变质量力学系统的Herglotz型Lagrange方程与Noether对称性和守恒量

蔡铭俣，张毅; Cai Mingyu; Zhang Yi

引 en

变质量力学系统的Herglotz型Lagrange方程与Noether对称性和守恒量^*

蔡铭俣¹
机构：
1. 苏州科技大学数学科学学院,苏州 215009
×
，张毅²
机构：
2. 苏州科技大学土木工程学院,苏州 215011
×

1. 苏州科技大学数学科学学院,苏州 215009；
2. 苏州科技大学土木工程学院,苏州 215011；

HERGLOTZ TYPE LAGRANGE EQUATIONS AND NOETHER SYMMETRY AND CONSERVED QUANTITY FOR MECHANICAL SYSTEMS WITH VARIABLE MASS^*

Cai Mingyu¹
Affiliation：
1. School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009 , China
×
，Zhang Yi²
Affiliation：
2. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011 , China
×

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

通讯作者:

张毅,E-mail:zhy@mail.usts.edu.cn

中图分类号:O316

文献标识码:A

文章编号:1672-6553-2022-20(6)-106-08

DOI:10.6052/1672-6553-2022-012

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参考文献
出版信息

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目录contents

摘要 Abstract
关键词 Keywords
1 变质量力学系统的Herglotz型Lagrange方程
2 Herglotz型Noether对称性
3 Herglotz型Noether定理
4 Herglotz型Noether逆定理
5 算例
6 结论
参考文献

摘要

Herglotz变分原理提供了非保守耗散问题的变分描述,同时变质量力学在自然界和工程领域有大量的应用,因此将Herglotz变分原理应用于变质量力学系统的Lagrange方程与守恒律研究,为研究变质量力学提供了一个新的途径.文中建立了变质量力学系统的Herglotz型广义变分原理,导出了变质量系统的Herglotz型Lagrange方程.定义了变质量力学系统的Herglotz型Noether对称性,建立并证明了Herglotz型Noether定理及其逆定理.文末给出两个变质量非保守系统的具体例子以说明结果的应用.

Abstract

Herglotz’s variational principle provides a variational description of non-conservative dissipation problems, and variable mass mechanics is widely used in nature and engineering. Therefore, it provides a new way to study variable mass mechanics by applying Herglotz’s variational principle to Lagrange equations and conservation laws of variable mass mechanics systems. In this paper, the Herglotz type generalized variational principle of mechanical systems with variable mass is established and the Herglotz type Lagrange equations of mechanical systems with variable mass are derived. Herglotz type Noether symmetry of variable mass mechanical systems is defined, and the Herglotz Noether theorem and its inverse theorem are established and proved. At the end of this paper, two concrete examples of non-conservative systems with variable mass are given to illustrate the application of the results.

关键词

变质量力学系统； Herglotz型广义变分原理； Noether定理；守恒量

Keywords

variable mass mechanical system ； Herglotz type generalized variational principle ； Noether theorem ； conserved quantity

引言
众所周知，对于非保守这样一类自然界广泛存在的现象，人们无法将其纳入哈密顿变分原理的经典框架之中^[1].德国数学家Herglotz提出的一类变分问题^[2]，不仅是对经典变分原理的推广，而且提供非保守和耗散过程的变分描述.Herglotz变分问题与经典变分问题的不同之处在于其微分方程不仅取决于时间、曲线及其导数，还取决于积分泛函本身.Donchev^[3]和Lazo等^[4，5]对若干重要的非保守经典和量子系统的变分描述进行了研究，如:黏性力下振动弦、非保守电磁理论、非保守薛定谔方程等.Georgieva和Guenther研究了Herglotz型Noether定理^[6，7].Santos等^[8，9]将 Herglotz变分问题推广到高阶微商情形.近年来，张毅等将Herglotz变分问题引入力学系统^[10]，建立了非保守系统^[11，12]、非完整系统^[13]和Birkhoff系统^[14，15]的Herglotz变分原理和Noether定理.但是，到目前为止关于Herglotz变分原理的研究还仅限于常质量系统.
变质量系统力学研究质量变化的物体的运动与作用在其上的力之间的关系.第一个系统地研究变质量力学的是Meshchersky^[16]，通过引入质量以不为零的相对速度分离或并入物体时所产生的冲击力，他建立了变质量质点的动力学基本方程.1989年，杨来伍和梅凤翔^[17]在他们的专著中系统地介绍了变质量系统的牛顿力学和变质量系统的分析力学.变质量系统在自然界和工程技术领域有大量应用实例^[18-20]，包括复杂系统，如火箭运动、移动机器人拾取或释放物体等，以及简单系统，如喷水系统或漏气的充气气球的运动等.守恒律是变质量系统力学研究的一个重要方面.Cveticanin利用d’Alembert-Lagrange原理构建了变质量系统的Noether守恒律^[21].梅凤翔在其著作^[1，22]中研究了变质量力学系统的对称性与守恒量.最近，姜文安等^[23-25]研究了一类变质量系统的Noether守恒律和Mei守恒律.
变质量力学系统的问题可分为两类^[26]:一类是质量变化不改变运动学性质; 另一类是质量的变化将引起系统运动学性质的改变.本文研究第一类问题.将Herglotz广义变分原理推广到变质量非保守力学系统，导出变质量系统的Herglotz型Lagrange方程，研究并证明其Noether定理及其逆定理.
1 变质量力学系统的Herglotz型Lagrange方程
假设变质量力学系统由N个质点组成.在时刻t，第i个质点的质量为 $m_{i} （ i = 1，2 ， \dots ， N ）$ ; 在时刻t +dt，由质点分离（或并入）的微粒质量为dm_i.设系统所受完整约束中不含质点的质量，质点系的位形由n个广义坐标 $q_{s} （ s = 1，2 ， \dots ， n ）$ 确定，并假设

m_{i} = m_{i} (t, q_{s}, {\dot{q}}_{s})

(1)

变质量力学系统的Herglotz变分问题可表述如下:
确定函数 $q_{s} （ t ） (t \in [t_{1} ， t_{2}])$ ，使得泛函 $z （ t_{2} ）$ 取得极值，即 $z （ t_{2} ）$ →extr.，其中 $z （ t ）$ 由微分方程

\dot{z} (t) = L (t, q_{s} (t), {\dot{q}}_{s} (t), z (t), m_{i} (t, q_{s}, {\dot{q}}_{s}))

(2)

定义，且满足端点条件

\begin{matrix} {q_{s} (t)|}_{t = t_{1}} = q_{s_{1}}, {q_{s} (t)|}_{t = t_{2}} = q_{s_{2}} \\ (s = 1,2, \dots, n) \end{matrix}

(3)

和初始条件

{z (t)|}_{t = t_{1}} = z_{1}

(4)

泛函 $z （ t ）$ 也称为Herglotz作用量.
令 $\frac{Д}{Д t} ， \frac{Д}{Д q_{s}}$ 以及 $\frac{Д}{Д {\dot{q}}_{s}}$ 分别表示将质量当作常数时对t，q_s以及 ${\dot{q}}_{s}$ 的偏导数，称为凝固偏导数; 符号DDt表示将质量当作常数时对时间的导数，称为凝固导数.
对方程（2）取等时变分，有

δ \dot{z} = \frac{Д L}{Д q_{s}} δ q_{s} + \frac{Д L}{Д \dot{q_{s}}} δ {\dot{q}}_{s} + \frac{\partial L}{\partial m_{i}} δ m_{i} + \frac{\partial L}{\partial z} δ z

(5)

其中

δ m_{i} = \frac{\partial m_{i}}{\partial q_{s}} δ q_{s} + \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} δ {\dot{q}}_{s}

(6)

由交换关系

δ \dot{z} = \frac{d}{d t} δ z

(7)

则方程（5）可以表示为

\frac{d}{d t} δ z = A + \frac{\partial L}{\partial z} δ z

(8)

其中

A = \frac{Д L}{Д q_{s}} δ q_{s} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial q_{s}} δ q_{s} + \frac{Д L}{Д \dot{q_{s}}} δ {\dot{q}}_{s} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} δ {\dot{q}}_{s}

(9)

方程（8）有解

\begin{matrix} δ z (t) e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) - δ z (t_{1}) = \\ \int_{t_{1}}^{t} A e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) d τ \end{matrix}

(10)

考虑到初始条件（4），且 $z (t_{2})$ 取得极值，因此有

δ z (t_{1}) = δ z (t_{2}) = 0

(11)

所以有

\int_{t_{1}}^{t_{2}} A e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) d τ = 0

(12)

将式（9）代入方程（12），并对其中含 $δ {\dot{q}}_{s}$ 的项进行分部积分，得

\begin{matrix} \int_{t_{1}}^{t_{2}} e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) \{\frac{Д L}{Д q_{s}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial q_{s}} - \frac{D}{D t} \frac{Д L}{Д \dot{q_{s}}} - \\ \frac{\partial L}{\partial m_{i}} \frac{Д L}{Д \dot{q_{s}}} {\dot{m}}_{i} - \frac{d}{d t} (\frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) + \frac{\partial L}{\partial z} \frac{Д L}{Д \dot{q_{s}}} + \\ \frac{\partial L}{\partial z} \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}\} δ q_{s} d t = 0 \end{matrix}

(13)

由于δq_s的独立性，根据变分学基本引理，可得

\begin{matrix} e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) (- \frac{D}{D t} \frac{Д L}{Д \dot{q_{s}}} + \frac{Д L}{Д q_{s}} + \\ \frac{\partial L}{\partial z} \frac{Д L}{Д q_{s}} + ψ_{s}) = 0 (s = 1,2, \dots, n) \end{matrix}

(14)

其中

\begin{matrix} ψ_{s} = \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial q_{s}} + \frac{\partial L}{\partial z} \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} - \frac{\partial}{\partial m_{i}} \frac{Д L}{Д \dot{q_{s}}} {\dot{m}}_{i} - \\ \frac{d}{d t} (\frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) \end{matrix}

(15)

方程（14）称为变质量力学系统用凝固导数表示的Herglotz型Lagrange方程.
容易证明下述关系:

\frac{d}{d t} \frac{Д L}{Д \dot{q_{s}}} = \frac{D}{D t} \frac{Д L}{Д \dot{q_{s}}} + \frac{\partial}{\partial m_{i}} \frac{Д L}{Д \dot{q_{s}}} {\dot{m}}_{i}

(16)

将式（16）代入式（14）可得:

\begin{matrix} e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) (- \frac{d}{d t} \frac{Д L}{Д \dot{q_{s}}} + \frac{Д L}{Д q_{s}} + \\ \frac{\partial L}{\partial z} \frac{Д L}{Д \dot{q_{s}}} + φ_{s}) = 0 (s = 1,2, \dots, n) \end{matrix}

(17)

其中

φ_{s} = \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial q_{s}} + \frac{\partial L}{\partial z} \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} - \frac{d}{d t} (\frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}})

(18)

方程（17）称为变质量力学系统用半凝固导数表示的Herglotz型Lagrange方程.
容易证明下述关系:

\frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{s}} = \frac{d}{d t} \frac{Д L}{Д \dot{q_{s}}} + \frac{d}{d t} (\frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}})

(19)

\frac{\partial L}{\partial q_{s}} = \frac{Д L}{Д q_{s}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial q_{s}}

(20)

将式（19）和式（20）代入式（17）可得

\begin{matrix} e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) (- \frac{d}{d t} \frac{\partial L}{\partial {\dot{q}}_{s}} + \frac{\partial L}{\partial q_{s}} + \frac{\partial L}{\partial z} \frac{\partial L}{\partial {\dot{q}}_{s}}) = \\ 0 (s = 1,2, \dots, n) \end{matrix}

(21)

方程（21）称为变质量力学系统用普通导数表示的Herglotz型Lagrange方程.
上述三类变质量系统的Herglotz型Lagrange方程是经典变质量完整力学系统广义坐标表示的Lagrange方程^[17]的推广.文献^[17]指出:当质点质量依赖于广义坐标、广义速度和时间时，应用凝固导数表示的方程最简便，应用普通导数表示的方程最繁杂.
2 Herglotz型Noether对称性
引入时间t和广义坐标q_s的r参数有限变换群G_r的无限小变换

\bar{t} = t + Δ t, {\bar{q}}_{s} (t) = q_{s} (t) + Δ q_{s} (s = 1,2, \dots, n)

(22)

或其展开式

\begin{matrix} \bar{t} = t + ε_{α} ξ_{0}^{α} (t, q_{s}, {\dot{q}}_{s}, m_{i}, z), \\ {\bar{q}}_{s} (t) = q_{s} (t) + ε_{α} ξ_{s}^{α} (t, q_{j}, {\dot{q}}_{j}, m_{i}, z), \\ (s, j = 1,2, \dots, n; α = 1,2, \dots, r; \\ i = 1,2, \dots, N) \end{matrix}

(23)

其中，ε_α为无限小的参数， $ξ_{0}^{α}$ 和 $ξ_{s}^{α}$ 称为该变换的生成元.在变换式（22）的作用下，Herglotz作用量z变成

\bar{z} (\bar{t}) = z (t) + Δ z (t)

(24)

全变分Δz是Herglotz作用量z在无限小变换中相对于ε_α的主线性部分的前后之差.对任意函数F（t），有如下关系

Δ F = δ F + \dot{F} Δ t

(25)

对于完整约束系统，有交换关系

\frac{d}{d t} δ F = δ \frac{d}{d t} F = δ \dot{F}

(26)

因此得到

Δ \dot{F} = \frac{d}{d t} Δ F - \dot{F} \frac{d}{d t} Δ t

(27)

对式（2）进行非等时变分:

Δ \dot{z} = \frac{Д L}{Д t_{}} Δ t + \frac{Д L}{Д q_{s}} Δ q_{s} + \frac{Д L}{Д \dot{q_{s}}} Δ {\dot{q}}_{s} + \frac{\partial L}{\partial z} Δ z + \frac{\partial L}{\partial m_{i}} Δ m_{i}

(28)

由于

Δ m_{i} = \frac{\partial m_{i}}{\partial t} Δ t + \frac{\partial m_{i}}{\partial q_{s}} Δ q_{s} + \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} Δ {\dot{q}}_{s}

(29)

将式（29）代入式（28），并利用关系式（27），得

\begin{matrix} \frac{d}{d t} Δ z = \frac{Д L}{Д t_{}} Δ t + \frac{Д L}{Д q_{s}} Δ q_{s} + \frac{Д L}{Д \dot{q_{s}}} Δ {\dot{q}}_{s} + \frac{\partial L}{\partial z} Δ z + \\ \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial t} Δ t + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial q_{s}} Δ q_{s} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} Δ {\dot{q}}_{s} + \\ L \frac{d}{d t} Δ t \end{matrix}

(30)

方程（30）的解为

\begin{matrix} Δ z (t) e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) - Δ z (t_{1}) = \\ \int_{t_{1}}^{t} \{e x p (- \int_{t_{1}}^{τ} \frac{\partial L}{\partial z} d θ) (\frac{Д L}{Д t_{}} Δ t + \\ \frac{Д L}{Д q_{s}} Δ q_{s} + \frac{Д L}{Д \dot{q_{s}}} Δ {\dot{q}}_{s} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial t} Δ t + \\ \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial q_{s}} Δ q_{s} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} Δ {\dot{q}}_{s} + L \frac{d}{d t} Δ t)\} d τ \end{matrix}

(31)

显然 $Δ z (t_{1}) = 0$ ，所以方程（31）还可以写为
$\begin{matrix} Δ z (t) e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) = \\ \int_{t_{1}}^{t} \{\frac{d}{d τ} \{[L Δ t + (\frac{Д L}{Д \dot{q_{s}}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) (Δ q_{s} - {\dot{q}}_{s} Δ t)] \end{matrix}$

\begin{matrix} \exp (- \int_{t}^{τ} \frac{\partial L}{\partial z} d θ)\} + \\ e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) (- \frac{D}{D t} \frac{Д L}{Д \dot{q_{s}}} + \frac{Д L}{Д q_{s}} + \\ \frac{\partial L}{\partial z} \frac{Д L}{Д \dot{q_{s}}} + ψ_{s}) (Δ q_{s} - {\dot{q}}_{s} Δ t)\} d τ = 0 \end{matrix}

(32)

由于

Δ t = ε_{α} ξ_{0}^{α}, Δ q_{s} = ε_{α} ξ_{s}^{α}

(33)

将式（33）代入式（31）可得

\begin{matrix} Δ z (t) e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) = \\ \int_{t_{1}}^{t} \{e x p (- \int_{t_{1}}^{τ} \frac{\partial L}{\partial z} d θ) [(\frac{Д L}{Д τ_{}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial τ}) ξ_{0}^{α} + \\ (\frac{Д L}{Д q_{s}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial q_{s}}) ξ_{s}^{α} + (\frac{Д L}{Д \dot{q_{s}}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) {\dot{ξ}}_{s}^{α} + \\ (L - \frac{Д L}{Д \dot{q_{s}}} {\dot{q}}_{s} - \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} {\dot{q}}_{s}) {\dot{ξ}}_{0}^{α}]\} ε_{α} d τ \end{matrix}

(34)

将式（33）代入式（32）可得

\begin{matrix} Δ z (t) e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) = \\ \int_{t_{1}}^{t} \{\frac{d}{d τ} \{[L ξ_{0}^{α} + (\frac{Д L}{Д \dot{q_{s}}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) {\bar{ξ}}_{s}^{α}] \\ \exp (- \int_{t_{1}}^{τ} \frac{\partial L}{\partial z} d θ)\} + \\ e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) (- \frac{D}{D t} \frac{Д L}{Д \dot{q_{s}}} + \frac{Д L}{Д q_{s}} + \\ \frac{\partial L}{\partial z} \frac{Д L}{Д \dot{q_{s}}} + ψ_{s}) {\bar{ξ}}_{s}^{α}\} ε_{α} d τ = 0 \end{matrix}

(35)

其中

{\bar{ξ}}_{s}^{α} = ξ_{s}^{α} - {\dot{q}}_{s} ξ_{0}^{α}

(36)

定义1 对于变质量力学系统（14），如果对于无限小变换（22）的每一个变换，等式

Δ z (t_{2}) = 0

(37)

成立，则称变换是该系统的Noether对称变换.
根据定义1以及方程（34），可以得到如下判据.
判据1 在变质量力学系统（14）中，对于无限小变换（23），假设生成元 $ξ_{0}^{α}$ ， $ξ_{s}^{α}$ 符合下列条件

\begin{matrix} (\frac{Д L}{Д t} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial t}) ξ_{0}^{α} + (\frac{Д L}{Д q_{s}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial q_{s}}) ξ_{s}^{α} + \\ (\frac{Д L}{Д \dot{q_{s}}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) {\dot{ξ}}_{s}^{α} + (L - \frac{Д L}{Д \dot{q_{s}}} {\dot{q}}_{s} - \\ \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} {\dot{q}}_{s}) {\dot{ξ}}_{0}^{α} = 0 (α = 1,2, \dots, r) \end{matrix}

(38)

则称变换是该系统的Noether对称变换.
当r =1时，式（38）给出

\begin{matrix} (\frac{Д L}{Д t} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial t}) ξ_{0} + (\frac{Д L}{Д q_{s}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial q_{s}}) ξ_{s} + \\ (\frac{Д L}{Д \dot{q_{s}}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) {\dot{ξ}}_{s} + (L - \frac{Д L}{Д \dot{q_{s}}} {\dot{q}}_{s} - \\ \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} {\dot{q}}_{s}) {\dot{ξ}}_{0} = 0 \end{matrix}

(39)

式（39）为变质量力学系统（14）的Noether等式.
3 Herglotz型Noether定理
定理1 对于变质量力学系统（14），若给定的有限变换群G_r的无限小变换（23）是Herglotz型Noether对称变换，则系统存在r个线性独立的守恒量

\begin{matrix} I_{N}^{α} = [L ξ_{0}^{α} + (\frac{Д L}{Д \dot{q_{s}}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) {\bar{ξ}}_{s}^{α}] e x p \\ (- \int_{t_{1}}^{τ} \frac{\partial L}{\partial z} d θ) = c_{α} \end{matrix}

(40)

证明由于无限小变换（23）是变质量力学系统（14）的Herglotz型Noether对称变换，根据定义1，有 $Δ z (t_{2}) = 0$ .利用公式（35），得到

\begin{matrix} \int_{t_{1}}^{t_{2}} {\frac{d}{d τ} \{[L ξ_{0}^{α} + (\frac{Д L}{Д \dot{q_{s}}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) {\bar{ξ}}_{s}^{α}] e x p \\ (- \int_{t_{1}}^{τ} \frac{\partial L}{\partial z} d θ)\} + e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) \\ (- \frac{D}{D t} \frac{Д L}{Д \dot{q_{s}}} + \frac{Д L}{Д q_{s}} + \frac{\partial L}{\partial z} \frac{Д L}{Д \dot{q_{s}}} + ψ_{s}) {\bar{ξ}}_{s}^{α}\} ε_{α} d τ = 0 \end{matrix}

(41)

由于区间[t₁，t₂]是任意的，ε_α是相互独立的，根据变分法的基本引理，得到

\begin{matrix} \frac{d}{d t} \{[L ξ_{0}^{α} + (\frac{Д L}{Д \dot{q_{s}}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) {\bar{ξ}}_{s}^{α}] e x p \\ (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ)\} + e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) \\ (- \frac{D}{D t} \frac{Д L}{Д \dot{q_{s}}} + \frac{Д L}{Д q_{s}} + \frac{\partial L}{\partial z} \frac{Д L}{Д \dot{q_{s}}} + ψ_{s}) {\bar{ξ}}_{s}^{α} = 0 \end{matrix}

(42)

对于变质量力学系统有Herglotz型Lagrange方程（14），因此得

\frac{d}{d t} \{[L ξ_{0}^{α} + (\frac{Д L}{Д \dot{q_{s}}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) {\bar{ξ}}_{s}^{α}] e x p (- \int_{t}^{τ} \frac{\partial L}{\partial z} d θ)\} = 0

(43)

积分便可得到守恒量（40）.
4 Herglotz型Noether逆定理
已知变质量力学系统（14）有r个线性独立的守恒量

I^{α} = I^{α} (t, q_{s}, {\dot{q}}_{s}, m_{i}, z) = c_{α} (α = 1,2, \dots, r)

(44)

求满足Herglotz型Noether对称性的生成函数 $ξ_{0}^{α}$ 和 $ξ_{s}^{α}$ .
将式（14）乘 ${\bar{ξ}}_{s}^{α}$ ，并对s求和，得到

\begin{matrix} e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) (- \frac{D}{D t} \frac{Д L}{Д \dot{q_{s}}} + \frac{Д L}{Д q_{s}} + \\ \frac{\partial L}{\partial z} \frac{Д L}{Д \dot{q_{s}}} + ψ_{s}) {\bar{ξ}}_{s}^{α} = 0 \end{matrix}

(45)

将式（44）对t求导数后与式（45）相加，展开得

\begin{matrix} \frac{Д I^{α}}{Д t} + \frac{Д I^{α}}{Д q_{s}} {\dot{q}}_{s} + \frac{Д I^{α}}{Д \dot{q_{s}}} {\ddot{q}}_{s} + \frac{\partial I^{α}}{\partial m_{i}} {\dot{m}}_{s} + \frac{\partial I^{α}}{\partial z} L + \\ e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) (- \frac{D}{D t} \frac{Д L}{Д \dot{q_{s}}} + \frac{Д L}{Д q_{s}} + \\ \frac{\partial L}{\partial z} \frac{Д L}{Д \dot{q_{s}}} + ψ_{s}) {\bar{ξ}}_{s}^{α} = 0 \end{matrix}

(46)

令含 ${\ddot{q}}_{k}$ 项的系数为0，可得

\begin{matrix} \frac{Д I^{α}}{Д \dot{q_{s}}} + \frac{\partial I^{α}}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} - {\bar{ξ}}_{k}^{α} e x p (- \int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) \\ [\frac{Д^{2} L}{Д \dot{q_{s}} Д \dot{q_{k}}} + \frac{\partial}{\partial m_{i}} \frac{Д L}{Д \dot{q_{k}}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}} + \frac{\partial}{\partial {\dot{q}}_{s}} (\frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{k}})] = \\ 0 (s = 1,2, \dots n; α = 1,2, \dots, r) \end{matrix}

(47)

再令式（44）等于式（40），可得

\begin{matrix} ξ_{0}^{α} = \frac{1}{L} [e x p (\int_{t_{1}}^{t} \frac{\partial L}{\partial z} d θ) I^{α} - \\ (\frac{Д L}{Д \dot{q_{s}}} + \frac{\partial L}{\partial m_{i}} \frac{\partial m_{i}}{\partial {\dot{q}}_{s}}) {\bar{ξ}}_{s}^{α}] (α = 1,2, \dots, r) \end{matrix}

(48)

由式（47）和式（48）可以找到无限小生成函数，它们对应变质量力学系统（14）的Noether对称变换.于是有:
定理2 对于变质量力学系统（14），如果已知r个线性独立的守恒量（44），那么生成函数由式（47）和式（48）确定的无限小变换相应于系统的Herglotz型Noether对称性.
定理2称为变质量力学系统的Herglotz型Noether逆定理.
5 算例
例1 假设一变质量质点在铅垂平面内运动，其质量为

\begin{matrix} m (t) = m_{0} e x p (- α t) (m_{0} = c o n s t .; \\ α = c o n s t ., α > 0) \end{matrix}

(49)

设质点的运动受黏性阻尼的作用，阻尼系数为c =const.，微粒分离的绝对速度为零.试研究该系统的Noether对称性和守恒量.
选取广义坐标 $q_{1} = x ， q_{2} = y$ ，则系统的Herglotz型Lagrange函数为

\begin{matrix} L = \frac{1}{2} m (t) ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2}) - m (t) g q_{2} - \\ [\frac{c}{m (t)} + α] z \end{matrix}

(50)

其中，泛函ｚ满足

\begin{matrix} \dot{z} = \frac{1}{2} m (t) ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2}) - m (t) g q_{2} - \\ [\frac{c}{m (t)} + α] z \end{matrix}

(51)

根据式（14）给出该系统的运动方程为

\begin{matrix} e x p [λ (t)] [- m (t) {\ddot{q}}_{1} - c {\dot{q}}_{1}] = 0 \\ e x p [λ (t)] [- m (t) {\ddot{q}}_{2} - m (t) g - c {\dot{q}}_{2}] = 0 \end{matrix}

(52)

其中， $λ （ t ） = \frac{c (e^{α t} - e^{α t_{1}})}{α m_{0}} + α (t - t_{1})$ .
Noether等式（39）给出

\begin{matrix} [- \frac{1}{2} α m (t) ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2}) + α m (t) g q_{2} - \\ \frac{α c}{m (t)} z] ξ_{0} - m (t) g ξ_{2} + m (t) {\dot{q}}_{1} {\dot{ξ}}_{1} + \\ m (t) {\dot{q}}_{2} {\dot{ξ}}_{2} - \{\frac{1}{2} m (t) ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2}) + m (t) g q_{2} + \\ [\frac{c}{m (t)} + α] z\} {\dot{ξ}}_{0} = 0 \end{matrix}

(53)

方程（53）有解

ξ_{0} = 0, ξ_{1} = 1, ξ_{2} = 0

(54)

ξ_{0} = 0, ξ_{1} = - \frac{c}{α m (t)} - l n {\dot{q}}_{1}, ξ_{2} = 0

(55)

由定理1，得到

I_{N} = e x p [λ (t)] m (t) {\dot{q}}_{1} = const.

(56)

I_{N} = e x p [λ (t)] m (t) {\dot{q}}_{1} [- \frac{c}{α m (t)} - l n {\dot{q}}_{1}] = const.

(57)

式（56）和式（57）分别为生成函数（54）、函数（55）相应的Noether守恒量.
根据Herglotz型Noether逆定理，由已知的守恒量求Noether对称性.假设系统有守恒量（56），则由式（47）和式（48），可以得到

\begin{matrix} e x p [λ (t)] m (t) (1 - {\bar{ξ}}_{1}) = 0 \\ - e x p [λ (t)] m (t) {\bar{ξ}}_{2} = 0 \end{matrix}

(58)

以及

ξ_{0} = \frac{m {\dot{q}}_{1} - m {\dot{q}}_{1} {\bar{ξ}}_{1} - m {\dot{q}}_{2} {\bar{ξ}}_{2}}{L}

(59)

由方程（58）和方程（59）解得

ξ_{0} = 0, ξ_{1} = 1, ξ_{2} = 0

(60)

例2 球形的雨滴在大气中下落，沿途受到水汽的充实.设雨滴的初始半径为r₀，由于凝聚，雨滴的质量以正比于其表面积的速率增加^[22]，比例系数为α.假设雨滴下落过程中受到与速度成正比的阻力作用，阻力系数为μ.试研究该系统的Noether对称性与守恒量.
因为

\frac{d m}{d t} = 4 π r^{2} α, r = r_{0} + α t

(61)

故有

m (t) = \frac{4}{3} π r^{3}

(62)

系统的Herglotz型Lagrange函数为

L = \frac{1}{2} m (t) ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2} + {\dot{q}}_{3}^{2}) - m g q_{3} - (\frac{μ}{m} - \frac{3 α}{r}) z

(63)

式中的 Herglotz作用量 $z （ t ）$ 满足方程

\dot{z} = \frac{1}{2} m (t) ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2} + {\dot{q}}_{3}^{2}) - m g q_{3} - (\frac{μ}{m} - \frac{3 α}{r}) z

(64)

根据式（14），该系统的运动微分方程为

\begin{matrix} e x p [λ (t)] (- m {\ddot{q}}_{1} - μ {\dot{q}}_{1}) = 0 \\ e x p [λ (t)] (- m {\ddot{q}}_{2} - μ {\dot{q}}_{2}) = 0 \\ e x p [λ (t)] (- m {\ddot{q}}_{3} - m g - μ {\dot{q}}_{3}) = 0 \end{matrix}

(65)

其中

\begin{matrix} λ (t) = 3 \ln (\frac{r_{0} + α t_{1}}{r}) + \frac{3 μ}{8 π α {(r_{0} + α t_{1})}^{2}} - \\ \frac{μ r}{2 α m} \end{matrix}

(66)

Noether等式（39）给出

\begin{matrix} [2 π r^{2} α ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2} + {\dot{q}}_{3}^{2}) - 4 π r^{2} α g q_{3} - \frac{3 α μ}{m r} z] ξ_{0} - \\ m (t) g ξ_{3} + m (t) {\dot{q}}_{1} {\dot{ξ}}_{1} + m (t) {\dot{q}}_{2} {\dot{ξ}}_{2} + \\ m (t) {\dot{q}}_{3} {\dot{ξ}}_{3} - [\frac{1}{2} m (t) ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2} + {\dot{q}}_{3}^{2}) + \\ m (t) g q_{3} + (\frac{μ}{m} - \frac{3 α}{r}) z] {\dot{ξ}}_{0} = 0 \end{matrix}

(67)

方程（67）有解

ξ_{0} = 0, ξ_{1} = 1, ξ_{2} = 1, ξ_{3} = 0

(68)

ξ_{0} = 0, ξ_{1} = \frac{3 μ}{8 π α r^{2}} - l n {\dot{q}}_{1}, ξ_{2} = 1, ξ_{3} = 0

(69)

由定理1，得到

I_{N} = e x p [λ (t)] [m (t) {\dot{q}}_{1} + m (t) {\dot{q}}_{2}] = con st.

(70)

\begin{matrix} I_{N} = e x p [λ (t)] [m {\dot{q}}_{2} + m {\dot{q}}_{1} (\frac{3 μ}{8 π α r^{2}} - \\ l n {\dot{q}}_{1})] = c o n s t . \end{matrix}

(71)

式（70）、式（71）分别为生成函数（68）、函数（69）所导致的Noether守恒量.
根据Herglotz型Noether逆定理，由已知守恒量求Noether对称性.假设系统有守恒量（70），则由式（47）和式（48），可以得到

\begin{matrix} e x p [λ (t)] m (t) (1 - {\bar{ξ}}_{1}) = 0 \\ e x p [λ (t)] m (t) (1 - {\bar{ξ}}_{2}) = 0 \\ - e x p [λ (t)] m (t) {\bar{ξ}}_{3} = 0 \end{matrix}

(72)

以及

ξ_{0} = \frac{m (t) ({\dot{q}}_{1} + {\dot{q}}_{2} - {\dot{q}}_{1} {\bar{ξ}}_{1} - {\dot{q}}_{2} {\bar{ξ}}_{2} - {\dot{q}}_{3} {\bar{ξ}}_{3})}{L}

(73)

由方程（72）、方程（73）解得

ξ_{0} = 0, ξ_{1} = 1, ξ_{2} = 1, ξ_{3} = 0

(74)

6 结论
变质量力学系统研究的是物体质量在不断变化的状态下，物体运动状态与作用在其上力的关系.由于变质量系统在自然界和工程领域普遍存在，因此变质量力学的研究具有重要意义.本文研究了变质量力学系统的Herglotz型Lagrange方程与Noether对称性及其守恒量，主要工作如下:提出了变质量力学系统中的Herglotz型广义变分原理，并基于该原理建立了分别用凝固导数、半凝固导数、普通导数表示的变质量力学系统的Herglotz型Lagrange方程（14）、式（17）和式（21）; 提出了变质量力学系统Herglotz型Noether对称性的定义及其判据方程（39）; 建立并证明了变质量力学系统Herglotz型Noether定理（定理1）及其逆定理（定理2）.
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- [26] 梅凤翔.高等分析力学.北京:北京理工大学出版社,1991(Mei F X.Advanced analytical mechanics.Beijing:Beijing Institute of Technology Press,1991(in Chinese))

基本信息

中图分类号: O316
文献标识码: A
DOI: 10.6052/1672-6553-2022-012
文章编号: 1672-6553-2022-20(6)-106-08

基金信息

国家自然科学基金赞助项目(11972241,11572212)；
江苏省自然科学基金资助项目(BK20191454)；
The project supported by the National Natural Science Foundation of China(11972241,11572212)；
the Natural Science Foundation of Jiangsu Province of China(BK20191454)；

引用信息

稿件历史

收稿日期: 2022-01-04

参考文献

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变质量力学系统的Herglotz型Lagrange方程与Noether对称性和守恒量*

HERGLOTZ TYPE LAGRANGE EQUATIONS AND NOETHER SYMMETRY AND CONSERVED QUANTITY FOR MECHANICAL SYSTEMS WITH VARIABLE MASS*

摘要

Abstract

关键词

Keywords

1 变质量力学系统的Herglotz型Lagrange方程

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2 Herglotz型Noether对称性

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3 Herglotz型Noether定理

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4 Herglotz型Noether逆定理

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5 算例

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变质量力学系统的Herglotz型Lagrange方程与Noether对称性和守恒量^*

HERGLOTZ TYPE LAGRANGE EQUATIONS AND NOETHER SYMMETRY AND CONSERVED QUANTITY FOR MECHANICAL SYSTEMS WITH VARIABLE MASS^*