不同积分变分原理的统一

Unification of different integral variational principles

依据定量因果原理的数学表示,统一地导出了Lagrange量中含坐标关于时间一阶、二阶导数的积分型的Hamilton原理、Voss原理、Hlder原理和Maupertuis-Lagrange原理等,给出了这些原理的本质联系和统一描述.得出f0=0并不是通常的保持Euler-Lagrange方程不变的结果,而是满足定量因果原理的结果.还得出Lagrange量的所有的积分型变分原理等价地对应于两类满足定量因果原理的不变形式.同时发现所有积分型变分原理的运动方程都是Euler-Lagrange方程,但不同条件的变分原理所对应的不同群G作用下的守恒量是不同的.从而可对过去众多零散的积分型变分原理有一个系统和深入的理解,并使这些变分原理自然地成为定量因果原理的推论.

In terms of a mathematical expression of the quantitative causal principle, this paper gives a unification of Hamilton, Voss, Holder, Maupertuis-Lagrange variational principles of integral style of the second-order Lagrangians, and finds the intrinsic relations among all the different integral variational principles. It is proved that f(0) = 0 is just the result satisfying the quantitative causal principle. The Noether conservation charges of Hamilton, Voss, Holder, Maupertuis-Lagrange variational principles are shown up, and the intrinsic relations among the Noether conservation charges of all the integral variational principles are discovered. Our investigations make the expressions of the past scrappy numerous variational principles be unified into a relative consistent system of all the variational principles in terms of the quantitative causal principle, and show that all the variational principles become deductions of the quantitative causal principle.