Mathematics

CLC number: O316

Integrating Factors and Conservation Laws of Herglotz Type for Birkhoffian Systems

Herglotz型Birkhoff系统的积分因子与守恒律

¹ School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
² College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, Jiangsu, China

^† Corresponding author. E-mail: zhy@mail.usts.edu.cn

Received: 1 June 2024

Abstract

The method of integrating factors is used to study the conservation laws of the Herglotz type Birkhoffian systems in this paper. Firstly, the definition of the integrating factors of the Herglotz type Birkhoffian systems is given. Secondly, the relationship between the integrating factors and conservation laws is studied, and the conservation theorems of Herglotz type Birkhoff's equations and their inverse theorems are established. Thirdly, two types of generalized Killing equations for calculating integrating factors are given. Finally, as an example, a linear damped oscillator is taken. This example can be transformed into a Herglotz type Birkhoffian system. The resulting conservation theorems are used to find the conserved quantities for this example.

摘要

采用积分因子方法研究Herglotz型Birkhoff系统的守恒律。首先，给出了Herglotz型Birkhoff系统的积分因子的定义。 其次，研究了积分因子与守恒律之间的关系，建立了Herglotz型Birkhoff方程的守恒定理及其逆定理。第三，给出了计算积分因子的两类广义Killing方程。最后以线性阻尼振子为例，将其化为Herglotz型Birkhoff系统，利用所得到的守恒定理找到了该例的守恒律。