Pfaff-Birkhoff Variational Problem And Its Symmetry Based On Non-standard Birkhoffian

Non-standard Lagrangian,also known as "non-natural Lagrangian",is different from standard Lagrangian in that non-standard Lagrangian is not expressed in the form of the difference between kinetic energy and potential energy.By studying its variational problem,non-standard Lagrangian can describe the nonlinear system which is difficult to be described by standard Lagrangian.Birkhoff mechanics,as a natural extension of Hamilton mechanics,marks a new development stage of analytical mechanics.If we extend the idea of nonstandard Lagrangian to Birkhoff systems,the study of the dynamics model based on the nonstandard Birkhoffian can provide a new idea for solving the non-conservative nonlinear dynamics problems.As a result,in this paper,the dynamics models based on non-standard Birkhoffians(including exponential Birkhoffian,power-low Birkhoffian,and logarithm Birkhoffian)are proposed,which are called non-standard Birkhoff systems.Furthermore,the symmetry theorem is studied on the non-standard Birkhoff system.The specific content is as follows:1.Noether symmetries and conserved quantities of dynamical systems based on nonstandard Birkhoffians are studied.The definitions and criteria of transformation in nonstandard Birkhoff systems are given,which mainly include Noether symmetric transformation and quasi-symmetric transformation.The Noether theorems of non-standard Birkhoff dynamics are proved,and the relations between Noether symmetry and conserved quantities of non-standard Birkhoff dynamics are established.2.Noether symmetries and conserved quantities of dynamical systems based on fractional non-standard Birkhoffians are studied.The definitions and criteria of transformation under nonstandard Birkhoff system on fractional derivative are given.The Noether theorems based on the non-standard Birkhoff dynamics on the fractional derivative are proved,and the relations between the Noether symmetry and the conserved quantities on the fractional derivative are established.3.Noether symmetries and conserved quantities of dynamical systems based on fractional non-standard Lagrangians are studied.The definitions and criterion of Noether symmetry under the transformation group of fractional non-standard Lagrange systems with time invariance and time variation are given,and the Noether theorems of the system are established.4.Lie symmetries and conserved quantities of dynamical systems based on nonstandard Birkhoffians are studied.The definite equations of Lie symmetry under nonstandard Birkhoff system are given,and the structure equations are further obtained.According to the structure equation,the corresponding Lie symmetry theorems are proved.