The Three-order Symmetry And Conserved Quantity

First,the development and formulations in rectangular, cylindrical and spherical coordinates of jerk are simply introduced in this paper, then development history, the recent progresses and definitions of three kinds of symmetries and the conserved quantities are discussed in detail. The requirement of Noether symmetry, Lie symmetry and form invariance are Hamilton principle, conformal invariance of differential equation and formal invariance of Lagrange equation separately, the conserved quantities are the Noether conserved quantity, the generalized Hojman conserved quantity and new type conserved quantity, then, nonholonomic mechanical system of chetave's type and holonomic mechanical system of variable mass are taken as example, the differential equation of the two systems are shown, the main difference of them is the form of generalized force, the symmetries and conserved quantities are proposed after that.In the second paragraph, using the principle of jerk and Newton second law, the three-order Lagrange equation is derived, it's the most important differential equation in this article, then the three-order Hamilton principle can be get according to two-order one, the three-order Noether symmetry and conserved quantity are obtained next. Its formulation and physics meaning are discussed under time and space translation.The three-order Lagrange equation is expressed as third order derivative of the generalized coordinate in the nonsingular system, and three-order derivative of the general coordinate under infinitesimal transformation is given, the three-order Lie symmetry is presented by using conformal invariance of differential equation afterward, later the three-order generalized Hojman conserved quantity is derived from the tenable condition of three-order Lie symmetry,it's also simply discussed on two special conditions.In the last part of second paragraph, the condition of the three-order form invariance is studied after giving its definition. Comparing the conditions of three-order Lie symmetry and form invariance, the relation between the two is given; harmonic oscillator is given as example to illustrate the application at last.