极坐标哈密顿体系约当型与弹性楔的佯谬解

讨论了极坐标弹性平面哈密顿体系的当型，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。

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

The classical two-dimensional solutions for the stress distribution inan elastic wedge subjected to a concentrated couple at the vertex becomeinfinite when the vertex angle .Similarly, the classical solutions for the stress distribution in anelastic wedge subjected to tractions proportional to r-1( 1) on the surfaces become also infinite when andconstant satisfy the definite relations. They are paradoxes in anelastic wedge. Looking from the analogy theory between computationalstructural mechanics and optimal control, the Hamiltonian system theorycan be introduced into the theory of elasticity. So much effectivemathematical physics methods as the separation of variables andeigenfunction expansion etc. Can be applied directly in elasticityinstead of the traditional semi-inverse solution. The new solutionsystem realizes a translation from Euclidean space to symplectic space.The plane elasticity in polar coordinate also can be derived toHamiltonian system by introducing the dual variables, so an elasticwedge can be solved directly in symplectic space. In this paper, theparadoxes in elastic wedge are restudied under Hamiltonian system inpolar coordinate. For the elastic wedge subjected to a concentratedcouple, paradox occurs as =-1 is a double eigenvalue, I.e. 2=2, the solution of the paradox just corresponds to Jordan formeigenfunction vector. On the other hand, for the elastic wedgesubjected to tractions proportional to r-1 ( 1) on thesurfaces, initial paradox or the secondary paradox occures as is asingle or double eigenvalue, of course, the solution of the initial orsecondary paradox just corresponds to first or second order Jordan form.These solutions can be solved directly and rationally by normalmathematical physics methods. This work shows that special paradox inEuclidean space under Lagrange system just is Jordan form solutions insymplectic space under Hamiltonian system, and the specificcharacteristics of paradox are due to the Jordan form for Hamiltoniansystem. This work not only provides an efficient method to solveparadox, but also demonstrates the further applications of Hamiltoniansystem.