张毅^,2)苏州科技大学 土木工程学院,江苏苏州 215011

NOETHER QUASI-SYMMETRY AND APPROXIMATE NOETHER CONSERVATION LAWS FOR WEAKLY NONLINEAR DYNAMICAL EQUATIONS ¹⁾

Zhang Yi^,2)College of Civil Engineering,Suzhou University of Science and Technology,Suzhou 215011,Jiangsu,China

通讯作者: 2) 张毅, 教授, 主要研究方向: 分析力学. E-mail:zhy@mail.usts.edu.cn

收稿日期:2020-07-6接受日期:2020-09-1网络出版日期:2020-11-18

基金资助:

1) 国家自然科学基金.11972241 国家自然科学基金.11572212 江苏省自然科学基金.BK20191454

Received:2020-07-6Accepted:2020-09-1Online:2020-11-18
作者简介 About authors


摘要
自然界和工程技术领域存在大量的非线性问题,它们通常需要用非线性微分方程来描述. 守恒量在微分方程的求解、约化和定性分析方面发挥重要作用. 因此,研究非线性动力学方程的近似守恒量具有重要意义. 文章利用 Noether 对称性方法研究弱非线性动力学方程的近似守恒量. 首先,将弱非线性动力学方程化为一般完整系统的 Lagrange 方程,在 Lagrange 框架下建立 Noether 准对称性的定义和广义 Noether 等式,给出近似 Noether 守恒量. 其次,将弱非线性动力学方程化为相空间中一般完整系统的 Hamilton 方程,在 Hamilton 框架下建立 Noether 准对称性的定义和广义 Noether 等式,给出近似 Noether 守恒量. 再次,将弱非线性动力学方程化为广义 Birkhoff 方程,在 Birkhoff 框架下建立 Noether 准对称性的定义和广义 Noether 等式,给出近似 Noether 守恒量. 最后,以著名的 van der Pol 方程,Duffing 方程以及弱非线性耦合振子为例,分析三个不同框架下弱非线性系统的 Noether 准对称性与近似 Noether 守恒量的计算. 结果表明：同一弱非线性动力学方程可以化为不同的一般完整系统或不同的广义 Birkhoff 系统;Hamilton 框架下的结果是 Birkhoff 框架的特例,而 Lagrange 框架下的结果与 Hamilton 框架的等价. 利用 Noether 对称性方法寻找弱非线性动力学方程的近似守恒量不仅方便有效,而且具有较大的灵活性.
关键词： 弱非线性动力学;近似 Noether 守恒量;Noether 准对称性;广义 Noether 等式

Abstract
There are a lot of nonlinear problems in nature and engineering technology, which need to be described by nonlinear differential equations. Conservation laws play an important role in solving, reducing and qualitative analysis of differential equations. Therefore, it is of great significance to study the approximate conservation laws of nonlinear dynamical equations. In this paper, we apply the Noether symmetry method to the study of approximate conservation laws of weakly nonlinear dynamical equations. Firstly, the weakly nonlinear dynamical equations are transformed into the Lagrange equations of general holonomic system. Under the Lagrangian framework, the definition of Noether quasi-symmetry and the generalized Noether identities are established, and the approximate Noether conservation laws are obtained. Secondly, the weakly nonlinear dynamical equations are transformed into the Hamilton equations of general holonomic system in phase space. Under the Hamiltonian framework, the definition of Noether quasi-symmetry and the generalized Noether identities are established, and the approximate Noether conservation laws are obtained. Thirdly, the weakly nonlinear dynamical equations are transformed into the generalized Birkhoff's equations. Under the Birkhoffian framework, the definition of Noether quasi-symmetry and the generalized Noether identities are established, and the approximate Noether conservation laws are obtained. Finally, taking the famous Van der Pol equation, the Duffing equation and the weakly nonlinear coupled oscillators as examples, the computation of Noether quasi-symmetries and approximate conservation laws for weakly nonlinear systems under three different frameworks is analyzed. The results show that the same weakly nonlinear dynamical equation can be reduced to different general holonomic systems or different generalized Birkhoff systems. The result under the Hamiltonian framework is a special case of the Birkhoffian framework, while the result under the Lagrangian framework is equivalent to that under the Hamiltonian framework. Using Noether symmetry method to find approximate conservation laws of weakly nonlinear dynamical equations is not only convenient and effective, but also has great flexibility.
Keywords：weakly nonlinear dynamics;approximate Noether conservation laws;Noether quasi-symmetry;generalized Noether identity


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本文引用格式
张毅. 弱非线性动力学方程的 Noether 准对称性与近似 Noether 守恒量¹⁾. 力学学报[J], 2020, 52(6): 1765-1773 DOI:10.6052/0459-1879-20-242
Zhang Yi. NOETHER QUASI-SYMMETRY AND APPROXIMATE NOETHER CONSERVATION LAWS FOR WEAKLY NONLINEAR DYNAMICAL EQUATIONS ¹⁾. Chinese Journal of Theoretical and Applied Mechanics[J], 2020, 52(6): 1765-1773 DOI:10.6052/0459-1879-20-242

引言

在自然界和工程技术领域存在大量的非线性问题,它们通常需要用非线性微分方程来描述. 守恒量或第一积分在微分方程求解、约化以及定性分析方面发挥重要作用^[1-3]. 利用对称性寻找守恒量是一个有效方法,如 Lie 理论^[4-7]、Noether 定理^[8-13]和 Mei 对称性^[14-17]. Lie 对称性是微分方程的不变性,因而在以微分方程表示的数学模型中 Lie 对称性方法得到普遍应用^[18-22].Noether 对称性依赖于作用量泛函,由于非线性微分方程一般不具有 Lagrange 结构,因此通过 Noether 对称性寻找微分方程的守恒量遇到了很大的困难.1998 年,Govinder 及其合作者基于 Lie 点变换提出了近似 Noether 对称性^[23]. 近年来,近似对称性方法和近似守恒量研究取得不少成果^[24-31].本文研究弱非线性动力学方程的 Noether 准对称性,将 Noether 对称性方法应用于具有小参数的非线性微分方程系统,分别基于 Lagrange 框架,Hamilton 框架和 Birkhoff 框架,证明了近似 Noether 守恒量定理.文末以著名的 van der Pol 方程,Duffing 方程,以及两自由度的弱非线性耦合振子为例,说明结果的应用.

1 Lagrange 框架下的近似 Noether 守恒量

设在 Lagrange 框架下,弱非线性动力学方程可化为一般完整系统的 Lagrange 方程,有

(1)$\dfrac{\text{d}}{\text{d} t}\dfrac{\partial L}{\partial \dot {q}_s } - \dfrac{\partial L}{\partial q_s } = Q_s \left( {t,{\pmb q},\dot{\pmb q},\varepsilon } \right) \left( {s = 1,2, \cdots ,n} \right)$
其中,$L = L\left( {t, {\pmb q}, \dot{\pmb q} } \right)$ 为 Lagrange 函数,$Q_s = Q_s \left( {t,{\pmb q}, \dot{\pmb q}, \varepsilon } \right)$ 为非势广义力,$\varepsilon $ 为小参数 $(\varepsilon \ll1)$. 取无限小变换

(2)$ \left. \bar {t} = t + \upsilon \tau \left( {t,{\pmb q},\dot{\pmb q}} \right) \\ \bar {q}_s \left( \bar {t} \right) = q_s \left( t \right) + \upsilon \xi _s \left( {t,{\pmb q},\dot{\pmb q}} \right) \ \ \left( {s = 1,2, \cdots ,n} \right) \right\}$
这里 $\tau $ 和 $\xi _s $ 是生成函数,$\upsilon $ 是无限小参数. 如果成立

(3)$ \Delta \int_{t_1 }^{t_2 } {L\left( {t,{\pmb q},\dot{\pmb q}} \right)} \text{d} t = - \int_{t_1 }^{t_2 } {\left[ {\dfrac{\text{d}}{\text{d} t}\left( {\Delta G} \right) + Q_s \delta q_s } \right]} \text{d} t$
其中,$\Delta G = \upsilon G$,而 $G = G\left( {t,{\pmb q}, \dot{\pmb q}}\right)$ 称为规范函数,则这种不变性称为系统 (1) 的 Noether 准对称性. 由式 (3) 可导出广义 Noether 等式

(4)$ L\dot {\tau } + X^{(1)}\left( L \right) + Q_s \left( {\xi _s - \dot {q}_s \tau } \right) + \dot {G} = 0$
其中

(5)$ X^{(1)} = \tau \dfrac{\partial }{\partial t} + \xi _s \dfrac{\partial }{\partial q_s } + \left( {\dot {\xi }_s - \dot {q}_s \dot {\tau }} \right)\dfrac{\partial }{\partial \dot {q}_s }$
不失一般性,设广义力为

(6)$ Q_s \left( {t,{\pmb q},\dot{\pmb q},\varepsilon } \right) = Q_s^0 \left( {t,{\pmb q},\dot{\pmb q}} \right) + \varepsilon Q_s^1 \left( {t,{\pmb q},\dot{\pmb q}} \right)$
相应地,设生成函数 $\tau $,$\xi _s $,以及规范函数 $G$ 为

(7)$ \left. \tau = \tau ^0 + \varepsilon \tau ^1 + o\left( \varepsilon \right) \\ \xi _s = \xi _s^0 + \varepsilon \xi _s^1 + o\left( \varepsilon \right) \\ G = G^0 + \varepsilon G^1 + o\left( \varepsilon \right) \right\}$
则广义 Noether 等式 (4) 成为

(8)$L\dot {\tau }^0 + X_0^{(1)} \left( L \right) + Q_s^0 \left( {\xi _s^0 - \dot {q}_s \tau ^0} \right) + \dot {G}^0 = 0$
(9)$L\dot {\tau }^1 + X_1^{(1)} \left( L \right) + Q_s^0 \left( {\xi _s^1 - \dot {q}_s \tau ^1} \right) + Q_s^1 \left( {\xi _s^0 - \dot {q}_s \tau ^0} \right) + \dot {G}^1 = 0$
其中

(10)$ X_k^{(1)} = \tau ^k\dfrac{\partial }{\partial t} + \xi _s^k \dfrac{\partial }{\partial q_s } + \left( {\dot {\xi }_s^k - \dot {q}_s \dot {\tau }^k} \right)\dfrac{\partial }{\partial \dot {q}_s } \left( {k = 0,1} \right)$
定义 1 对于弱非线性动力学系统 (1),如果沿着方程 (1) 的所有解曲线,有

(11)$\dfrac{\text{d}}{\text{d} t}I = O\left( {\varepsilon ^2} \right)$
其中 $I = I_0 + \varepsilon I_1 $,则称 $I$ 为系统 (1) 的近似守恒量. 于是有

定理 1 对于弱非线性动力学系统 (1),如果广义Noether 等式 (8) 和 (9) 有解,则系统存在近似 Noether 守恒量

(12)$I_{N} = \sum_{k = 0}^1 \varepsilon ^k\left[ {L\tau ^k + \dfrac{\partial L}{\partial \dot {q}_s }\left( {\xi _s^k - \dot {q}_s \tau ^k} \right) + G^k} \right]$
证明 由于

(13)$\begin{array}{l} \dfrac{\text{d}}{\text{d} t}I_{N} = \sum_{k = 0}^1 {\varepsilon ^k\left[ {\left( {\dfrac{\partial L}{\partial t} + \dfrac{\partial L}{\partial q_s }\dot {q}_s + \dfrac{\partial L}{\partial \dot {q}_s }\ddot {q}_s } \right)\tau ^k} \right.} +\\ L\dot {\tau }^k + \dfrac{\text{d}}{\text{d} t}\dfrac{\partial L}{\partial \dot {q}_s }\left( {\xi _s^k - \dot {q}_s \tau ^k} \right) +\\ \left. { \dfrac{\partial L}{\partial \dot {q}_s }\left( {\ddot {\xi }_s^k - \ddot {q}_s \tau ^k - \dot {q}_s \dot {\tau }^k} \right) + \dot {G}^k} \right] \end{array}$
将式 (8) 和式 (9) 代入式 (13),并利用方程 (1),可得

(14)$ \dfrac{\text{d}}{\text{d} t}I_{N} = \varepsilon ^2Q_s^1 \left( {\xi _s^1 - \dot {q}_s \tau ^1} \right)$
因此,式 (12) 是系统 (1) 的近似 Noether 守恒量. 证毕.

2 Hamilton 框架下的近似 Noether 守恒量

设在 Hamilton 框架下,弱非线性动力学方程可化为相空间中一般完整系统的 Hamilton 方程,有

(15)$ \dot {q}_s = \dfrac{\partial H}{\partial p_s } , \ \ \dot {p}_s = - \dfrac{\partial H}{\partial q_s } + \tilde {Q}_s \left( {t,{\pmb q},{\pmb p},\varepsilon } \right) \\ \left( {s = 1,2, \cdots ,n} \right)$
其中 $p_s = \dfrac{\partial L}{\partial \dot {q}_s }$ 为广义动量,$H\left( t,{\pmb q},{\pmb p} \right)$ 为 Hamilton 函数,$\tilde {Q}_s =\tilde {Q}_s \left( {t,{\pmb q},{\pmb p},\varepsilon } \right)$ 为非势广义力,$\varepsilon$ 为小参数 ($\varepsilon \ll 1)$. 在相空间中,取无限小变换

(16)$\left.\begin{array}{l} \bar {t} = t + \upsilon \tau \left( {t,{\pmb q},{\pmb p}} \right) \\ \bar {q}_s \left( \bar {t} \right) = q_s \left( t \right) + \upsilon \xi _s \left( {t,{\pmb q},{\pmb p}} \right) \\ \bar {p}_s \left( \bar {t} \right) = p_s \left( t \right) + \upsilon \eta _s \left( {t,{\pmb q},{\pmb p}} \right)\ \ \left( {s = 1,2, \cdots ,n} \right)\end{array} \right\} $
这里 $\tau $,$\xi _s $ 和 $\eta _s $ 是生成函数,$\upsilon$ 是无限小参数. 如果成立

(17)$ \Delta \int_{t_1 }^{t_2 } {\left[ {p_s \dot {q}_s - H\left( {t,{\pmb q},{\pmb p}} \right)} \right]} \text{d} t =\\ - \int_{t_1 }^{t_2 } \left( {\dfrac{\text{d}}{\text{d} t}\left( {\Delta G} \right) + \tilde {Q}_s \delta q_s } \right) \text{d} t$
其中 $\Delta G = \upsilon G$,$G = G\left( {t,{\pmb q},{\pmb p}} \right)$ 为规范函数,则这种不变性称为系统 (15) 的 Noether 准对称性. 由式 (17) 可导出广义 Noether 等式为

(18)$ p_s \dot {\xi }_s + \dot {q}_s \eta _s - X^{\left( 0 \right)}\left( H \right) - H\dot {\tau } + Q_s \left( {\xi _s - \dot {q}_s \tau } \right) + \dot {G} = 0$
其中

(19)$ X^{ ( 0 )} = \tau \dfrac{\partial }{\partial t} + \xi _s \dfrac{\partial }{\partial q_s } + \eta _s \dfrac{\partial }{\partial p_s }$
设广义力为

(20)$\tilde {Q}_s \left( {t,{\pmb q},{\pmb p},\varepsilon } \right) = \tilde {Q}_s^0 \left( {t,{\pmb q},{\pmb p}} \right) + \varepsilon \tilde {Q}_s^1 \left( {t,{\pmb q},{\pmb p}} \right)$
相应地,生成函数为

(21)$\left.\begin{array}{l} \tau = \tau ^0 + \varepsilon \tau ^1 + o\left( \varepsilon \right) \\ \xi _s = \xi _s^0 + \varepsilon \xi _s^1 + o\left( \varepsilon \right) \\ \eta _s = \eta _s^0 + \varepsilon \eta _s^1 + o\left( \varepsilon \right)\end{array} \right\}$
则广义 Noether 等式 (18) 成为

(22)$p_s \dot {\xi }_s^0 + \dot {q}_s \eta _s^0 - X_0^{\left( 0 \right)} \left( H \right) - H\dot {\tau }^0 + \tilde {Q}_s^0 \left( {\xi _s^0 - \dot {q}_s \tau ^0} \right) + \dot {G}^0 = 0$
(23)$p_s \dot {\xi }_s^1 + \dot {q}_s \eta _s^1 - X_1^{\left( 0 \right)} \left( H\right) - H\dot {\tau }^1 + \tilde {Q}_s^0 \left( {\xi _s^1 - \dot {q}_s \tau ^1} \right)+ \tilde {Q}_s^1 \left( {\xi _s^0 - \dot {q}_s \tau ^0} \right) + \dot {G}^1 = 0 $
其中

(24)$ X_k^{\left( 0 \right)} = \tau ^k\dfrac{\partial }{\partial t} + \xi _s^k \dfrac{\partial }{\partial q_s } + \eta _s^k \dfrac{\partial }{\partial p_s } \left( {k = 0,1} \right)$
定义 2 对于弱非线性动力学系统 (15),如果沿着方程 (15) 的所有解曲线,有

(25)$\dfrac{\text{d}}{\text{d} t}I = O\left( {\varepsilon ^2} \right)$
其中 $I = I_0 + \varepsilon I_1 $,则称 $I$ 为系统 (15) 的近似守恒量. 于是有定理 2.

定理 2 对于弱非线性动力学系统 (15),如果广义Noether 等式 (22) 和式 (23) 有解,则系统存在近似 Noether 守恒量

(26)$ I_{N} = \sum_{k = 0}^1 {\varepsilon ^k\left\{ {p_s \xi _s^k - H\tau ^k + G^k} \right\}}$
证明 由于

(27)$ \dfrac{\text{d}}{\text{d} t}I_{N} = \sum_{k = 0}^1 {\varepsilon ^k\left\{ {\dot {p}_s \xi _s^k + p_s \dot {\xi }_s^k - \dfrac{\partial H}{\partial t}\tau ^k} \right.}-\\ \left. { \dfrac{\partial H}{\partial q_s }\dot {q}_s \tau ^k - \dfrac{\partial H}{\partial p_s }\dot {p}_s \tau ^k - H\dot {\tau }^k + \dot {G}^k} \right\} =\\ \varepsilon ^2Q_s^1 \left( {\xi _s^1 - \dot {q}_s \tau ^1} \right)$
因此,式 (26) 是系统 (15) 的近似 Noether 守恒量. 证毕.

3 Birkhoff 框架下的近似 Noether 守恒量

设在 Birkhoff 框架下,弱非线性动力学方程可化为广义 Birkhoff 方程,有

(28)$ \left( {\dfrac{\partial R_\nu }{\partial a^\mu } - \dfrac{\partial R_\mu }{\partial a^\nu }} \right)\dot {a}^\nu - \dfrac{\partial B}{\partial a^\mu } - \dfrac{\partial R_\mu }{\partial t} = \\ - \varLambda _\mu \left( {t,{\pmb a},\varepsilon } \right) \ \ \ \left( {\mu ,\nu = 1,2, \cdots ,2n} \right)$
其中,$R_\mu = R_\mu \left( {t,{\pmb a}} \right)$ 是 Birkhoff 函数组,$B = B\left( {t,{\pmb a}}\right)$ 是 Birkhoff 函数,$\varLambda _\mu = \varLambda _\mu \left( {t,{\pmb a},\varepsilon }\right)$ 称为附加项,$\varepsilon $ 为小参数 ($\varepsilon \ll 1)$.

取无限小变换

(29)$ \bar {t} = t + \upsilon \tau \left( {t,{\pmb a}} \right), \bar {a}^\mu \left( \bar {t} \right) = \\ a^\mu \left( t \right) + \upsilon \xi _\mu \left( {t,{\pmb a}} \right) \ \ \ \left( {\mu = 1,2, \cdots ,2n} \right)$
这里 $\tau $ 和 $\xi _\mu $ 是生成函数,$\upsilon $ 是无限小参数. 如果成立

(30)$ \Delta \int_{t_1 }^{t_2 } {\left\{ {R_\mu \left( {t,{\pmb a}} \right)\dot {a}^\mu - B\left( {t,{\pmb a}} \right)} \right\}} \text{d} t =\\ - \int_{t_1 }^{t_2 } {\left[ {\dfrac{\text{d}}{\text{d} t}\left( {\Delta G} \right) + \varLambda _\mu \delta a^\mu } \right]} \text{d} t$
其中$\Delta G = \upsilon G$,$G = G\left( {t,{\pmb a}}\right)$ 为规范函数,则这种不变性称为系统 (28) 的 Noether 准对称性.由式 (30) 可导出广义 Noether 等式为

(31)$ - B\dot {\tau } + R_\mu \dot {\xi }_\mu + X^{\left( 0 \right)}\left( {R_\mu } \right)\dot {a}^\mu - X^{\left( 0 \right)}\left( B \right) +\\ \varLambda _\mu \left( {\xi _\mu - \dot {a}^\mu \tau } \right) + \dot {G} = 0$
其中

(32)$X^{\left( 0 \right)} = \tau \dfrac{\partial }{\partial t} + \xi _\mu \dfrac{\partial }{\partial a^\mu }$
设附加项 $\varLambda _\mu $ 为

(33)$\varLambda _\mu \left( {t,{\pmb a},\varepsilon } \right) = \varLambda _\mu ^0 \left( {t,{\pmb a}} \right) + \varepsilon \varLambda _\mu ^1 \left( {t,{\pmb a}} \right)$
相应地,生成函数为

(34)$\tau = \tau ^0 + \varepsilon \tau ^1 + o\left( \varepsilon \right) , \ \ \xi _\mu = \xi _\mu ^0 + \varepsilon \xi _\mu ^1 + o\left( \varepsilon \right)$
则广义 Noether 等式 (31) 成为

(35)$ - B\dot {\tau }^0 + R_\mu \dot {\xi }_\mu ^0 + X_0^{\left( 0 \right)} \left( {R_\mu } \right)\dot {a}^\mu - X_0^{\left( 0 \right)} \left( B \right)+ \\ \varLambda _\mu ^0 \left( {\xi _\mu ^0 - \dot {a}^\mu \tau ^0} \right) + \dot {G}^0 = 0$
(36)$ - B\dot {\tau }^1 + R_\mu \dot {\xi }_\mu ^1 + X_1^{\left( 0 \right)} \left( {R_\mu } \right)\dot {a}^\mu - X_1^{\left( 0 \right)} \left( B \right) +\\ \varLambda _\mu ^0 \left( {\xi _\mu ^1 - \dot {a}^\mu \tau ^1} \right) + \varLambda _\mu ^1 \left( {\xi _\mu ^0 - \dot {a}^\mu \tau ^0} \right) + \dot {G}^1 = 0$
其中

(37)$ X_k^{\left( 0 \right)} = \tau ^k\dfrac{\partial }{\partial t} + \xi _\mu ^k \dfrac{\partial }{\partial a^\mu } \ \ \ \left( {k = 0,1} \right)$
定义 3 对于弱非线性动力学系统 (28),如果沿着方程 (28) 的所有解曲线,有

(38)$\dfrac{\text{d}}{\text{d} t}I = O\left( {\varepsilon ^2} \right)$
其中 $I = I_0 + \varepsilon I_1 $,则称 $I$ 为系统 (28) 的近似守恒量. 于是有

定理 3 对于弱非线性动力学系统 (28),如果广义Noether 等式 (35) 和 (36) 有解,则系统存在近似 Noether 守恒量

(39)$ I_{N} = \sum_{k = 0}^1 {\varepsilon ^k\left( {R_\mu \xi _\mu ^k - B\tau ^k + G^k} \right)}$
证明 由于

(40)$ \dfrac{\text{d}}{\text{d} t}I_{N} = \sum_{k = 0}^1 {\varepsilon ^k\left[ {\left( {\dfrac{\partial R_\mu }{\partial t} + \dfrac{\partial R_\mu }{\partial a^\nu }\dot {a}^\nu } \right)\xi _\mu ^k + R_\mu \dot {\xi }_\mu ^k } \right.} -\\ \left. { B\dot {\tau }^k - \left( {\dfrac{\partial B}{\partial t} + \dfrac{\partial B}{\partial a^\mu }\dot {a}^\mu } \right)\tau ^k + \dot {G}^k} \right] =\\ \varepsilon ^2\varLambda _\mu ^1 \left( {\xi _\mu ^1 - \dot {a}^\mu \tau ^1} \right)$
因此,式 (39) 是系统 (28) 的近似 Noether 守恒量.

4 讨论

首先,Hamilton 框架是 Birkhoff 框架的特例.

实际上,若取

(41)$\left.\begin{array}{l} a^\mu = \left\{ \!\!\begin{array}{ll} {q_\mu } , & \mu = 1,2, \cdots ,n \\ p_{\mu - n} , & \mu = n + 1,n + 2, \cdots ,2n \end{array} \right. \\ R_\mu = \left\{\!\! \begin{array}{ll} {p_\mu } , & \mu = 1,2, \cdots ,n \\ 0 , & \mu = n + 1,n + 2, \cdots ,2n \end{array} \right.\\ \varLambda _\mu = \left\{ \!\! \begin{array}{ll} {\tilde {Q}_\mu } , & \mu = 1,2, \cdots ,n \\ 0 , & \mu = n + 1,n + 2, \cdots ,2n \end{array} \right. \\ B = H \end{array}\right\} $
则 Birkhoff 框架下的广义 Birkhoff 方程 (28),广义Noether 等式 (35) 和 (36),近似 Noether 守恒量 (39)退化为 Hamilton 框架下的 Hamilton 方程 (15),广义 Noether 等式 (22) 和 (23),近似 Noether 守恒量 (26).

其次,Lagrange 框架与 Hamilton 框架等价.

实际上,令

(42)$\left.\begin{array}{l} p_s = \dfrac{\partial L}{\partial \dot {q}_s } \\ H\left( {t,{\pmb q},{\pmb p}} \right) = \left. {\left[ {p_s \dot {q}_s - L\left( {t,{\pmb q},\dot{\pmb q}} \right)} \right]} \right|_{\dot{q} = \dot{q}\left( {t,{q},{p}} \right)} \\ \tilde {Q}_s \left( {t,{\pmb q},{\pmb p},\varepsilon } \right) = Q_s \left( {t,{\pmb q},\dot{\pmb q}\left( {t,{\pmb q},{\pmb p}} \right),\varepsilon } \right) \end{array}\right\} $
则容易验证 Lagrange 框架下的 Lagrange 方程 (1),广义 Noether 等式 (8) 和 (9),近似 Noether 守恒量 (12)等 价于 Hamilton 框架下的 Hamilton 方程 (15),广义 Noether 等式 (22) 和 (23),近似 Noether 守恒量 (26).

5 算例

5.1 van der Pol 方程的近似守恒量

著名的 van der Pol 方程为^[32]

(43)$\ddot {q} + q - \varepsilon \left( {1 - q^2} \right)\dot {q} = 0$
试研究此系统的近似 Noether 守恒量.

方程 (43) 可化为一般完整系统的 Lagrange 方程,有

(44)$L = \dfrac{1}{2}\dot {q}^2 - \dfrac{1}{2}q^2 , \ \ Q^0 = 0 , \ \ Q^1 = \left( {1 - q^2} \right)\dot {q}$
广义 Noether 等式 (8) 和 (9) 给出

(45)$\dfrac{1}{2}\left( {\dot {q}^2 - q^2} \right)\dot {\tau }^0 + \dot {q}\left( {\dot {\xi }^0 - \dot {q}\dot {\tau }^0} \right) - q\xi ^0 + \dot {G}^0 = 0$
(46)$\dfrac{1}{2}\left( {\dot {q}^2 - q^2} \right)\dot {\tau }^1 + \dot {q}\left( {\dot {\xi }^1 - \dot {q}\dot {\tau }^1} \right) - q\xi ^1+ \\ \left( {1 - q^2} \right)\dot {q}\left( {\xi ^0 - \dot {q}\tau ^0} \right) + \dot {G}^1 = 0$
联立方程 (45) 和 (46),有如下解

(47)$\tau ^0 = 0 , \ \ \tau ^1 = - 1 , \ \ \xi ^0 = 0 , \ \ \xi ^1 = 0 , \ \ G^0 = 0 , \ \ G^1 = 0$
(48)$\left.\begin{array}{l} \!\!\! \tau ^0 = 0 , \ \ \tau ^1 = - \cos 2t , \ \ \xi ^0 = 0 , \ \ \xi ^1 = q\sin 2t \\ G^0 = 0 , \ \ G^1 = - q^2\cos 2t \end{array}\right\} $
(49)$\left.\begin{array}{l} \!\!\! \tau ^0 = 0 , \ \ \tau ^1 = \sin 2t , \ \ \xi ^0 = 0 , \ \ \xi ^1 = q\cos 2t \\ G^0 = 0 , \ \ G^1 = q^2\sin 2t\end{array}\right\} $
生成函数 (47) $\sim $ (49) 相应于 van der Pol 方程 (43) 的 Noether 准对称性,由定理 1,可以得到

(50)$I_{N}^1 = \varepsilon \dfrac{1}{2}\left( {\dot {q}^2 + q^2} \right)$
(51)$I_{N}^2 = \varepsilon \left[ {\dfrac{1}{2}\left( {\dot {q}^2 - q^2} \right)\cos 2t + q\dot {q}\sin 2t} \right]$
(52)$I_{N}^3 = \varepsilon \left[ { - \dfrac{1}{2}\left( {\dot {q}^2 - q^2} \right)\sin 2t + q\dot {q}\cos 2t} \right]$
式 (50) $\sim $ 式 (52) 是 van der Pol 方程 (43) 的近似 Noether 守恒量.

方程 (43) 也可化为其他形式的一般完整系统. 例如

(53)$ L = \dfrac{1}{2}\dot {q}^2 , \ \ Q^0 = - q , \ \ Q^1 = \left( {1 - q^2} \right)\dot {q}$
此时,广义 Noether 等式为

(54)$\dfrac{1}{2}\dot {q}^2\dot {\tau }^0 + \dot {q}\left( {\dot {\xi }^0 - \dot {q}\dot {\tau }^0} \right) - q\left( {\xi ^0 - \dot {q}\tau ^0} \right) + \dot {G}^0 = 0$
(55)$\dfrac{1}{2}\dot {q}^2\dot {\tau }^1 + \dot {q}\left( {\dot {\xi }^1 - \dot {q}\dot {\tau }^1} \right) - q\left( {\xi ^1 - \dot {q}\tau ^1} \right)+\\ \left( {1 - q^2} \right)\dot {q}\left( {\xi ^0 - \dot {q}\tau ^0} \right) + \dot {G}^1 = 0$
取

(56)$\left.\begin{array}{l} \tau ^0 = 0 , \ \ \tau ^1 = - 1 , \ \ \xi ^0 = 0 , \ \ \xi ^1 = 0\\ G^0 = 0 , \ \ G^1 = \dfrac{1}{2}q^2 \end{array}\right\} $
则有近似 Noether 守恒量

(57)$ I_{N} = \varepsilon \dfrac{1}{2}\left( {\dot {q}^2 + q^2} \right)$
如取

(58)$\left.\begin{array}{l} \tau ^0 = 1 , \ \ \tau ^1 = - \cos 2t , \ \ \xi ^0 = \dot {q} , \ \ \xi ^1 = q\sin 2t \\ G^0 = - \dfrac{1}{2}\dot {q}^2 , \ \ G^1 = - \dfrac{1}{2}q^2\cos 2t \end{array}\right\} $
则有近似 Noether 守恒量

(59)$ I_{N} = \varepsilon \left[ {\dfrac{1}{2}\left( {\dot {q}^2 - q^2} \right)\cos 2t + q\dot {q}\sin 2t} \right]$
结果表明,同一弱非线性动力学方程可以化为不同的完整非保守系统,同一近似 Noether 守恒量可以相应于不同的 Noether 准对称性. 因此,利用 Noether 准对称性方法求近似守恒量具有较大的灵活性.

此外,也可以在 Hamilton 框架和 Birkhoff 框架下计算 van der Pol 方程的近似 Noether 守恒量.

在 Hamilton 框架下,方程 (43) 可化为相空间中一般完整系统. 例如,取 Hamilton 函数和广义力为

(60)$ H = \dfrac{1}{2}p^2 , \ \ \tilde {Q}^0 = - q , \ \ \tilde {Q}^1 = \left( {1 - q^2} \right)p$
则广义 Noether 等式为

(61)$p\dot {\xi }^0 + \dot {q}\eta ^0 - \eta ^0p - \dfrac{1}{2}p^2\dot {\tau }^0 - q\left( {\xi ^0 - \dot {q}\tau ^0} \right) + \dot {G}^0 = 0$
(62)$p\dot {\xi }^1 + \dot {q}\eta ^1 - \eta ^1p - \dfrac{1}{2}p^2\dot {\tau }^1 - q\left( {\xi ^1 - \dot {q}\tau ^1} \right)+\\ \left( {1 - q^2} \right)p\left( {\xi ^0 - \dot {q}\tau ^0} \right) + \dot {G}^1 = 0$
方程 (61) 和 (62) 有解

(63)$\left.\begin{array}{l} \tau ^0 = 0 , \ \ \tau ^1 = - 1 , \ \ \xi ^0 = 0 , \ \ \xi ^1 = 0 \\ \eta ^0 = 0 , \ \ \eta ^1 = 0 , \ \ G^0 = 0 , \ \ G^1 = \dfrac{1}{2}q^2 \end{array}\right\}$
(64)$\left.\begin{array}{l} \tau ^0 = 0 , \ \ \tau ^1 = - \cos 2t , \ \ \xi ^0 = 0 , \ \ \xi ^1 = q\sin 2t \\ \eta ^0 = 0 , \ \ \eta ^1 = 2q\cos 2t - p\sin 2t \\ G^0 = 0 , \ \ G^1 = - \dfrac{1}{2}q^2\cos 2t \end{array}\right\}$
(65)$\left.\begin{array}{l} \tau ^0 = 0 , \ \ \tau ^1 = \sin 2t , \ \ \xi ^0 = 0 , \ \ \xi ^1 = q\cos 2t \\ \eta ^0 = 0 , \ \ \eta ^1 = - 2q\sin 2t - p\cos 2t \\ G^0 = 0 , \ \ G^1 = \dfrac{1}{2}q^2\sin 2t \end{array}\right\}$
由定理 2,得到

(66)$I_{N}^1 = \varepsilon \left( {\dfrac{1}{2}p^2 + \dfrac{1}{2}q^2} \right)$
(67)$I_{N}^2 = \varepsilon \left( {pq\sin 2t + \dfrac{1}{2}p^2\cos 2t - \dfrac{1}{2}q^2\cos 2t} \right)$
(68)$I_{N}^3 = \varepsilon \left( {pq\cos 2t - \dfrac{1}{2}p^2\sin 2t + \dfrac{1}{2}q^2\sin 2t} \right)$
这是 van der Pol 方程 (43) 在相空间中的近似 Noether 守恒量.

在 Birkhoff 框架下,方程 (43) 可化为广义 Birkhoff 系统. 例如,取

(69)$\left.\begin{array}{l} B = \dfrac{1}{2}\left( {a^1} \right)^2 + \dfrac{1}{2}\left( {a^2} \right)^2 , \ \ R_1 = a^2 , \ \ R_2 = 0 \\ \varLambda _1^0 = 0 , \ \ \varLambda _1^1 = \left[ {1 - \left( {a^1} \right)^2} \right]a^2 , \ \ \varLambda _2^0 = \varLambda _2^1 = 0 \end{array}\right\} $
则广义 Noether 等式为

(70)$ - \dfrac{1}{2}\left( {a^1} \right)^2\dot {\tau }^0 - \dfrac{1}{2}\left( {a^2} \right)^2\dot {\tau }^0 + a^2\dot {\xi }_1^0 + \xi _2^0 \dot {a}^1 -\\ \xi _1^0 a^1 - \xi _2^0 a^2 + \dot {G}^0 = 0$
(71)$ - \dfrac{1}{2}\left( {a^1} \right)^2\dot {\tau }^1 - \dfrac{1}{2}\left( {a^2} \right)^2\dot {\tau }^1 + a^2\dot {\xi }_1^1 + \xi _2^1 \dot {a}^1 - \xi _1^1 a^1-\\ \xi _2^1 a^2 + \left[ {1 - \left( {a^1} \right)^2} \right]a^2\left( {\xi _1^0 - \dot {a}^1\tau ^0} \right) + \dot {G}^1 = 0$
方程 (70) 和 (71) 有解

(72)$\left.\begin{array}{l} \tau ^0 = 0 , \ \ \tau ^1 = - 1 , \ \ \xi _1^0 = 0 , \ \ \xi _1^1 = 0 \\ \xi _2^0 = 0 , \ \ \xi _2^1 = 0 , \ \ G^0 = 0 , \ \ G^1 = 0 \end{array}\right\} $
(73)$\left.\begin{array}{l} \tau ^0 = 0 , \ \ \tau ^1 = - \cos 2t , \ \ \xi _1^0 = 0 , \ \ \xi _1^1 = a^1\sin 2t \\ \xi _2^0 = 0 , \ \ \xi _2^1 = 2a^1\cos 2t - a^2\sin 2t \\ G^0 = 0 , \ \ G^1 = - \left( {a^1} \right)^2\cos 2t \end{array}\right\} $
(74)$\left.\begin{array}{l} \tau ^0 = 0 , \ \ \tau ^1 = \sin 2t , \ \ \xi _1^0 = 0 , \ \ \xi _1^1 = a^1\cos 2t \\ \xi _2^0 = 0 , \ \ \xi _2^1 = - 2a^1\sin 2t - a^2\cos 2t \\ G^0 = 0 , \ \ G^1 = \left( {a^1} \right)^2\sin 2t \end{array}\right\} $
由定理 3,得

(75)$I_{N}^1 = \dfrac{1}{2}\varepsilon \left[ {\left( {a^1} \right)^2 + \left( {a^2} \right)^2} \right]$
(76)$I_{N}^2 = \varepsilon \left\{ {a^2a^1\sin 2t - \dfrac{1}{2}\cos 2t\left[ {\left( {a^1} \right)^2 - \left( {a^2} \right)^2} \right]} \right\}$
(77)$I_{N}^3 = \varepsilon \left\{ {a^2a^1\cos 2t + \dfrac{1}{2}\sin 2t\left[ {\left( {a^1} \right)^2 - \left( {a^2} \right)^2} \right]} \right\}$
这是 van der Pol 方程 (43) 在 Birkhoff 框架下的近似 Noether 守恒量.

显然,在 3 种不同框架下可以得到 van der Pol 方程相同的近似 Noether 守恒量.

5.2 Duffing 方程的近似守 恒量

著名的 Duffing 方程为^[32]

(78)$\ddot {q} + q + \varepsilon q^3 = 0$
试研究其近似 Noether 守恒量.

方程 (78) 可化为相空间中一般完整系统的 Hamilton 方程,有

(79)$ H = \dfrac{1}{2}\left( {p^2 + q^2} \right), Q^0 = 0, Q^1 = - q^3$
其中 $p = \dfrac{\partial L}{\partial \dot {q}} = \dot {q}$. 广义 Noether 等式 (22) 和式 (23)给出

(80)$p\dot {\xi }^0 + \dot {q}\eta ^0 - \xi ^0q - \eta ^0p - \dfrac{1}{2}\left( {p^2 + q^2} \right)\dot {\tau }^0 + \dot {G}^0 = 0$
(81)$p\dot {\xi }^1 + \dot {q}\eta ^1 - \xi ^1q - \eta ^1p - \dfrac{1}{2}\left( {p^2 + q^2} \right)\dot {\tau }^1 - q^3\left( {\xi ^0 - \dot {q}\tau ^0} \right) + \dot {G}^1 = 0$
方程 (80) 和 (81) 有解

(82)$\left.\begin{array}{l} \tau ^0 = 0 , \ \ \tau ^1 = 1 , \ \ \xi ^0 = \xi ^1 = 0 \\ \eta ^0 = \eta ^1 = 0 , \ \ G^0 = 0 , \ \ G^1 = - \dfrac{1}{4}q^4 \end{array}\right\} $
(83)$\left.\begin{array}{l} \tau ^0 = 0 , \ \ \tau ^1 = 0 , \ \ \xi ^0 = 0 , \ \ \xi ^1 = \cos t , \ \ \eta ^0 = 0 \\ \eta ^1 = - \sin t , \ \ G^0 = 0 , \ \ G^1 = q\sin t\end{array}\right\} $
由定理 2,可得到

(84)$I_{N}^1 = - \dfrac{1}{2}\left( {p^2 + q^2} \right) - \varepsilon \left[ {\dfrac{1}{2}\left( {p^2 + q^2} \right) + \dfrac{1}{4}q^4} \right]$
(85)$I_{N}^2 = \varepsilon \left( {p\cos t + q\sin t} \right)$
式 (84) 和式 (85) 是 Duffing 振子 (78) 的近似 Noether 守恒量.

5.3 弱非线性耦合振子的 近似守恒量

两自由度的弱非线性耦合振子方程为^[33]

(86)$ \left.\begin{array}{l} {\ddot {x} = - \omega _1^2 x + \varepsilon \alpha _1 y^2} \\ {\ddot {y} = - \omega _2^2 y + 2\varepsilon \alpha _1 xy} \end{array} \right \}$
试研究其近似 Noether 守恒量.

令

(87)$a^1 = x , \ \ a^2 = y , \ \ a^3 = \dot {x} , \ \ a^4 = \dot {y}$
方程 (86) 可化为广义 Birkhoff 方程,有

(88)$\left.\begin{array}{l} \!\! B = \dfrac{1}{2}\left[ {\left( {a^3} \right)^2 + \left( {a^4} \right)^2} \right] + \dfrac{1}{2}\omega _1^2 \left( {a^1} \right)^2 + \dfrac{1}{2}\omega _2^2 \left( {a^2} \right)^2 \\ R_1 = a^3 , \ \ R_2 = a^4 , \ \ R_3 = 0 , \ \ R_4 = 0 \\ \varLambda _1^0 = 0 , \ \ \varLambda _1^1 = \alpha _1 \left( {a^2} \right)^2 , \ \ \varLambda _2^0 = 0 , \ \ \varLambda _2^1 = 2\alpha _1 a^1a^2, \\ \varLambda _3^0 = \varLambda _3^1 = 0 , \ \ \varLambda _4^0 = \varLambda _4^1 = 0 \!\!\end{array}\right\} $
广义 Noether 等式 (35) 和 (36) 给出

(89)$ - B\dot {\tau }^0 + a^3\dot {\xi }_1^0 + a^4\dot {\xi }_2^0 + \xi _3^0 \dot {a}^1 + \xi _4^0 \dot {a}^2 - \xi _1^0 \omega _1^2 a^1 -\\ \xi _2^0 \omega _2^2 a^2 - \xi _3^0 a^3 - \xi _4^0 a^4 + \dot {G}^0 = 0$
(90)$\begin{array}{l} - B\dot {\tau }^1 + a^3\dot {\xi }_1^1 + a^4\dot {\xi }_2^1 + \xi _3^1 \dot {a}^1 + \xi _4^1 \dot {a}^2 - \xi _1^1 \omega _1^2 a^1 -\\ \xi _2^1 \omega _2^2 a^2 - \xi _3^1 a^3 - \xi _4^1 a^4 + \alpha _1 \left( {a^2} \right)^2\left( {\xi _1^0 - \dot {a}^1\tau ^0} \right) +\\ + 2\alpha _1 a^1a^2\left( {\xi _2^0 - \dot {a}^2\tau ^0} \right) + \dot {G}^1 = 0\end{array}$
方程 (89) 和 (90) 有解

(91)$\left.\begin{array}{l} \tau ^0 = 1 , \ \ \tau ^1 = 1 , \ \ \xi _1^0 = \xi _1^1 = \xi _2^0 = \xi _2^1 = 0 , \ \ \xi _3^0 = \xi _3^1 = 0 \\ \xi _4^0 = \xi _4^1 = 0 , \ \ G^0 = 0 , \ \ G^1 = \alpha _1 a^1\left( {a^2} \right)^2 \end{array}\right\} $
(92)$\left.\begin{array}{l} \tau ^0 = \tau ^1 = 0 , \ \ \xi _1^0 = 0 , \ \ \xi _1^1 = \sin \omega _1 t \\ \xi _2^0 = \xi _2^1 = \xi _3^0 = \xi _3^1 = \xi _4^0 = \xi _4^1 = 0 \\ G^0 = 0 , \ \ G^1 = - \omega _1 a^1\cos \omega _1 t\end{array}\right\} $
(93)$\left.\begin{array}{l} \tau ^0 = - 1 , \ \ \tau ^1 = 0 , \ \ \xi _1^0 = \xi _1^1 = 0 , \ \ \xi _2^0 = 0 \\ \xi _2^1 = \cos \omega _2 t , \ \ \xi _3^0 = \xi _3^1 = \xi _4^0 = \xi _4^1 = 0 \\ G^0 = 0 , \ \ G^1 = - \alpha _1 a^1\left( {a^2} \right)^2 + \omega _2 a^2\sin \omega _2 t\end{array}\right\} $
生成函数 (91) $\sim $ (93) 相应于振子方程 (86) 的 Noether 准对称性,由定理 3,可得到

(94)$\begin{array}{l}I_{N}^1 = - \dfrac{1}{2}\left[ {\left( {a^3} \right)^2 + \left( {a^4} \right)^2} \right] - \dfrac{1}{2}\omega _1^2 \left( {a^1} \right)^2-\\ \dfrac{1}{2}\omega _2^2 \left( {a^2} \right)^2 - \dfrac{1}{2} \varepsilon\left[ {\left( {a^3} \right)^2 + \left( {a^4} \right)^2} \right]-\\ \varepsilon \left[ {\dfrac{1}{2}\omega _1^2 \left( {a^1} \right)^2 + \dfrac{1}{2}\omega _2^2 \left( {a^2} \right)^2 - \alpha _1 a^1\left( {a^2} \right)^2} \right]\end{array}$
(95)$I_{N}^2 = \varepsilon \left\{ {a^3\sin \omega _1 t - \omega _1 a^1\cos \omega _1 t} \right\}$
(96)$I_{N}^3 = \dfrac{1}{2}\left[ {\left( {a^3} \right)^2 + \left( {a^4} \right)^2} \right] + \dfrac{1}{2}\omega _1^2 \left( {a^1} \right)^2 + \dfrac{1}{2}\omega _2^2 \left( {a^2} \right)^2 +\\ \varepsilon \left[ {a^4\cos \omega _2 t + \omega _2 a^2\sin \omega _2 t - \alpha _1 a^1\left( {a^2} \right)^2} \right]$
式 (94) $\sim $ 式 (96) 是弱非线性耦合振子系统 (86) 的近似 Noether 守恒量.

6 结论

用分析力学的方法研究非线性微分方程的动力学性质具有重要的理论和实际意义. 通常寻找微分方程的守恒量可采用 Lie 对称性方法. 实际上,也可以利用 Noether 对称性方法. 文章通过分析Noether 准对称性来探寻弱非线性动力学方程的近似 Noether 守恒量. 分别基于 Lagrange 框架、Hamilton 框架和 Birkhoff 框架,建立了 Noether 准对称性的定义和广义 Noether 等式,证明了近似 Noether 守恒量定理,并通过三个经典问题,即 vanderPol 方程、Duffing 方程和弱非线性耦合振子方程,展示了弱非线性动力学方程的 Noether 准对称性与近似 Noether 守恒量的计算.研究表明：Hamilton 框架下的近似 Noether 守恒量是 Birkhoff 框架下的近似 Noether 守恒量的特例,而 Lagrange 框架下的近似 Noether 守恒量等价于 Hamilton 框架下的近似 Noether 守恒量. 此外,由于将弱非线性动力 学方程化为一般完整系统的 Lagrange 方程 (1)或 Hamilton 方程 (15) 或广义 Birkhoff 方程 (28) 时,其动力学函数的选取是不唯一的,因此利用Noether 对称性方法寻找弱非线性动力学方程的近似守恒量不仅方便有效,而且具有较大的灵活性.本文方法和结果可进一步推广应用于含有两个或更多小参数的或含有多个独立变量的非线性动力学系统.

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DOIURL
Compared with the classical variational principle, the generalized variational principle of Herglotz based upon the action defined by differential equations gives a variational description of nonconservative dynamical system. The principle can describe all dynamical processes and nonconservative or dissipative systems. In the present study, the principle is extended to phase space, and the generalized variational principle of Herglotz type for non-conservative mechanical system in phase space is given and Noether's theorem and its inverse of the system are studied. Firstly, the generalized variational principle of Herglotz type in phase space is presented, a variational description of non-conservative system in phase space is given, and the corresponding Hamilton canonical equations are deduced. Secondly, based upon the relation between non-isochronal variation and isochronal variation, two basic formulae for the variation of Hamilton-Herglotz action in phase space are obtained. Thirdly, the definition and the criterion of Noether symmetry are given, and Noether's theorem and its inverse of nonconservative system for the variational problem of Herglotz type in phase space are proposed and proved, and the inner relation between the Noether symmetry and the conserved quantity for mechanical systems in phase space is revealed. The generalized variational principle of Herglotz type reduces to the classical variational principle under classical conditions, and Noether's theorem for the variational problem of Herglotz type reduces to the classical Noether's theorem of Hamilton system. In the end of the paper, we take the famous Emden equation and damping oscillator with second power as examples to illustrate the application of the results.

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