions for the minimum path are that the first order variation is zero and the second order variation is positive. The second conditions are based on the arbitrary solution of Riccati’s differential equation and also the known Legender’s and Jacobi criteria for minimum for the case of fixed end-points. Two examples are solved: the problem of the minimal length of a curve joining two fixed boundary curves and problem of motion of a particle between variable boundaries for which the Hamilton action integral is minimal.