Nielsen方程的三重组合梯度表示及其稳定性分析

尹明旭，陈向炜; Yin Mingxu; Chen Xiangwei

引 en

Nielsen方程的三重组合梯度表示及其稳定性分析^*

尹明旭¹
机构：
1. 苏州科技大学数学科学学院, 苏州 215009
×
，陈向炜²
机构：
2. 商丘师范学院电子电气工程学院, 商丘 476000
×

1. 苏州科技大学数学科学学院, 苏州 215009；
2. 商丘师范学院电子电气工程学院, 商丘 476000；

The Triple Combination Gradient Representation Of Nielsen Equation and Its Stability Analysis^*

Yin Mingxu¹
Affiliation：
1. Collegr of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009 , China
×
，Chen Xiangwei²
Affiliation：
2. College of Electronic and Electrical Engineering, Shangqiu Normal University, Shangqiu 476000 , China
×

1. Collegr of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009 , China；
2. College of Electronic and Electrical Engineering, Shangqiu Normal University, Shangqiu 476000 , China；

通讯作者:

陈向炜,E-mail∶hnchenxw@163.com

中图分类号:O316

文献标识码:A

文章编号:1672-6553-2023-21(2)-024-009

DOI:10.6052/1672-6553-2022-025

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出版信息

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目录contents

摘要 Abstract
关键词 Keywords
1 三重组合梯度系统及其性质
1.1 三重组合梯度系统Ⅰ
1.2 三重组合梯度系统Ⅱ
1.3 三重组合梯度系统Ⅲ
1.4 三重组合梯度系统Ⅳ
2 Nielsen方程的三重组合梯度表示
3 算例
4 结论
参考文献

摘要

研究Nielsen方程的三重组合梯度表示以及方程零解稳定性.首先给出4类三重组合梯度系统及其性质.其次,给出完整系统和非完整系统的Nielsen方程转化成三重组合梯度系统的条件;将完整和非完整两类Nielsen方程分别化为三重组合梯度系统并研究方程的零解稳定性.最后,举例验证结果的应用.

Abstract

The triple combination gradient representation of Nielsen equation and the stability of the zero solution of the equation are studied. First, four kinds of triple combination gradient systems and their properties are given. Secondly, the conditions for transforming Nielsen equations of complete and nonholonomic systems into triple combination gradient systems are given; The two types of Nielsen equations are respectively transformed into triple combined gradient systems and the stability of the zero solution of the equations is studied. Finally, examples are given to verify the application of the results.

关键词

Nielsen方程；组合梯度系统；稳定性

Keywords

Nielsen equation ； combination gradient system ； stability

引言
非完整系统动力学是动力学研究中的重要方向之一，《非完整系统力学基础》对非完整系统动力学进行了完整介绍^[1].Nielsen方程是八类一阶非完整约束系统的方程中的一类^[2]. 梅凤翔教授在著作^[1]提出了Nielsen算子，由此将八类一阶非完整约束系统的方程归并为三大体系，方程中出现Nielsen算子的归为Nielsen体系.Nielsen体系与Euler-Lagrange体系的方程具有等价性，但在方程计算和约化的过程中，Nielsen方程可能更加方便.近年来，对Nielsen方程的研究取得了很多重要成果^[3-8]，对Nielsen方程开展进一步深入研究很有必要也很有意义.
力学系统稳定性是分析力学中一个重要的研究方向.梯度系统的稳定性非常适合用Lyapunov函数进行研究^[9-10].将力学系统转化为梯度系统，通过梯度系统的性质研究力学系统稳定性，问题就会变得简便.有关约束力学的梯度表示，梅凤翔等研究了Lagrange系统、Hamilton系统和Birkhoff系统的梯度系统方法和斜梯度系统方法^[11-15]; 陈向炜等研究了Chetaev型非完整系统的广义梯度表示以及定常Chetaev型非完整系统的组合梯度表示^[16-17]; 王嘉航、张毅给出了非自治广义Birkhoff系统的半负定矩阵梯度系统表示^[18]; 崔金超等研究了Appell方程的广义梯度表示和稳定性^[19]; 李彦敏、梅凤翔用两类广义梯度表示研究了Tzénoff方程解的稳定性^[20].有关三重组合梯度系统，王嘉航、张毅给出了自治广义Birkhoff系统的三重组合梯度表示^[21]; 章婷婷等给出了判定定常Chetaev型非完整系统稳定性的三重组合梯度方法^[22]; 董孟峰等通过三重组合梯度方法判定了广义Birkhoff系统稳定性^[23].Nielsen方程解的稳定性是Nielsen方程研究中的一个重要方向，对比梅凤翔等通过广义斜梯度系统和具有对称负定矩阵的广义梯度系统进行了研究^[24]，其中梯度系统都是非组合的形式，且需要判定Lyapunov函数的全导数是否负定.本文将给出Nielsen方程的4类三重组合梯度的表示，通过三重组合梯度系统的性质研究Nielsen方程的零解稳定性，此时，Lyapunov函数的全导数一定是负定的，给出完整和非完整系统的Nielsen方程的算例，说明研究结果的应用.
1 三重组合梯度系统及其性质
1.1 三重组合梯度系统Ⅰ
其微分方程有形式^[11]

\begin{matrix} {\dot{x}}_{i} = \frac{\partial V (t, x)}{\partial x_{i}} + b_{i j} \frac{\partial V (t, x)}{\partial x_{i}} + \\ s_{i j} \frac{\partial V (t, x)}{\partial x_{i}}, (i, j = 1,2, \dots, m) \end{matrix}

(1)

其中 $x = (x_{1} ， x_{2} ， \dots ， x_{m}) ， b_{i j} （ x ）$ 为反对称矩阵，s_ij（x）为对称负定负定矩阵，V为势函数.
根据（1）求 $\dot{V}$ ，可得

\dot{V} = - \frac{\partial V}{\partial x_{i}} \frac{\partial V}{\partial x_{i}} + \frac{\partial V}{\partial x_{i}} s_{i j} \frac{\partial V}{\partial x_{j}}

(2)

其中 $\dot{V}$ 负定，若V正定，则解渐进稳定.
1.2 三重组合梯度系统Ⅱ
其微分方程有形式^[11]

\begin{matrix} {\dot{x}}_{i} = - \frac{\partial V (t, x)}{\partial x_{i}} + b_{i j} (x) \frac{\partial V (t, x)}{\partial x_{j}} + \\ a_{i j} (x) \frac{\partial V (t, x)}{\partial x_{j}}, (i, j = 1,2, \dots, m) \end{matrix}

(3)

其中 $x = (x_{1} ， x_{2} ， \dots ， x_{m}) ， b_{i j} （ x ）$ 为反对称矩阵， $a_{i j} （ x ）$ 为半负定矩阵，V为势函数.
根据式（3）求 $\dot{V}$ ，可得

\dot{V} = - \frac{\partial V}{\partial x_{i}} \frac{\partial V}{\partial x_{i}} + \frac{\partial V}{\partial x_{i}} a_{i j} \frac{\partial V}{\partial x_{j}}

(4)

其中， $\dot{V}$ 负定，若V正定，则解渐进稳定.
1.3 三重组合梯度系统Ⅲ
其微分方程有形式^[11]

\begin{matrix} {\dot{x}}_{i} = - \frac{\partial V (t, x)}{\partial x_{i}} + s_{i j} (x) \frac{\partial V (t, x)}{\partial x_{j}} + \\ a_{i j} (x) \frac{\partial V (t, x)}{\partial x_{j}}, (i, j = 1,2, \dots, m) \end{matrix}

(5)

其中 $x = (x_{1} ， x_{2} ， \dots ， x_{m}) ， s_{i j} （ x ）$ 为对称负定矩阵， $a_{i j} （ x ）$ 为半负定矩阵，V为势函数.
根据式（5）求 $\dot{V}$ ，可得

\dot{V} = - \frac{\partial V}{\partial x_{i}} \frac{\partial V}{\partial x_{i}} + \frac{\partial V}{\partial x_{i}} s_{i j} \frac{\partial V}{\partial x_{j}} + \frac{\partial V}{\partial x_{i}} a_{i j} \frac{\partial V}{\partial x_{j}}

(6)

其中， $\dot{V}$ 负定，若V正定，则解渐进稳定.
1.4 三重组合梯度系统Ⅳ
其微分方程有形式^[11]

\begin{matrix} {\dot{x}}_{i} = b_{i j} (x) \frac{\partial V (t, x)}{\partial x_{i}} + s_{i j} (x) \frac{\partial V (t, x)}{\partial x_{j}} + \\ a_{i j} (x) \frac{\partial V (t, x)}{\partial x_{j}}, (i, j = 1,2, \dots, m) \end{matrix}

(7)

其中， $x = (x_{1} ， x_{2} ， \dots ， x_{m}) ， b_{i j} （ x ）$ 为反对称矩阵， $s_{i j} （ x ）$ 为对称负定矩阵， $a_{i j} （ x ）$ 为半负定矩阵，V为势函数.
根据式（7）求 $\dot{V}$ ，可得

\dot{V} = \frac{\partial V}{\partial x_{i}} s_{i j} \frac{\partial V}{\partial x_{j}} + \frac{\partial V}{\partial x_{i}} a_{i j} \frac{\partial V}{\partial x_{j}}

(8)

其中， $\dot{V}$ 负定，若V正定，则解渐进稳定.
2 Nielsen方程的三重组合梯度表示
完整系统的Nielsen方程为

\frac{\partial {\dot{T}}^{2}}{\partial {\dot{q}}_{s}} - 2 \frac{\partial T}{\partial q_{s}} = Q_{s}, (s = 1,2, \dots, n)

(9)

其中， $q_{s} （ s = 1，2 ， \dots ， n ）$ 为确定力学系统位形的n个广义坐标， $T = T (q_{s} ， {\dot{q}}_{s} ， t)$ 为动能， $\dot{T}$ 为动能对t的导数， $Q_{s} = Q_{s} (q_{k} ， {\dot{q}}_{k} ， t)$ 为广义力.所有的广义加速度可由式（9）解出，记为

{\ddot{q}}_{s} = α_{s} (q_{k}, {\dot{q}}_{k}, t), (s, k = 1,2, \dots, n)

(10)

将方程（10）降阶，令

(11)

可得

{\dot{a}}^{μ} = F_{μ} (t, a^{ν}), (μ, ν = 1,2, \dots, 2 n)

(12)

其中，

(13)

非完整系统的Nielsen方程为

\frac{\partial \dot{T}}{\partial {\dot{q}}_{s}} - 2 \frac{\partial T}{\partial q_{s}} = Q_{s} + λ_{β} \frac{\partial f_{β}}{\partial {\dot{q}}_{s}}, (s = 1,2, \dots, n)

(14)

此时，系统受到g个双面理想Chetaev型非完整约束:

f_{β} (t, q_{s}, {\dot{q}}_{s}), (β = 1,2, \dots, g)

(15)

其中λ_β为约束乘子，可由式（14）和式（15）在运动微分方程积分之前求得:

λ_{β} = λ_{β} (t, q, \dot{q})

(16)

将式（16）代入式（14）可以求得

\frac{\partial \dot{T}}{\partial {\dot{q}}_{s}} - 2 \frac{\partial T}{\partial q_{s}} = Q_{s} + Λ_{s}, (s = 1,2, \dots, n)

(17)

其中

Λ_{s} = Λ_{s} (t, q, \dot{q}) = λ_{β} (t, q, \dot{q}) \frac{\partial f_{β}}{\partial {\dot{q}}_{s}}

(18)

式（18）是非完整系统（14）和（15）相对应的完整系统的方程.只需要研究式（18），如果运动初始条件满足式（15）的约束方程，此时式（18）的解就给出相应的非完整系统的运动. 所有的广义加速度可由式（18）解出，记为

{\ddot{q}}_{s} = γ_{s} (q_{k}, {\dot{q}}_{k}, t), (s, k = 1,2, \dots, n)

(19)

将方程（19）降阶，令

(20)

可得

{\dot{a}}^{μ} = G_{μ} (t, a^{ν}), (μ, ν = 1,2, \dots, 2 n)

(21)

其中，

(22)

降阶后的方程（12）若要能成为三重组合梯度系统，需存在矩阵b_μν，s_μν，a_μν和函数V=V（t，a）满足以下条件之一:

\begin{matrix} F_{μ} = - \frac{\partial V}{\partial a^{ν}} + b_{μ ν} \frac{\partial V}{\partial a^{ν}} + s_{μ ν} \frac{\partial V}{\partial a^{ν}}, (μ, ν = 1,2, \dots 2 n) \\ F_{μ} = - \frac{\partial V}{\partial a^{ν}} + b_{μ ν} \frac{\partial V}{\partial a^{ν}} + a_{μ ν} \frac{\partial V}{\partial a^{ν}}, (μ, ν = 1,2, \dots 2 n) \\ F_{μ} = - \frac{\partial V}{\partial a^{ν}} + s_{μ ν} \frac{\partial V}{\partial a^{ν}} + a_{μ ν} \frac{\partial V}{\partial a^{ν}}, (μ, ν = 1,2, \dots 2 n) \\ F_{μ} = b_{μ ν} \frac{\partial V}{\partial a^{ν}} + s_{μ ν} \frac{\partial V}{\partial a^{ν}} + a_{μ ν} \frac{\partial V}{\partial a^{ν}}, (μ, ν = 1,2, \dots 2 n) \end{matrix}

(23)

降阶后的方程（21）若要能成为三重组合梯度系统，需存在矩阵b_μν，s_μν，a_μν和函数V=V（t，a）满足以下条件之一:

\begin{matrix} G_{μ} = - \frac{\partial V}{\partial a^{ν}} + b_{μ ν} \frac{\partial V}{\partial a^{ν}} + s_{μ ν} \frac{\partial V}{\partial a^{ν}}, (μ, ν = 1,2, \dots 2 n) \\ G_{μ} = - \frac{\partial V}{\partial a^{ν}} + b_{μ ν} \frac{\partial V}{\partial a^{ν}} + a_{μ ν} \frac{\partial V}{\partial a^{ν}}, (μ, ν = 1,2, \dots 2 n) \\ G_{μ} = - \frac{\partial V}{\partial a^{ν}} + s_{μ ν} \frac{\partial V}{\partial a^{ν}} + a_{μ ν} \frac{\partial V}{\partial a^{ν}}, (μ, ν = 1,2, \dots 2 n) \\ G_{μ} = b_{μ ν} \frac{\partial V}{\partial a^{ν}} + s_{μ ν} \frac{\partial V}{\partial a^{ν}} + a_{μ ν} \frac{\partial V}{\partial a^{ν}}, (μ, ν = 1,2, \dots 2 n) \end{matrix}

(24)

3 算例
对完整和非完整Nielsen方程的4类三重组合梯度表示的8类情形分别用算例说明结果的应用.
例1 完整Nielsen方程的三重组合梯度系统Ⅰ表示. 单自由度系统的动能和广义力分别为

\begin{matrix} T = \frac{1}{2} (\dot{q} t + q)^{2} \\ Q = - 3 \dot{q} t^{2} - 6 q t^{3} - 2 \dot{q} t^{3} - 2 q t^{2} + 2 \dot{q} t \end{matrix}

(25)

建立Nielsen方程，并研究零解的稳定性.
计算得

\begin{matrix} \dot{T} = (\ddot{q} t + 2 \dot{q}) (\dot{q} t + q) \\ \frac{\partial \dot{T}}{\partial \dot{q}} = t (\ddot{q} t + 2 \dot{q}) + 2 (\dot{q} t + q) \\ \frac{\partial T}{\partial q} = (\dot{q} t + q) \end{matrix}

(26)

Nielsen方程

\frac{\partial \dot{T}}{\partial \dot{q}} - 2 \frac{\partial T}{\partial q} = Q

(27)

给出

\ddot{q} = - 3 \dot{q} - 6 q t - 2 \dot{q} t - 2 q

(28)

(29)

则有

\begin{matrix} {\dot{a}}^{1} = - 2 a^{1} t + 2 a^{2} \\ {\dot{a}}^{2} = - 3 a^{2} \end{matrix}

(30)

可表示为

(\begin{matrix} {\dot{a}}^{1} \\ {\dot{a}}^{2} \end{matrix}) = [(\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) + (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 2 \end{matrix})] (\begin{matrix} \frac{\partial V}{\partial a^{1}} \\ \frac{\partial V}{\partial a^{2}} \end{matrix})

(31)

其中

V = \frac{1}{2} {(a^{1})}^{2} t + \frac{1}{2} {(a^{2})}^{2}

(32)

这是三重组合梯度系统Ⅰ的形式.V在 $a^{1} = a^{2} = 0$ 的邻域内正定.因此，零解 $a^{1} = a^{2} = 0$ 是渐进稳定的.
例2 完整Nielsen方程的三重组合梯度系统Ⅱ表示. 单自由度系统的动能和广义力分别为

\begin{matrix} T = \frac{1}{2} [\dot{q} (1 + t) + q] \\ Q = - 6 \dot{q} (1 + t)^{2} - 8 q (1 + t)^{2} - 4 q s i n t (1 + t)^{2} - \\ 2 q c o s t (1 + t) 2 - 2 \dot{q} s i n t (1 + t)^{2} + 2 \dot{q} (1 + t) \end{matrix}

(33)

建立Nielsen方程，并研究零解的稳定性.
计算得

\begin{matrix} \dot{T} = [\ddot{q} (1 + t) + 2 \dot{q}] [\dot{q} (1 + t) + q] \\ \frac{\partial \dot{T}}{\partial \dot{q}} = (1 + t) [\ddot{q} (1 + t) + 2 \dot{q}] + 2 [\dot{q} (1 + t) + q] \\ \frac{\partial T}{\partial q} = [\dot{q} (1 + t) + q] \end{matrix}

(34)

Nielsen方程

\frac{\partial \dot{T}}{\partial \dot{q}} - 2 \frac{\partial T}{\partial q} = Q

(35)

给出

\ddot{q} = - 6 \dot{q} - 8 q - 4 q s i n t - 2 q c o s t - 2 \dot{q} s i n t

(36)

(37)

则有

\begin{matrix} {\dot{a}}^{1} = - 2 (2 + s i n t) a^{1} + 2 a^{2} \\ {\dot{a}}^{2} = - 2 a^{2} \end{matrix}

(38)

可表示为

(\begin{matrix} {\dot{a}}^{1} \\ {\dot{a}}^{2} \end{matrix}) = [(\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) + (\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 1 \end{matrix})] (\begin{matrix} \frac{\partial V}{\partial a^{1}} \\ \frac{\partial V}{\partial a^{2}} \end{matrix})

(39)

其中

V = \frac{1}{2} {(a^{1})}^{2} (2 + s i n t) + \frac{1}{2} {(a^{2})}^{2}

(40)

这是三重组合梯度系统Ⅱ的形式. V在 $a^{1} = a^{2} = 0$ 的邻域内正定.因此，零解 $a^{1} = a^{2} = 0$ 是渐进稳定的.
例3 完整Nielsen方程的三重组合梯度系统Ⅲ表示. 单自由度系统的动能和广义力分别为

\begin{matrix} T = \frac{1}{2} (\dot{q} t + q)^{2} \\ Q = - 11 \dot{q} t^{2} - 43 q t - 8 q t^{3} - 3 \dot{q} t^{3} + 2 \dot{q} t \end{matrix}

(41)

建立Nielsen方程，并研究零解的稳定性.
计算得

\begin{matrix} \dot{T} = (\ddot{q} t + 2 \dot{q}) (\dot{q} t + q) \\ \frac{\partial \dot{T}}{\partial \dot{q}} = t (\ddot{q} t + 2 \dot{q}) + 2 (\dot{q} t + q) \\ \frac{\partial T}{\partial q} = (\dot{q} t + q) \end{matrix}

(42)

Nielsen方程

\frac{\partial \dot{T}}{\partial \dot{q}} - 2 \frac{\partial T}{\partial q} = Q

(43)

给出

\ddot{q} = - 11 \dot{q} - 43 q - 8 q t - 3 \dot{q} t

(44)

(45)

则有

\begin{matrix} {\dot{a}}^{1} = - 5 a^{1} - 3 a^{1} t + 5 a^{2} \\ {\dot{a}}^{2} = - 2 a^{1} + 2 a^{1} t - 6 a^{2} \end{matrix}

(46)

可表示为

(\begin{matrix} {\dot{a}}^{1} \\ {\dot{a}}^{2} \end{matrix}) = [(\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 2 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 1 \end{matrix})] (\begin{matrix} \frac{\partial V}{\partial a^{1}} \\ \frac{\partial V}{\partial a^{2}} \end{matrix})

(47)

其中

V = \frac{1}{2} {(a^{1})}^{2} (1 + t) + \frac{1}{2} {(a^{2})}^{2} - a^{1} a^{2}

(48)

这是三重组合梯度系统Ⅲ的形式.V在 $a^{1} = a^{2} = 0$ 的邻域内正定.因此，零解 $a^{1} = a^{2} = 0$ 是渐进稳定的.
例4 完整Nielsen方程的三重组合梯度系统Ⅳ表示. 单自由度系统的动能和广义力分别为

\begin{matrix} T = \frac{1}{2} {(\dot{q} e^{t} + q e^{t})}^{2} \\ Q = - 3 \dot{q} e^{3 t} - 3 q e^{3 t} + 3 q e^{2 t} + \dot{q} e^{2 t} \end{matrix}

(49)

建立Nielsen方程，并研究零解的稳定性.
计算得

\begin{matrix} \dot{T} = (\ddot{q} e^{t} + 2 \dot{q} e^{t} + q e^{t}) (\dot{q} e^{t} + q e^{t}) \\ \frac{\partial \dot{T}}{\partial \dot{q}} = e^{t} (\ddot{q} e^{t} + 2 \dot{q} e^{t} + q e^{t}) + 2 e^{t} (\dot{q} e^{t} + q e^{t}) \\ \frac{\partial T}{\partial q} = (\dot{q} e^{t} + q e^{t}) e^{t} \end{matrix}

(50)

Nielsen方程

\frac{\partial \dot{T}}{\partial \dot{q}} - 2 \frac{\partial T}{\partial q} = Q

(51)

给出

\ddot{q} = - 3 \dot{q} e^{t} - 3 q e^{t} + 2 q - \dot{q}

(52)

(53)

则有

\begin{matrix} {\dot{a}}^{1} = - 2 a^{1} + a^{2} e^{t} \\ {\dot{a}}^{2} = 3 a^{1} - 3 a^{2} e^{t} \end{matrix}

(54)

可表示为

(\begin{matrix} {\dot{a}}^{1} \\ {\dot{a}}^{2} \end{matrix}) = [(\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 2 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 1 \end{matrix})] (\begin{matrix} \frac{\partial V}{\partial a^{1}} \\ \frac{\partial V}{\partial a^{2}} \end{matrix})

(55)

其中

V = \frac{1}{2} {(a^{1})}^{2} + \frac{1}{2} {(a^{2})}^{2} e^{t}

(56)

这是三重组合梯度系统Ⅳ的形式. V在 $a^{1} = a^{2} = 0$ 的邻域内正定.因此，零解 $a^{1} = a^{2} = 0$ 是渐进稳定的.
例5 非完整Nielsen方程的三重组合梯度系统Ⅰ表示. 系统的动能、广义力和非完整约束分别为

\begin{matrix} T = \frac{1}{2} ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2}) \\ Q_{1} = - 14 {\dot{q}}_{1} - 16 q_{1} e^{t} - 4 {\dot{q}}_{1} e^{t} \\ Q_{2} = - {\dot{q}}_{2} \\ f = {\dot{q}}_{1} + {\dot{q}}_{2} + q_{2} = 0 \end{matrix}

(57)

建立Nielsen方程，并研究零解的稳定性.
非完整Nielsen方程有形式

\frac{\partial \dot{T}}{\partial {\dot{q}}_{s}} - 2 \frac{\partial T}{\partial q_{s}} = Q_{s} + λ_{β} \frac{\partial f_{β}}{\partial {\dot{q}}_{s}}, (s = 1,2)

(58)

计算可得

\dot{T} = {\dot{q}}_{1} {\ddot{q}}_{1} + {\dot{q}}_{2} {\ddot{q}}_{2}

(59)

(60)

由方程可得

\begin{matrix} {\ddot{q}}_{1} = - 14 {\dot{q}}_{1} - 16 q_{1} e^{t} - 4 {\dot{q}}_{1} e^{t} + λ \\ {\ddot{q}}_{2} = - {\dot{q}}_{2} + λ \end{matrix}

(61)

解得

λ = 2 {\dot{q}}_{1} + q_{1} t^{2} + {\dot{q}}_{1} t^{2} + q_{1} t

(62)

代入得

{\ddot{q}}_{1} = 7 {\dot{q}}_{1} + 8 q_{1} e^{t} + 2 {\dot{q}}_{1} e^{t}

(63)

(64)

则有

\begin{matrix} {\dot{a}}^{1} = 2 a^{2} - 2 a^{1} (1 + e^{t}) \\ {\dot{a}}^{2} = 5 a^{1} - 5 a^{2} - 3 a^{2} e^{t} \end{matrix}

(65)

可表示为

(\begin{matrix} {\dot{a}}^{1} \\ {\dot{a}}^{2} \end{matrix}) = [(\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) + (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 2 \end{matrix})] (\begin{matrix} \frac{\partial V}{\partial a^{1}} \\ \frac{\partial V}{\partial a^{2}} \end{matrix})

(66)

其中

V = \frac{1}{2} {(a^{1})}^{2} + \frac{1}{2} {(a^{2})}^{2} (1 + e^{t}) - a^{1} a^{2}

(67)

这是三重组合梯度系统Ⅰ的形式. V在 $a^{1} = a^{2} = 0$ 的邻域内正定.因此，零解 $a^{1} = a^{2} = 0$ 是渐进稳定的.
例6 非完整Nielsen方程的三重组合梯度系统Ⅱ表示. 系统的动能、广义力和非完整约束分别为
$\begin{matrix} T = \frac{1}{2} ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2}) \\ Q_{1} = - \frac{15}{2} {\dot{q}}_{1} - \frac{5}{2} {\dot{q}}_{1} t - 5 q_{1} t \end{matrix}$

\begin{matrix} Q_{2} = - \frac{1}{2} {\dot{q}}_{2} \\ f = {\dot{q}}_{1} + 2 {\dot{q}}_{2} + q_{2} = 0 \end{matrix}

(68)

建立Nielsen方程，并研究零解的稳定性.
非完整Nielsen方程有形式

\frac{\partial \dot{T}}{\partial {\dot{q}}_{s}} - 2 \frac{\partial T}{\partial q_{s}} = Q_{s} + λ_{β} \frac{\partial f_{β}}{\partial {\dot{q}}_{s}}, (s = 1,2)

(69)

计算可得

\dot{T} = {\dot{q}}_{1} {\ddot{q}}_{1} + {\dot{q}}_{2} {\ddot{q}}_{2}

(70)

(71)

由方程可得

\begin{matrix} {\ddot{q}}_{1} = - \frac{15}{2} {\dot{q}}_{1} - \frac{5}{2} {\dot{q}}_{1} t - 5 q_{1} t + λ \\ {\ddot{q}}_{2} = - \frac{1}{2} {\dot{q}}_{2} + λ \end{matrix}

(72)

解得

λ = \frac{1}{2} {\dot{q}}_{1} t + \frac{3}{2} {\dot{q}}_{1} + q_{1} t

(73)

代入得

\begin{matrix} {\ddot{q}}_{1} = - 6 {\dot{q}}_{1} - 2 {\dot{q}}_{1} t - 4 q_{1} t \\ {\ddot{q}}_{2} = - \frac{1}{2} {\dot{q}}_{2} + \frac{3}{2} {\dot{q}}_{1} + \frac{1}{2} {\dot{q}}_{1} t + q_{1} t \end{matrix}

(74)

(75)

则有

\begin{matrix} {\dot{a}}^{1} = - 2 a^{1} + 2 a^{2} \\ {\dot{a}}^{2} = 4 a^{1} - 4 a^{2} - 2 a^{2} t \end{matrix}

(76)

可表示为

(\begin{matrix} {\dot{a}}^{1} \\ {\dot{a}}^{2} \end{matrix}) = [(\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) + (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 1 \end{matrix})] (\begin{matrix} \frac{\partial V}{\partial a^{1}} \\ \frac{\partial V}{\partial a^{2}} \end{matrix})

(77)

其中

V = \frac{1}{2} {(a^{1})}^{2} + \frac{1}{2} {(a^{2})}^{2} (1 + t) - a^{1} a^{2}

(78)

这是三重组合梯度系统Ⅱ的形式. V在 $a^{1} = a^{2} = 0$ 的邻域内正定.因此，零解 $a^{1} = a^{2} = 0$ 是渐进稳定的.
例7 非完整Nielsen方程的三重组合梯度系统Ⅲ表示. 系统的动能、广义力和非完整约束分别为

\begin{matrix} T = \frac{1}{2} ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2}) \\ Q_{1} = - 16 q_{1} \frac{1}{t^{2}} - 8 {\dot{q}}_{1} \frac{1}{t^{2}} - 4 {\dot{q}}_{1} \frac{1}{t} - 6 {\dot{q}}_{1} - 12 q_{1} \frac{1}{t} \\ Q_{2} = - {\dot{q}}_{2} \\ f = {\dot{q}}_{1} + {\dot{q}}_{2} + q_{2} = 0 \end{matrix}

(79)

建立Nielsen方程，并研究零解的稳定性.
非完整Nielsen方程有形式

\frac{\partial {\dot{T}}^{2}}{\partial {\dot{q}}_{s}} - 2 \frac{\partial T}{\partial q_{s}} = Q_{s} + λ_{β} \frac{\partial f_{β}}{\partial {\dot{q}}_{s}}, (s = 1,2)

(80)

计算可得

\dot{T} = {\dot{q}}_{1} {\ddot{q}}_{1} + {\dot{q}}_{2} {\ddot{q}}_{2}

(81)

(82)

由方程可得

\begin{matrix} {\ddot{q}}_{1} = - 16 q_{1} \frac{1}{t^{2}} - 8 {\dot{q}}_{1} \frac{1}{t^{2}} - 4 {\dot{q}}_{1} \frac{1}{t} - \\ 6 {\dot{q}}_{1} - 12 q_{1} \frac{1}{t} + λ \\ {\ddot{q}}_{2} = - {\dot{q}}_{2} + λ \end{matrix}

(83)

解得

λ = 8 q_{1} \frac{1}{t^{2}} + 4 {\dot{q}}_{1} \frac{1}{t^{2}} + 2 {\dot{q}}_{1} \frac{1}{t} + 3 {\dot{q}}_{1} + 6 q_{1} \frac{1}{t}

(84)

代入得

\begin{matrix} {\ddot{q}}_{1} = - 8 q_{1} \frac{1}{t^{2}} - 4 {\dot{q}}_{1} \frac{1}{t^{2}} - 2 {\dot{q}}_{1} \frac{1}{t} - 3 {\dot{q}}_{1} - 6 q_{1} \frac{1}{t} \\ {\ddot{q}}_{2} = 8 q_{1} \frac{1}{t^{2}} + 4 {\dot{q}}_{1} \frac{1}{t^{2}} + 2 {\dot{q}}_{1} \frac{1}{t} + 3 {\dot{q}}_{1} + 6 q_{1} \frac{1}{t} - {\dot{q}}_{2} \end{matrix}

(85)

(86)

则有

\begin{matrix} {\dot{a}}^{1} = - 3 a^{1} + 2 a^{2} \frac{1}{t^{2}} \\ {\dot{a}}^{2} = 2 a^{1} - 4 a^{2} \frac{1}{t^{2}} \end{matrix}

(87)

可表示为

(\begin{matrix} {\dot{a}}^{1} \\ {\dot{a}}^{2} \end{matrix}) = [(\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 2 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 1 \end{matrix})] (\begin{matrix} \frac{\partial V}{\partial a^{1}} \\ \frac{\partial V}{\partial a^{2}} \end{matrix})

(88)

其中

V = \frac{1}{2} {(a^{1})}^{2} + \frac{1}{2} {(a^{2})}^{2} \frac{1}{t^{2}}

(89)

这是三重组合梯度系统Ⅲ的形式. V在 $a^{1} = a^{2} = 0$ 的邻域内正定.因此，零解 $a^{1} = a^{2} = 0$ 是渐进稳定的.
例8 非完整Nielsen方程的三重组合梯度系统Ⅳ表示. 系统的动能、广义力和非完整约束分别为

\begin{matrix} T = \frac{1}{2} ({\dot{q}}_{1}^{2} + {\dot{q}}_{2}^{2}) \\ Q_{1} = - 6 {\dot{q}}_{1} - 4 {\dot{q}}_{1} e^{t} - 22 q_{1} e^{t} \\ Q_{2} = - {\dot{q}}_{2} s i n t - q_{2} c o s t \\ f = {\dot{q}}_{1} + {\dot{q}}_{2} + q_{2} s i n t = 0 \end{matrix}

(90)

建立Nielsen方程，并研究零解的稳定性.
非完整Nielsen方程有形式

\frac{\partial {\dot{T}}_{1}}{\partial {\dot{q}}_{s}} - 2 \frac{\partial T}{\partial q_{s}} = Q_{s} + λ_{β} \frac{\partial f_{β}}{\partial {\dot{q}}_{s}}, (s = 1,2)

(91)

计算可得

\dot{T} = {\dot{q}}_{1} {\ddot{q}}_{1} + {\dot{q}}_{2} {\ddot{q}}_{2}

(92)

(93)

由方程可得

\begin{matrix} {\ddot{q}}_{1} = - 6 {\dot{q}}_{1} - 4 {\dot{q}}_{1} e^{t} - 22 q_{1} e^{t} + λ \\ {\ddot{q}}_{2} = - {\dot{q}}_{2} s i n t - q_{2} c o s t + λ \end{matrix}

(94)

解得

λ = 3 {\dot{q}}_{1} + 2 {\dot{q}}_{1} e^{t} + 11 q_{1} e^{t}

(95)

代入得

\begin{matrix} {\ddot{q}}_{1} = - 3 {\dot{q}}_{1} - 2 {\dot{q}}_{1} e^{t} - 11 q_{1} e^{t} \\ {\ddot{q}}_{2} = 3 {\dot{q}}_{1} + 2 {\dot{q}}_{1} e^{t} + 11 q_{1} e^{t} - {\dot{q}}_{2} s i n t - q_{2} c o s t \end{matrix}

(96)

(97)

则有

\begin{matrix} {\dot{a}}^{1} = - 2 a^{1} e^{t} + a^{2} \\ {\dot{a}}^{2} = - 3 a^{1} e^{t} - 3 a^{2} \end{matrix}

(98)

可表示为

(\begin{matrix} {\dot{a}}^{1} \\ {\dot{a}}^{2} \end{matrix}) = [(\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 2 \end{matrix}) + (\begin{matrix} - 1 & 1 \\ 1 & - 1 \end{matrix})] (\begin{matrix} \frac{\partial V}{\partial a^{1}} \\ \frac{\partial V}{\partial a^{2}} \end{matrix})

(99)

其中

V = \frac{1}{2} {(a^{1})}^{2} e^{t} + \frac{1}{2} {(a^{2})}^{2}

(100)

这是三重组合梯度系统Ⅳ的形式. V在 $a^{1} = a^{2} = 0$ 的邻域内正定.因此，零解 $a^{1} = a^{2} = 0$ 是渐进稳定的.
4 结论
本文通过4类三重组合梯度系统研究Nielsen方程的零解稳定性.给出了4类三重组合梯度系统的形式和性质以及Nielsen方程转化成三重组合梯度系统的条件.利用三重组合梯度系统方法研究系统稳定性时，只需要判定Lyapunov函数是否正定，这为Nielsen方程零解稳定性研究提供了新的梯度方法.该方法也可推广应用于其他约束力学系统.
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- [14] 梅凤翔,吴惠彬.广义Birkhoff系统与一类组合梯度系统 [J].物理学报,2015,64(18)∶370-374.MEI F X,WU H B.Generalized Birkhoff system and a kind of combined gradient system [J].Acta Physica Sinica,2015,64(18)∶370-374.(in Chinese)
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- [17] 陈向炜,梅凤翔.定常Chetaev型非完整系统的组合梯度表示 [J].力学季刊,2016,37(01)∶45-55.CHEN X W,MEI F X.Combined gradient representations of stationary nonholonomic system of Chetaev’s type [J].Chinese Quarterly of Mechanics,2016,37(01)∶45-55.(in Chinese)
- [18] 王嘉航,张毅.非自治广义Birkhoff系统的半负定矩阵梯度系统表示 [J].中山大学学报(自然科学版),2018,57(03)∶60-63.WANG J H,ZHANG Y.A semi-negative definite matrix gradient system representation for non-autonomous generalized Birkhoff system [J].Acta Scientarum Naturalium Universiatis Sunyatseni,2018,57(03)∶60-63.(in Chinese)
- [19] 崔金超,廖翠萃,王勇.Appell方程的广义梯度表示及其稳定性分析 [J].北京理工大学学报,2017,37(02)∶216-220.CUI J C,LIAO C P,WANG C.Generalized gradient representations of Appell equatians and it’s stability analysis [J].Transactions of Beijing Institute of Technology,2017,37(02)∶216-220.(in Chinese)
- [20] 李彦敏,梅凤翔.Tzénoff方程解的稳定性 [J].云南大学学报(自然科学版),2018,40(05)∶897-902.LI Y M,MEI F X.On the stability of solution of Tzénoff equations [J].Journal of Yunnan University,2018,40(05)∶897-902.(in Chinese)
- [21] 王嘉航,张毅.自治广义Birkhoff系统的三重组合梯度系统表示 [J].云南大学学报(自然科学版),2019,41(03)∶497-502.WANG J H,ZHANG.Triple combined gradient system representations for autonomousgeneralized Birkhoffian system [J].Journal of Yunnan University(Natural Science Edition),2019,41(03)∶497-502.(in Chinese)
- [22] 章婷婷,张毅,张成璞,等.判定定常Chetaev型非完整系统稳定性的三重组合梯度方法 [J].动力学与控制学报,2019,17(04)∶306-312.ZHANG T T,ZHANG Y,ZHANG C P,et al.Triple combined gradient method for judging the the stability of nonholonomic systems of Chetaev’s type [J].Journal of Dynamics and Control,2019,17(04)∶306-312.(in Chinese)
- [23] 董孟峰,陈向炜.判定广义Birkhoff系统稳定性的三重组合梯度方法 [J].力学季刊,2019,40(03)∶543-548.DONG M F,CHEN X W.Triple combined gradient method for determining the stability of generalized Birkhoff systems [J].Chinese Quarterly of Mechanics,2019,40(03)∶543-548.(in Chinese)
- [24] 梅凤翔,吴惠彬,李彦敏.Nielsen方程的两类广义梯度表示 [J].北京大学学报(自然科学版),2016,52(04)∶588-591.MEI F X,WU H B,LI Y M.Two kinds of gradient representations for Nielsen equations [J].Acta Scientiarum Naturalium Universitatis Pekinensis,2016,52(04)∶588-591.(in Chinese)

基本信息

中图分类号: O316
文献标识码: A
DOI: 10.6052/1672-6553-2022-025
文章编号: 1672-6553-2023-21(2)-024-009

基金信息

国家自然科学基金资助项目(11372169)；
National Natural Science Foundation of China(11372169)；

引用信息

稿件历史

收稿日期: 2022-03-14

参考文献

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[12] 梅凤翔.关于梯度系统—分析力学札记之十八 [J].力学与实践,2012,34(01)∶89-90.MEI F X.On gradient systems-notes on analytical mechanics ⅩⅧ [J].Mechanics in Engineering,2012,34(01)∶89-90.(in Chinese)
[13] 梅凤翔,吴惠彬.广义Birkhoff系统的梯度表示 [J].动力学与控制学报,2012,10(04)∶289-292.MEI F X,WU H B.A gradient representation for generalized Birkhoff system [J].Journal of Dynamics and Control,2012,10(04)∶289-292.(in Chinese)
[14] 梅凤翔,吴惠彬.广义Birkhoff系统与一类组合梯度系统 [J].物理学报,2015,64(18)∶370-374.MEI F X,WU H B.Generalized Birkhoff system and a kind of combined gradient system [J].Acta Physica Sinica,2015,64(18)∶370-374.(in Chinese)
[15] 梅凤翔,吴惠彬.Birkhoff系统的广义斜梯度表示 [J].动力学与控制学报,2015,13(05)∶329-331.MEI F X,WU H B.Generalized skew-gradient representation for Birkhoff system [J].Journal of Dynamics and Control,2015,13(05)∶329-331.(in Chinese)
[16] 陈向炜,曹秋鹏,梅凤翔.切塔耶夫型非完整系统的广义梯度表示 [J].力学学报,2016,48(03)∶684-691.CHEN X W,CAO Q P,MEI F X.Generalized gradient representation of nonholonomic system of Chetaev’s type [J].Acta Mechanica Sinica 2016,48(03)∶ 684-691.(in Chinese)
[17] 陈向炜,梅凤翔.定常Chetaev型非完整系统的组合梯度表示 [J].力学季刊,2016,37(01)∶45-55.CHEN X W,MEI F X.Combined gradient representations of stationary nonholonomic system of Chetaev’s type [J].Chinese Quarterly of Mechanics,2016,37(01)∶45-55.(in Chinese)
[18] 王嘉航,张毅.非自治广义Birkhoff系统的半负定矩阵梯度系统表示 [J].中山大学学报(自然科学版),2018,57(03)∶60-63.WANG J H,ZHANG Y.A semi-negative definite matrix gradient system representation for non-autonomous generalized Birkhoff system [J].Acta Scientarum Naturalium Universiatis Sunyatseni,2018,57(03)∶60-63.(in Chinese)
[19] 崔金超,廖翠萃,王勇.Appell方程的广义梯度表示及其稳定性分析 [J].北京理工大学学报,2017,37(02)∶216-220.CUI J C,LIAO C P,WANG C.Generalized gradient representations of Appell equatians and it’s stability analysis [J].Transactions of Beijing Institute of Technology,2017,37(02)∶216-220.(in Chinese)
[20] 李彦敏,梅凤翔.Tzénoff方程解的稳定性 [J].云南大学学报(自然科学版),2018,40(05)∶897-902.LI Y M,MEI F X.On the stability of solution of Tzénoff equations [J].Journal of Yunnan University,2018,40(05)∶897-902.(in Chinese)
[21] 王嘉航,张毅.自治广义Birkhoff系统的三重组合梯度系统表示 [J].云南大学学报(自然科学版),2019,41(03)∶497-502.WANG J H,ZHANG.Triple combined gradient system representations for autonomousgeneralized Birkhoffian system [J].Journal of Yunnan University(Natural Science Edition),2019,41(03)∶497-502.(in Chinese)
[22] 章婷婷,张毅,张成璞,等.判定定常Chetaev型非完整系统稳定性的三重组合梯度方法 [J].动力学与控制学报,2019,17(04)∶306-312.ZHANG T T,ZHANG Y,ZHANG C P,et al.Triple combined gradient method for judging the the stability of nonholonomic systems of Chetaev’s type [J].Journal of Dynamics and Control,2019,17(04)∶306-312.(in Chinese)
[23] 董孟峰,陈向炜.判定广义Birkhoff系统稳定性的三重组合梯度方法 [J].力学季刊,2019,40(03)∶543-548.DONG M F,CHEN X W.Triple combined gradient method for determining the stability of generalized Birkhoff systems [J].Chinese Quarterly of Mechanics,2019,40(03)∶543-548.(in Chinese)
[24] 梅凤翔,吴惠彬,李彦敏.Nielsen方程的两类广义梯度表示 [J].北京大学学报(自然科学版),2016,52(04)∶588-591.MEI F X,WU H B,LI Y M.Two kinds of gradient representations for Nielsen equations [J].Acta Scientiarum Naturalium Universitatis Pekinensis,2016,52(04)∶588-591.(in Chinese)

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Nielsen方程的三重组合梯度表示及其稳定性分析*

The Triple Combination Gradient Representation Of Nielsen Equation and Its Stability Analysis*

摘要

Abstract

关键词

Keywords

1 三重组合梯度系统及其性质

1.1 三重组合梯度系统Ⅰ

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1.2 三重组合梯度系统Ⅱ

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1.3 三重组合梯度系统Ⅲ

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2 Nielsen方程的三重组合梯度表示

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3 算例

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Nielsen方程的三重组合梯度表示及其稳定性分析^*

The Triple Combination Gradient Representation Of Nielsen Equation and Its Stability Analysis^*