经典质点分析力学中三个转折点及其数学物理分析——逆散射问题研究引出的讨论

THREE TURING POINTS IN CLASSICAL ANALYTICAL MECHANICS AND ITS MATHEMATICAL AND PHYSICAL ANALYSIS-A INVESTIGATION RESULTING FROM RESEARCHES ON INVERSE SCATTERING PROBLEMS

经典质点分析力学有三个转折点 ,即虚功原理 ,Legendre变换和变分原理 .虚位移定义为满足虚功原理的位移 ,它可使有约束系统物理和数学模型完整化 ;Legendre变换是一种自变量和函数同时改变的变换 ,它在几何上是曲面的切平面 (或法方向 )与曲面上点之间的变换 ,在物理上是 (广义 )速度、Lagrange函数和 (广义 )动量、Hamilton函数之间的变换 ,这种变换可能将只对一阶偏微商非线性的一阶偏微分方程线性化 ,可将二阶偏微分方程如 Lagrange方程化为对称的一阶方程如 Hamilton正则方程 ;本文引入变分积分的全变分 。

There are three turning points in classical analytical mechanics, those are, the principle of virtual work, Legendre transformation and variational principles. The virtual displacement is defined as a displacement that satisfies the principle of virtual work and it makes the physical and mathematical model for a constrained mechanic system complete. The Legendre transformation is a transformation that transforms variables and functions. In geometry, the Legendre transformation is a correspondence between the points and the tangent planes (or the normal unit vectors) of a hypersurface, and in physics, it is a correspondence between (generalized) velocity and Lagrangian function, and (generalized) momentum and Hamiltonian function. By this transformation, the first order partial differential equations nonlinear only in the first partial derivatives may go over to linear ones and the second order partial differential equations (for example, Lagrange's equations of motion) may go over to a symmetric system of the first order partial equations (for example, Hamilton's canonical equations of motion). The noncontemporaneous variation of variational integral is introduced in this paper and it simplifies the proof of the conversion of the equations of motion in differential form to those in integral form for a mechanic system and vice versa.$$$$