极坐标哈密顿体系约当型与弹性楔的佯谬解

JORDAN SOLUTIONS FOR POLAR COORDINATE HAMILTONIAN SYSTEM AND SOLUTIONS OF PARADOXES IN ELASTIC WEDGE

讨论了极坐标弹性平面哈密顿体系约当型，并通过约当型的求解，直接给出了相关弹性 楔体佯谬问题的解．从理论上阐明了经典弹性力学中某些佯谬问题的出现是由于其对应的是哈 密顿体系中特殊的约当型解，同时也很自然地为该类问题提供了一个通用、有效的求解方法．

The classical two-dimensional solutions for the stress distribution in an elastic wedge subjected to a concentrated couple at the vertex become infinite when the vertex angle 20 = 2α (tg2α = α). Similarly, the classical solutions for the stress distribution in an elastic wedge subjected to tractions proportional to ru-1 (u≥ 1) on the surfaces become also infinite when or and constant u satisfy the definite relations. They are paradoxes in an elastic wedge. Looking from the analogy theory between computational structural mechanics and optimal control, the Hamiltonian system theory can be introduced into the theory of elasticity. So much effective mathematical physics methods as the separation of variables and eigenfunction expajnsion etc. can be applied directly in elasticity instead of the traditional semi-inverse solution. The new solution system realizes a translation from Euclidean space to symplectic space. The plane elasticity in polar coordinate also can be derived to Hamiltonian system by introducing the dual variables, so an elastic wedge can be solved directly in symplectic space. In this paper, the paradoxes in elastic wedge are restudied under Hamiltonian system in polar coordinate. For the elastic wedge subjected to a concentrated couple, paradox occurs as μ =-1 is a double eigenvalue, i.e. 20 = 2α, the solution of the paradox just corresponds to Jordan form eigenfunction vector. On the other hand, for the elastic wedge subjected to tractions proportional to ru-1 (μ≥1) on the surfaces, initial paradox or the secondajry paxadox occures as p is a single or double eigenvalue, of course, the solution of the initial or secondary paradox just corresponds to first or second order Jordajn form. These solutions can be solved directly and rationally by normal mathematical physics methods. This work shows that special paradox in Euclidean space under Lagrange system just is Jordajn form solutions in symplectic space under Hajrniltonian system, and the specific characteristics of paradox are due to the Jordan form for Hamiltonian system. This work not only provides an efficient method to solve paradox, but also demonstrates the further applications of Hamiltonian system.