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Maxwell strata in the Euler elastic problem. (English) Zbl 1203.49004

Summary: The classical Euler problem on stationary configurations of elastic rod in the plane is studied in detail by geometric control techniques as a left-invariant optimal control problem on the group of motions of a two-dimensional plane \(E(2)\). The attainable set is described, the existence and boundedness of optimal controls are proved. Extremals are parametrized by the Jacobi elliptic functions of natural coordinates induced by the flow of the mathematical pendulum on fibers of the cotangent bundle of \(E(2)\). The group of discrete symmetries of the Euler problem generated by reflections in the phase space of the pendulum is studied. The corresponding Maxwell points are completely described via the study of fixed points of this group. As a consequence, an upper bound on cut points in the Euler problem is obtained.

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
49Q10 Optimization of shapes other than minimal surfaces
93C10 Nonlinear systems in control theory
74B20 Nonlinear elasticity
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
65D07 Numerical computation using splines

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References:

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