﻿ 切塔耶夫型非完整系统的广义梯度表示
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 力学学报  2016, Vol. 48 Issue (3): 684-691  DOI: 10.6052/0459-1879-15-268 0

### 引用本文 [复制中英文]

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Chen Xiangwei, Cao Qiupeng, Mei Fengxiang. GENERALIZED GRADIENT REPRESENTATION OF NONHOLONOMIC SYSTEM OF CHETAEV'S TYPE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(3): 684-691. DOI: 10.6052/0459-1879-15-268.
[复制英文]

### 文章历史

2015-07-20 收稿
2016-01-04 录用
2016-01-29 网络版发表.

1. 商丘师范学院物理与电气信息学院, 商丘 476000;
2. 苏州科技学院数理学院, 苏州 215009;
3. 北京理工大学宇航学院, 北京 100081

0 引言

1 广义梯度系统 1.1 广义梯度系统Ⅰ

 ${{\dot{x}}_{i}}=-\frac{\partial V}{\partial {{x}_{i}}},i=1,2,\cdots ,m$ (1)

 $\dot{V}=\frac{\partial V}{\partial t}-\frac{\partial V}{\partial {{x}_{i}}}\frac{\partial V}{\partial {{x}_{i}}}$ (2)

1.2 广义梯度系统Ⅱ

 ${{\dot{x}}_{i}}={{b}_{ij}}\frac{\partial V}{\partial {{x}_{j}}},i,j=1,2,\cdots ,m$ (3)

 $\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial {{x}_{i}}}{{b}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}=\frac{\partial V}{\partial t}$ (4)

1.3 广义梯度系统Ⅲ

 ${{\dot{x}}_{i}}={{S}_{ij}}\frac{\partial V}{\partial {{x}_{j}}},i,j=1,2,\cdots ,m$ (5)

 $\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial {{x}_{i}}}{{S}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}$ (6)

1.4 广义梯度系统Ⅳ

 ${{\dot{x}}_{i}}={{a}_{ij}}\frac{\partial V}{\partial {{x}_{j}}},i,j=1,2,\cdots ,m$ (7)

 $\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial {{x}_{i}}}{{a}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}$ (8)

1.5 广义梯度系统 Ⅴ

 ${{\dot{x}}_{i}}=-\frac{\partial V}{\partial {{x}_{i}}}+{{b}_{ij}}\frac{\partial V}{\partial {{x}_{j}}},\quad i,j=1,2,\cdots ,m$ (9)

 $\dot{V}=\frac{\partial V}{\partial t}-\frac{\partial V}{\partial {{x}_{i}}}\frac{\partial V}{\partial {{x}_{i}}}$ (10)

1.6 广义梯度系统Ⅵ

 ${{\dot{x}}_{i}}=-\frac{\partial V}{\partial {{x}_{i}}}+{{S}_{ij}}\frac{\partial V}{\partial {{x}_{j}}},\quad i,j=1,2,\cdots ,m$ (11)

 $\dot{V}=\frac{\partial V}{\partial t}-\frac{\partial V}{\partial {{x}_{i}}}\frac{\partial V}{\partial {{x}_{i}}}+\frac{\partial V}{\partial {{x}_{i}}}{{S}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}$ (12)

1.7 广义梯度系统Ⅶ

 ${{\dot{x}}_{i}}=-\frac{\partial V}{\partial {{x}_{i}}}+{{a}_{ij}}\frac{\partial V}{\partial {{x}_{j}}},\quad i,j=1,2,\cdots ,m$ (13)

 $\dot{V}=\frac{\partial V}{\partial t}-\frac{\partial V}{\partial {{x}_{i}}}\frac{\partial V}{\partial {{x}_{i}}}+\frac{\partial V}{\partial {{x}_{i}}}{{a}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}$ (14)

1.8 广义梯度系统 Ⅷ

 $\dot{x}={{b}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}+{{S}_{ij}}\frac{\partial V}{\partial {{x}_{j}}},i,j=1,2,\cdots ,m$ (15)

 $\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial {{x}_{i}}}{{S}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}$ (16)

1.9 广义梯度系统Ⅸ

 ${{\dot{x}}_{i}}={{b}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}+{{a}_{ij}}\frac{\partial V}{\partial {{x}_{j}}},i,j=1,2,\cdots ,m$ (17)

 $\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial {{x}_{i}}}{{a}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}$ (18)

1.10 广义梯度系统Ⅹ

 ${{\dot{x}}_{i}}={{a}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}+{{S}_{ij}}\frac{\partial V}{\partial {{x}_{j}}},i,j=1,2,\cdots ,m$ (19)

 $\dot{V}=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial {{x}_{i}}}{{a}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}+\frac{\partial V}{\partial {{x}_{i}}}{{S}_{ij}}\frac{\partial V}{\partial {{x}_{j}}}$ (20)

2 切塔耶夫型非完整系统的广义梯度表示

 ${{f}_{\beta }}\left( t,q,\dot{q} \right)=0,\quad \beta =1,2,\cdots ,g$ (21)

 $\left. \begin{matrix} \frac{d}{dt}\frac{\partial L}{\partial {{{\dot{q}}}_{s}}}-\frac{\partial L}{\partial {{q}_{s}}}={{Q}_{s}}+{{\lambda }_{\beta }}\frac{\partial {{f}_{\beta }}}{\partial {{{\dot{q}}}_{s}}} \\ s=1,2,\cdots ,n;\beta =1,2,\cdots ,g \\ \end{matrix} \right\}$ (22)

 $\det \left( \frac{{{\partial }^{2}}L}{\partial {{{\dot{q}}}_{s}}\partial {{{\dot{q}}}_{k}}} \right)\ne 0$ (23)

 \begin{align} & \\ & \left. \begin{matrix} \frac{d}{dt}\frac{\partial L}{\partial {{{\dot{q}}}_{s}}}-\frac{\partial L}{\partial {{q}_{s}}}={{Q}_{s}}+{{\Lambda }_{s}} \\ s=1,2,\cdots ,n \\ \end{matrix} \right\} \\ \end{align} (24)

 ${{\Lambda }_{s}}=\text{ }{{\Lambda }_{s}}(t,q,\dot{q})={{\lambda }_{\beta }}\left( t,q,\dot{q} \right)\frac{\partial {{f}_{\beta }}}{\partial {{{\dot{q}}}_{s}}}$ (25)

 ${{\ddot{q}}_{s}}={{\alpha }_{s}}\left( t,q,\dot{q} \right),s=1,2,\cdots ,n$ (26)
 ${{a}^{s}}={{q}_{s}},{{a}^{n+s}}={{\dot{q}}_{s}}$ (27)

 ${{\dot{a}}^{\mu }}={{F}_{\mu }}\left( t,a \right),\mu =1,2,\cdots ,2n$ (28)

 ${{F}_{s}}={{a}^{n+s}},\quad {{F}_{n+s}}={{\alpha }_{s}}\left( t,a \right)$ (29)

 ${{p}_{s}}=\frac{\partial L}{\partial {{{\dot{q}}}_{s}}},\quad H={{p}_{s}}{{\dot{q}}_{s}}-L$ (30)

 ${{\dot{a}}^{\mu }}={{\omega }^{\mu v}}\frac{\partial H}{\partial {{a}^{\nu }}}+{{P}_{\mu }},\mu ,v=1,2,\cdots ,2n$ (31)

 \left. \begin{align} & {{a}^{s}}={{q}_{s}} \\ & {{a}^{n+s}}={{p}_{s}} \\ & H=H\left( t,a \right) \\ & {{\omega }^{\mu \nu }}=\left[ \begin{matrix} {{\mathbf{0}}_{n\times n}} & {{\mathbf{1}}_{n\times n}} \\ -{{\mathbf{1}}_{n\times n}} & {{\mathbf{0}}_{n\times n}} \\ \end{matrix} \right] \\ & {{P}_{s}}=0{{P}_{n+s}}={{{\tilde{Q}}}_{s}}+{{{\tilde{\Lambda }}}_{s}} \\ \end{align} \right\} (32)

 ${{F}_{\mu }}=-\frac{\partial V}{\partial {{a}^{\mu }}}$ (33)
 ${{F}_{\mu }}={{b}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (34)
 ${{F}_{\mu }}={{S}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (35)
 ${{F}_{\mu }}={{a}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (36)
 ${{F}_{\mu }}=-\frac{\partial V}{\partial {{a}^{\mu }}}+{{b}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (37)
 ${{F}_{\mu }}=-\frac{\partial V}{\partial {{a}^{\mu }}}+{{S}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (38)
 ${{F}_{\mu }}=-\frac{\partial V}{\partial {{a}^{\mu }}}+{{a}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (39)
 ${{F}_{\mu }}={{b}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}+{{S}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (40)
 ${{F}_{\mu }}={{b}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}+{{a}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (41)
 ${{F}_{\mu }}={{a}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}+{{S}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (42)

 ${{\omega }^{\mu \nu }}\frac{\partial H}{\partial {{a}^{v}}}+{{P}_{\mu }}=-\frac{\partial V}{\partial {{a}^{\mu }}}$ (43)
 ${{\omega }^{\mu \nu }}\frac{\partial H}{\partial {{a}^{v}}}+{{P}_{\mu }}={{b}_{\mu \nu }}\frac{\partial V}{\partial {{a}^{\nu }}}$ (44)
 ${{\omega }^{\mu \nu }}\frac{\partial H}{\partial {{a}^{v}}}+{{P}_{\mu }}={{S}_{\mu \nu }}\frac{\partial V}{\partial {{a}^{\nu }}}$ (45)
 ${{\omega }^{\mu \nu }}\frac{\partial H}{\partial {{a}^{v}}}+{{P}_{\mu }}={{a}_{\mu \nu }}\frac{\partial V}{\partial {{a}^{\nu }}}$ (46)
 ${{\omega }^{\mu \nu }}\frac{\partial H}{\partial {{a}^{v}}}+{{P}_{\mu }}=-\frac{\partial V}{\partial {{a}^{\mu }}}+{{b}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (47)
 ${{\omega }^{\mu \nu }}\frac{\partial H}{\partial {{a}^{v}}}+{{P}_{\mu }}=-\frac{\partial V}{\partial {{a}^{\mu }}}+{{S}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (48)
 ${{\omega }^{\mu \nu }}\frac{\partial H}{\partial {{a}^{v}}}+{{P}_{\mu }}=-\frac{\partial V}{\partial {{a}^{\mu }}}+{{a}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (49)
 ${{\omega }^{\mu \nu }}\frac{\partial H}{\partial {{a}^{v}}}+{{P}_{\mu }}={{b}_{\mu \nu }}\frac{\partial V}{\partial {{a}^{\nu }}}+{{S}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (50)
 ${{\omega }^{\mu \nu }}\frac{\partial H}{\partial {{a}^{v}}}+{{P}_{\mu }}={{b}_{\mu \nu }}\frac{\partial V}{\partial {{a}^{\nu }}}+{{a}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (51)
 ${{\omega }^{\mu \nu }}\frac{\partial H}{\partial {{a}^{v}}}+{{P}_{\mu }}={{a}_{\mu \nu }}\frac{\partial V}{\partial {{a}^{\nu }}}+{{S}_{\mu v}}\frac{\partial V}{\partial {{a}^{v}}}$ (52)

3 应用举例

 \left. \begin{align} & L=\frac{1}{2}(\dot{q}_{1}^{2}+\dot{q}_{2}^{2}) \\ & {{Q}_{1}}=-5[{{q}_{1}}{{\text{e}}^{t}}+{{{\dot{q}}}_{1}}(2+{{\text{e}}^{t}})] \\ & {{Q}_{2}}=-{{{\dot{q}}}_{2}} \\ & f=2{{{\dot{q}}}_{1}}+{{{\dot{q}}}_{2}}+{{q}_{2}}=0 \\ \end{align} \right\} (53)

 \begin{align} & L=\frac{1}{2}\left( \dot{q}_{1}^{2}+\dot{q}_{2}^{2} \right) \\ & {{Q}_{1}}=-8{{q}_{1}}\left[ 1+{{\text{e}}^{-t}} \right]-2{{{\dot{q}}}_{1}}{{\text{e}}^{-t}} \\ & {{Q}_{2}}=-{{{\dot{q}}}_{2}} \\ & f={{{\dot{q}}}_{1}}+{{{\dot{q}}}_{2}}+{{q}_{2}}=0 \\ \end{align} (54)

 $\left. \begin{array}{*{35}{l}} L=\frac{1}{2}(\dot{q}_{1}^{2}+\dot{q}_{2}^{2}) \\ {{Q}_{1}}=-4{{{\dot{q}}}_{1}}\left( 7+2\sin t \right)- \\ 8{{q}_{1}}\left( 10+5\sin t+\cos t \right) \\ {{Q}_{2}}=-{{q}_{1}}{{q}_{2}}-t{{{\dot{q}}}_{1}}{{q}_{2}}-t{{q}_{1}}{{{\dot{q}}}_{2}} \\ f={{{\dot{q}}}_{1}}+{{{\dot{q}}}_{2}}+t{{q}_{1}}{{q}_{2}}=0 \\ \end{array} \right\}$ (55)

 $\left. \begin{array}{*{35}{l}} L=\frac{1}{2}\left( \dot{q}_{1}^{2}+\dot{q}_{2}^{2}+\dot{q}_{3}^{2} \right) \\ {{Q}_{1}}=-{{{\dot{q}}}_{1}} \\ {{Q}_{2}}=-{{{\dot{q}}}_{2}} \\ {{Q}_{3}}=-\frac{4{{q}_{3}}}{2+\cos t}\left( 3-\sin t \right)- \\ {{{\dot{q}}}_{3}}\left( \frac{12+4\cos t-\sin t}{2+\cos t} \right) \\ f={{{\dot{q}}}_{1}}+{{{\dot{q}}}_{2}}{{q}_{3}}=0 \\ \end{array} \right\}$ (56)

 $\left. \begin{array}{*{35}{l}} L=\frac{1}{2}\left( \dot{q}_{1}^{2}+\dot{q}_{2}^{2} \right) \\ f={{{\dot{q}}}_{1}}+{{{\dot{q}}}_{2}}+{{q}_{1}}=0 \\ {{Q}_{1}}=-4{{{\dot{q}}}_{1}}\left( 7-2\sin t \right)- \\ 8{{q}_{1}}\left( 10-5\sin t-\cos t \right) \\ {{Q}_{2}}=-{{{\dot{q}}}_{1}} \\ \end{array} \right\}$ (57)

4 结论

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GENERALIZED GRADIENT REPRESENTATION OF NONHOLONOMIC SYSTEM OF CHETAEV'S TYPE
Chen Xiangwei, Cao Qiupeng, Mei Fengxiang
1. Department of Physics and Information Engineering, Shangqiu Normal University, Shangqiu 476000, China;
2. School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China;
3. School of Aerospace, Beijing Institute of Technology, Beijing 100081, China
Abstract: It is an important and di cult problem to study the stability of the non-steady and nonholonomic mechanical systems, and it is di cult to construct the Lyapunov function directly from the di erential equation. This paper gives an indirect method. The ten kinds of generalized gradient systems are proposed and the di erential equations of the ten kinds of generalized gradient systems are given respectively. Furthermore, the generalized gradient representations of a nonholonomic system of Chetaev's type are studied. The condition under which a nonholonomic system can be considered as a generalized gradient system is obtained, so the nonholonomic system of Chetaev's type is transformed into each generalized gradient systems. The characteristic of the generalized gradient systems can be used to study the stability of the nonholonomic system. This method appears to be more e ective when it is di cult to construct the Lyapunov function directly. Some examples are given to illustrate the application of the result.
Key words: nonholonomic system    generalized gradient system    stability