Research On Approximate Symmetries And Conserved Quantities Of Lagrange System

Lagrange system is very important and relatively simple, it can describe thedynamics system only use a function of L in Mechanical problems. If some systemcan be converted into Lagrange system, they will be able to discuss and solve moreconvenient. Seek symmetries and conserved quantities of system, it is important notonly in mathematics, but also reflects the profound physical nature. The principle ofsymmetry in physics is higher level of law, all conserved quantity are derived fromthe hidden symmetry behind them.This article is based on the classical analytical mechanics and modernsymmetry theory, it made a more detailed discussion of the theory of symmetriesand conserved quantities of Lagrange system, especially it studied for a class ofLagrange system with a relatively small parameter, it tried to improve andsupplement the existing theory. The main work of the full text is as follows:First, it discussed and summarized the theory of symmetries and conservedquantities of Lagrange system. In Lagrange system, Noether symmetry is a specialLie symmetry, so it pointed out that solving the symmetry of the system can beunified with the Lie symmetry method. The example1(Harmonic oscillator)andexample2(Kepler problem) which the article cited are two typical models ofclassical mechanics, this article use the Lie symmetry method to solve them, andgave all independent of the conserved quantity, and analyzed the physical meaningof the conserved quantity. Especially for example2, it provided a method which isrelatively new, and it can obtain all independent conserved quantities of Keplerproblem.Second, it introduced approximate symmetry theory by the idea of perturbationmethod combined with symmetry theory, it researched a Lagrange systemcontaining a small parameter, and established the theory of approximate symmetriesand approximate conserved quantities of such system. And the approximate symmetry and the approximate conserved quantities were divided, it pointed outthat stable approximate symmetry and stable approximate conserved quantities hadpractical significance. Combined with example1and example3, it shows themeaning of approximate symmetries and approximate conserved quantities. Inexample4, it analyzed and discussed the results obtained by the approximatesymmetry of the theory very detailed. It pointed out the advantage of theapproximate symmetry method, it can reduce the difficulty of solving the symmetryof such Lagrange system containing a small parameter, it can recursive solve theperturbed system approximate symmetry by no perturbation symmetry of thesystem step by step, and it will apprach precise symmetry with the improvement ofthe order. This is the mathematical nature of this method, so that we have a moreprofound understanding of this approximate symmetry theory.Finally, it summarized the article and looked future of the research work.