Conjugate Points in the Euler Elastic Problem

For the classical Euler elastic problem, conjugate points are described. Inflexional elasticas admit the first conjugate point between the first and third inflexion points. All other elasticas do not have conjugate points. As a result, the problem of stability of Euler elasticas is solved.

[1]  G. M.,et al.  A Treatise on the Mathematical Theory of Elasticity , 1906, Nature.

[2]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[3]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions , 1920, Nature.

[4]  Leonhard Euler.  Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti , 2013, 1307.7187.

[5]  V. I. Arnol'd,et al.  Characteristic class entering in quantization conditions , 1967 .

[6]  A V Saryčev,et al.  THE INDEX OF THE SECOND VARIATION OF A CONTROL SYSTEM , 1982 .

[7]  Stephen Wolfram,et al.  Mathematica: a system for doing mathematics by computer (2nd ed.) , 1991 .

[8]  D. F. Lawden.  Elliptic Functions and Applications , 1989 .

[9]  Lawrence S. Kroll.  Mathematica--A System for Doing Mathematics by Computer. , 1989 .

[10]  Kai Cieliebak,et al.  Symplectic Geometry , 1992, New Spaces in Physics.

[11]  J. Canny,et al.  Nonholonomic Motion Planning , 1992 .

[12]  L. Dai,et al.  Non-holonomic Kinematics and the Role of Elliptic Functions in Constructive Controllability , 1993 .

[13]  V. Jurdjevic.  Geometric control theory , 1996 .

[14]  A. Agrachev,et al.  Control Theory from the Geometric Viewpoint , 2004 .

[15]  A. Agrachev.  Geometry of Optimal Control Problems and Hamiltonian Systems , 2005, math/0506197.

[16]  Yurii Leonidovich Sachkov.  Complete description of the Maxwell strata in the generalized Dido problem , 2006 .

[17]  Юрий Леонидович Сачков,et al.  Полное описание стратов Максвелла в обобщенной задаче Дидоны@@@Complete description of the Maxwell strata in the generalized Dido problem , 2006 .

[18]  Yu. L. Sachkov,et al.  Maxwell strata in Euler's elastic problem , 2007, 0705.0614.

[19]  Y. Sachkov.  Maxwell strata in the Euler elastic problem , 2008 .