PERTURBATION TO LIE SYMMETRIES FOR GENERALIZED BIRKHOFFIAN SYSTEMS ON TIME SCALES
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摘要: 近几年, 在时间尺度框架下研究对称性与守恒量日益引发关注. 然而, 因该领域尚处于探索阶段, 其研究成果的可靠性仍在进一步考察中, 且对称性的摄动与绝热不变量的研究也正是基于此项工作之上, 因此对于该领域的深入探讨具有重要意义. 首先, 在时间尺度框架下, 给出了精确不变量与绝热不变量的定义, 对于未受扰动力作用和受扰动力作用的广义Birkhoff系统, 分别建立了Lie对称性及其摄动的确定方程和结构方程, 并基于此得到了该系统Lie对称性导致的精确不变量及其摄动导致的绝热不变量, 并给出相应证明. 其次, 考虑受约束Birkhoff系统, 对于未受扰动和受扰动的时间尺度上约束Birkhoff系统及其相应自由Birkhoff系统, 分别给出了Lie对称性导致精确不变量及其摄动导致绝热不变量的条件. 相应小节末尾分别给出算例并对所得守恒量进行了数值模拟, 直观地验证了结论的有效性. 取时间尺度为实数集和整数集, 所有结论可退化到经典连续型和离散型动力学系统. 本论文的方法与成果对时间尺度动力学系统对称性及其摄动理论研究具有一定的指导意义, 可应用和拓展到非迁移系统, 对偶系统, 分数阶时间尺度相结合系统, nabla导数情形等.
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关键词:
- 时间尺度 /
- Lie对称性 /
- 摄动与绝热不变量 /
- Birkhoff系统
Abstract: In recent years, the study of symmetry and conserved quantity under time-scale framework has increasingly attracted attention. Nevertheless, as this field is still in the exploratory stage, the reliability of its research results is still under further investigation, and the study of symmetry perturbations and adiabatic invariants is also based on this work. Therefore, in-depth exploration of this field is of great significance. First of all, under time-scale framework, the definitions of exact invariants and adiabatic invariants are given. For both undisturbed and disturbed generalized Birkhoffian systems, the determining equations and structural equations for Lie symmetry and its perturbations are established, respectively. Based on these, the exact invariants caused by Lie symmetry and the adiabatic invariants caused by perturbations to Lie symmetry in the both systems are obtained, and the corresponding proofs are given. Secondly, regarding the Birkhoffian systems under constrained conditions on time scales, we explore the conditions under which the Lie symmetry of the constrained Birkhoffian system and its corresponding free Birkhoffian system leads to exact invariants, as well as the conditions under which the perturbation to Lie symmetry of the corresponding disturbed systems leads to adiabatic invariants. At the end of the corresponding sections, examples are provided and numerical simulations are conducted on the obtained conserved quantities, which intuitively verified the effectiveness of the conclusions in this paper. Taking the time scale as the set of real numbers and integer, all conclusions in this paper can be degraded to classical continuous and discrete dynamical systems. The methods and research results of this paper have certain reference and guiding significance for the study of symmetry and perturbation theory of dynamical systems on time scales, and can be applied and extended to nonshifted systems, dual systems, the combination of fractional and time-scale systems, and nabla derivative cases, and so on. -
引 言
时间尺度是实数集的任意非空闭子集, 其理论的提出可以统一连续和离散分析. 对于时间尺度上动力学方程的研究, 在时间尺度$ \mathbb{T} $上可统一讨论以避免两次证明结果, 取$ \mathbb{T} = \mathbb{R} $和$ \mathbb{T} = \mathbb{Z} $即可得到相应的特殊结果. 此外, 根据实际情境, 也可选取许多其他时间尺度, 例如$ \mathbb{T} = {q^{{\mathbb{N}_0}}}\left( {q > 1} \right) $或$ \mathbb{T} = {q^\mathbb{Z}} \cup \left\{ 0 \right\} $等. 时间尺度理论在提出之初并未受到重视, 而是在实际应用中展现出其巨大的潜力后, 迅速引起广泛关注并应用于数学、力学和生物学等众多领域, 其在约束力学系统的重要地位也逐步得到认同[1-2]. 在时间尺度理论发展完善的过程中, Bohner等[3-4]先后出版了两本时间尺度专著, 全面细致地概括分析了时间尺度微积分的定义、性质及运算法则等, 为时间尺度领域的深度探索奠定了坚实基础.
寻求守恒量是约束力学系统动力学的主要任务之一, 也是近代分析力学的重要发展方向之一. 对称性是一种寻求守恒量的方法, 按提出顺序包括Noether对称性方法, Lie对称性方法和Mei对称性方法, 本文应用的是Lie对称性方法. Lie对称性方法在力学系统中的应用始于1979年Lutzky[5]的研究, 研究发现除了通过Hamilton作用量在无限小变换下的不变性——Noether对称性外, 还可以通过微分方程在时间和广义坐标无限小群变换下的不变性来寻求守恒量, 具体是建立并求解确定方程与结构方程进而找到Lie对称性导致的守恒量. Bluman等[6]认为Lie对称性方法是对称方法对微分方程适用性的扩展, 是一种高级算法, 且肯定了其对于构造微分方程显式解的技巧优势. 多年来, 学者们分别在经典情况[7-8]和分数阶框架下[9-12]深入研究, 至于时间尺度框架下对称性的研究, 始于Bartosiewicz等学者提出的delta导数下的Noether定理[13]和第二Euler-Lagrange方程[14]. 然而, 现阶段时间尺度上的对称性研究还尚未成熟, 主要集中在Noether对称性上[15-19],直至近期方才涌现一些时间尺度上Lie对称性的研究, 且所得的守恒量仍以Noether型为主[20-22].
在一个带参数的小干扰力作用下, 系统对称性及相应守恒量的变化分别用对称性摄动与绝热不变量来描述. 其中, 绝热不变量是一个相对于该参数的量级而言改变更缓慢的物理量, 而未受扰动系统的不变量则称为精确不变量, 即守恒量. 对称性摄动与绝热不变量的研究与动力学系统的可积性之间关系密切, 因此其在时间尺度上的研究目前也是一个新兴课题且已取得了一些进展[23-28].
Birkhoff系统是一类更广泛的约束力学系统, 可应用于强子物理、统计力学和工程等诸多领域, 形式简洁的Hamilton力学是其特殊情形. Birkhoff方程的构造往往存在技术上的困难, 但广义Birkhoff方程的出现不仅降低了构造的难度还增加了自由度. 1993年梅凤翔教授称右端增加一个附加项的Birkhoff方程为广义Birkhoff方程并研究了其Noether对称性[29], 2013年又出版了一本著作全面地介绍广义Birkhoff系统[30], 因此其与时间尺度结合的相关研究也值得重视.
由于Anerot等[31]围绕Noether定理[13]和第二Euler-Lagrange方程[14]研究时, 用一种他们认为更合理的方法得到了时间尺度上Lagrange系统的Noether定理, 因此本工作拟采用Anerot等的方法, 研究时间尺度上广义Birkhoff系统和受约束Birkhoff系统的Lie对称性摄动与绝热不变量, 以期为时间尺度上约束力学系统摄动理论研究提供参考数据.
1. 时间尺度上广义Birkhoff系统
1.1 建立方程
从Pfaff作用量
$$ {A_B}\left( {{a_\nu }\left( \cdot \right)} \right) = \int_{{t_{_1}}}^{{t_2}} {\left[ {{R_l}\left( {t,a_\nu ^\sigma } \right)a_l^\Delta - B\left( {t,a_\nu ^\sigma } \right)} \right]\Delta t} $$ (1) 出发, 时间尺度上广义Pfaff-Birkhoff原理可表示为
$$ \int_{{t_1}}^{{t_2}} {\left\{ {\delta \left[ {{R_l}\left( {t,a_\nu ^\sigma } \right)a_l^\Delta - B\left( {t,a_\nu ^\sigma } \right)} \right] + {\varLambda _l}\left( {t,a_\nu ^\sigma } \right)\delta a_l^\sigma } \right\}} \Delta t = 0 $$ (2) 满足端点条件和交换关系
$$\qquad\qquad \delta {a_l}\left| {_{t = {t_1}}} \right. = \delta {a_l}\left| {_{t = {t_2}}} \right. = 0 $$ (3) $$ \qquad\qquad \delta {a}_{l}^{\Delta } = {\left(\delta {a}_{l}\right)}^{\Delta }\text{, }\;\;\delta {a}_{l}^{\sigma } = {\left(\delta {a}_{l}\right)}^{\sigma } $$ (4) 其中, $ B,{R}_{l},{\varLambda }_{l}\left(t,{a}_{\nu }^{\sigma }\right): \mathbb{T}\times {{\mathbb{R}}}^{2 n}\to {\mathbb{R}} $分别为Birkhoff函数, Birkhoff函数组和附加项, $ \sigma :\mathbb{T}\to \mathbb{T} $为前跳算子且$ \sigma \left( t \right) = \inf \left\{ {s \in \mathbb{T}:s > t} \right\} $, $ \rho :\mathbb{T}\to \mathbb{T} $为后跳算子且$ \rho \left( t \right) = \sup \left\{ {s \in \mathbb{T}:s < t} \right\} $. 此外$ a_l^\Delta \left( t \right) = {{\Delta {a_l}\left( t \right)} \mathord{\left/ {\vphantom {{\Delta {a_l}\left( t \right)} {\Delta t}}} \right. } {\Delta t}} $, $ a_l^\sigma \left( t \right) = \left( {{a_l} \circ \sigma } \right)\left( t \right) $, $ t \in {\left[ {{t_1},{t_2}} \right]^\kappa } = \left[ {{t_1},{t_2}} \right]\backslash \left( {\rho \left( {{t_2}} \right),{t_2}} \right] $, $ l,\nu = 1,2, \cdots , 2 n $, $ n \in \mathbb{N} $.
利用式(2) ~ 式(4)及Dubois-Reymond引理, 可得到
$$ \frac{{\partial {R_\nu }\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }} - R_l^\Delta + {\varLambda _l}\left( {t,a_\varpi ^\sigma } \right) = 0 $$ (5) 式(5)称为时间尺度上广义Birkhoff方程, 其中$ t\in {\left[{t}_{1},{t}_{2}\right]}^{\kappa }, l,\nu ,\varpi = 1,2,\cdots ,2 n, n\in {N} $.
1.2 Lie对称性和精确不变量
首先, 给出精确不变量的定义.
定义1 如果时间尺度上力学系统的物理量$ {I_0}\left( {t,{a_\nu },a_\nu ^\sigma ,a_\nu ^\Delta } \right) $满足$ {{\Delta {I_0}} \mathord{\left/ {\vphantom {{\Delta {I_0}} {\Delta t}}} \right. } {\Delta t}} = 0 $, 则称$ {I_0} $为该系统的精确不变量.
其次, 引进无限小变换
$$ \left.\begin{split} &\bar t = t + \upsilon \xi _{B0}^0\left( {t,{a_\varpi }} \right) + o\left( \upsilon \right) \\ &{\bar a_l}\left( {\bar t} \right) = {a_l}\left( t \right) + \upsilon \xi _{Bl}^0\left( {t,{a_\varpi }} \right) + o\left( \upsilon \right)\end{split}\right\} $$ (6) 其中, $ \upsilon $是无限小参数, $ \xi _{B0}^0 $和$ \xi _{Bl}^0 $是无限小生成元. 时间尺度上广义Birkhoff系统的Lie对称性确定方程为
$$ \begin{split} &X_0^{\left( 1 \right)}\left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}} \right)a_\nu ^\Delta + \left( {\xi _{B\nu }^{0\Delta } - a_\nu ^\Delta \xi _{B0}^{0\Delta }} \right)\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }} - X_0^{\left( 1 \right)}\left( {\frac{{\partial B}}{{\partial a_l^\sigma }}} \right) - \\ &\qquad X_0^{\left( 1 \right)}\left( {R_l^\Delta } \right) + X_0^{\left( 1 \right)}\left( {{\varLambda _l}} \right) = 0 \end{split}$$ (7) 其中无限小生成元向量和它的一次扩展为
$$\left. \begin{split} &X_0^{\left( 0 \right)} = \xi _{B0}^0\frac{\partial }{{\partial t}} + \xi _{Bl}^0\frac{\partial }{{\partial {a_l}}} \\ &X_0^{\left( 1 \right)} = X_0^{\left( 0 \right)} + \left( {\xi _{Bl}^{0\Delta } - a_l^\Delta \xi _{B0}^{0\Delta }} \right)\frac{\partial }{{\partial a_l^\Delta }}\end{split} \right\}$$ (8) 定理1 对于时间尺度上广义Birkhoff系统(方程(5)), 如果存在规范函数$ {G^0}\left( {t,a_\varpi ^\sigma } \right) $使得Lie对称性的生成元$ \xi _{B0}^0 $, $ \xi _{Bl}^0 $满足结构方程
$$ \begin{split} &\left( {{R_l}a_l^\Delta - B} \right)\xi _{B0}^{0\Delta } + X_0^{\left( 1 \right)}\left( {{R_l}a_l^\Delta - B} \right) + {G^{0\Delta }} + \\ &\qquad \mu \left( t \right)\left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }}} \right)a_l^\Delta \xi _{B0}^{0\Delta } + {\varLambda _l}\left( {\xi _{Bl}^{0\sigma } - a_l^{\Delta \sigma }\xi _{B0}^{0\sigma }} \right) = 0 \end{split}$$ (9) 则Lie对称性导致精确不变量
$$ \begin{split} &{I_0} = {R_l}\xi _{Bl}^0 - \mu \left( t \right)\left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)\xi _{B0}^0 - B\xi _{B0}^0 + {G^0} +\\ &\qquad \int_{{t_1}}^t \Bigg\{ {{\Bigg[ {B + \mu \left( \tau \right)\left( {\frac{{\partial {R_l}}}{{\partial \tau }}a_l^\Delta - \frac{{\partial B}}{{\partial \tau }}} \right)} \Bigg]}^\Delta } + \\ &\qquad \frac{{\partial {R_l}}}{{\partial \tau }}a_l^\Delta - \frac{{\partial B}}{{\partial \tau }} - {\varLambda _l}a_l^{\Delta \sigma } \Bigg\}\xi _{B0}^{0\sigma }\Delta \tau \end{split} $$ (10) 其中, $ a_l^{\Delta \sigma } = {\left( {a_l^\Delta } \right)^\sigma } $, $ \mu \left( t \right) = \sigma \left( t \right) - t $, $ t \in \mathbb{T} $.
证明 根据定义1有
$$ \begin{split} &{{\Delta {I_0}} /{\Delta t}} = R_l^\Delta \xi _{Bl}^{0\sigma } + {R_l}\xi _{Bl}^{0\Delta } - \mu \left( t \right)\left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)\xi _{B0}^{0\Delta } - \\ &\qquad B\xi _{B0}^{0\Delta } + {G^{0\Delta }} + \frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta \xi _{B0}^{0\sigma } - \frac{{\partial B}}{{\partial t}}\xi _{B0}^{0\sigma } - {\varLambda _l}a_l^{\Delta \sigma }\xi _{B0}^{0\sigma } = \\ &\qquad R_l^\Delta \xi _{Bl}^{0\sigma } + {R_l}\xi _{Bl}^{0\Delta } + \left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)\xi _{B0}^0 - B\xi _{B0}^{0\Delta } +\\ &\qquad {G^{0\Delta }} - {\varLambda _l}a_l^{\Delta \sigma }\xi _{B0}^{0\sigma }\\[-1pt]\end{split} $$ (11) 注意到
$$ \begin{split} &X_0^{\left( 1 \right)}\left( {{R_l}a_l^\Delta - B} \right) = X_0^{\left( 0 \right)}\left( {{R_l}a_l^\Delta - B} \right) + \\ &\qquad \left( {\xi _{Bl}^{0\Delta } - a_l^\Delta \xi _{B0}^{0\Delta }} \right)\left( {{R_l} + a_\nu ^\Delta \frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}\frac{{\partial a_l^\sigma }}{{\partial a_l^\Delta }} - \frac{{\partial B}}{{\partial a_l^\sigma }}\frac{{\partial a_l^\sigma }}{{\partial a_l^\Delta }}} \right) = \\ &\qquad \left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)\xi _{B0}^0 + \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }}} \right)\xi _{Bl}^{0\sigma } +\\ &\qquad {R_l}\left( {\xi _{Bl}^{0\Delta } - a_l^\Delta \xi _{B0}^{0\Delta }} \right) - \mu a_l^\Delta \xi _{B0}^{0\Delta }\left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }}} \right) \end{split}$$ (12) 将式(9)代入式(11), 再先后利用式(12)和式(5)可得
$$ \begin{split} &{{\Delta {I_0}} /{\Delta t}} = R_l^\Delta \xi _{Bl}^{0\sigma } + {R_l}\xi _{Bl}^{0\Delta } + \left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)\xi _{B0}^0 -\\ &\qquad {R_l}a_l^\Delta \xi _{B0}^{0\Delta } - X_0^{\left( 1 \right)}\left( {{R_l}a_l^\Delta - B} \right) - \mu a_l^\Delta \xi _{B0}^{0\Delta }\left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }}} \right) - \end{split}$$ $$ \begin{split} &\qquad {\varLambda _l}\xi _{Bl}^{0\sigma } = R_l^\Delta \xi _{Bl}^{0\sigma } - \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }}} \right)\xi _{Bl}^{0\sigma } - {\varLambda _l}\xi _{Bl}^{0\sigma } =\\ &\qquad - \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }} - R_l^\Delta + {\varLambda _l}} \right)\xi _{Bl}^{0\sigma } = 0 \end{split}$$ 注1 定理1中的$ \mu \left(t\right):\mathbb{T}\to \left[0,\infty \right) $称为步差函数, 详细可参考文献[4].
1.3 Lie对称性摄动和绝热不变量
首先, 给出绝热不变量的定义.
定义2 若$ {I_z}\left( {t,{a_\nu },a_\nu ^\sigma ,a_\nu ^\Delta ,\varepsilon } \right) $是时间尺度上力学系统的一个含有小参数$ \varepsilon $的最高次幂为$ z $的物理量, 且$ {{\Delta {I_z}} \mathord{\left/ {\vphantom {{\Delta {I_z}} {\Delta t}}} \right. } {\Delta t}} $与$ {\varepsilon ^{z + 1}} $成比例关系, 则称$ {I_z} $为该系统的$ z $阶绝热不变量.
受小干扰力$ \varepsilon {W_{Bl}}\left( {t,a_\varpi ^\sigma } \right) $作用的广义Birkhoff系统运动微分方程为
$$ \frac{{\partial {R_\nu }\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }} - R_l^\Delta + {\varLambda _l}\left( {t,a_\varpi ^\sigma } \right) = \varepsilon {W_{Bl}}\left( {t,a_\varpi ^\sigma } \right) $$ (13) 此时, 若考虑扰动系统的生成元和规范函数都是在未受扰动状态基础上发生的相应改变, 则有
$$\left.\begin{split} &{\xi _{B0}} = \xi _{B0}^0 + \varepsilon \xi _{B0}^1 + {\varepsilon ^2}\xi _{B0}^2 + \cdots = {\varepsilon ^m}\xi _{B0}^m \\ &{\xi _{Bl}} = \xi _{Bl}^0 + \varepsilon \xi _{Bl}^1 + {\varepsilon ^2}\xi _{Bl}^2 + \cdots = {\varepsilon ^m}\xi _{Bl}^m \\ &G = {G^0} + \varepsilon {G^1} + {\varepsilon ^2}{G^2} + \cdots = {\varepsilon ^m}{G^m}\end{split}\right\} $$ (14) 其中, $ m = 0,1,2, \cdots $. 此时无限小变换可表示为
$$\left. \begin{split} &\bar t = t + \upsilon {\xi _{B0}}\left( {t,{a_\varpi }} \right) + o\left( \upsilon \right) \\ &{\bar a_l}\left( {\bar t} \right) = {a_l}\left( t \right) + \upsilon {\xi _{Bl}}\left( {t,{a_\varpi }} \right) + o\left( \upsilon \right)\end{split} \right\}$$ (15) 且Lie对称性确定方程为
$$ \begin{split} &{X^{\left( 1 \right)}}\left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}} \right)a_\nu ^\Delta + \left( {\xi _{B\nu }^\Delta - a_\nu ^\Delta \xi _{B0}^\Delta } \right)\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }} - {X^{\left( 1 \right)}}\left( {\frac{{\partial B}}{{\partial a_l^\sigma }}} \right) -\\ &\qquad {X^{\left( 1 \right)}}\left( {R_l^\Delta } \right) + {X^{\left( 1 \right)}}\left( {{\varLambda _l}} \right) = \varepsilon {X^{\left( 1 \right)}}\left( {{W_{Bl}}} \right)\end{split} $$ (16) 其中
$$ \left.\begin{split} &{X^{\left( 0 \right)}} = {\xi _{B0}}\frac{\partial }{{\partial t}} + {\xi _{Bl}}\frac{\partial }{{\partial {a_l}}} = {\varepsilon ^m}\left( {\xi _{B0}^m\frac{\partial }{{\partial t}} + \xi _{Bl}^m\frac{\partial }{{\partial {a_l}}}} \right) = {\varepsilon ^m}X_m^{\left( 0 \right)} \\ &{X^{\left( 1 \right)}} = {X^{\left( 0 \right)}} + \left( {\xi _{Bl}^\Delta - a_l^\Delta \xi _{B0}^\Delta } \right)\frac{\partial }{{\partial a_l^\Delta }} =\\ &\qquad {\varepsilon ^m}\left[ {X_m^{\left( 0 \right)} + \left( {\xi _{Bl}^{m\Delta } - a_l^\Delta \xi _{B0}^{m\Delta }} \right)\frac{\partial }{{\partial a_l^\Delta }}} \right] = {\varepsilon ^m}X_m^{\left( 1 \right)}\\[-1pt]\end{split}\right\} $$ (17) 将式(17)和式(14)代入式(16)并保留$ {\varepsilon ^m} $项的系数, 则有
$$ \begin{split} &X_m^{\left( 1 \right)}\left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}} \right)a_\nu ^\Delta + \left( {\xi _{B\nu }^{m\Delta } - a_\nu ^\Delta \xi _{B0}^{m\Delta }} \right)\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }} - X_m^{\left( 1 \right)}\left( {\frac{{\partial B}}{{\partial a_l^\sigma }}} \right) - \\ &\qquad X_m^{\left( 1 \right)}\left( {R_l^\Delta } \right) + X_m^{\left( 1 \right)}\left( {{\varLambda _l}} \right) = X_{m - 1}^{\left( 1 \right)}\left( {{W_{Bl}}} \right)\end{split} $$ (18) 其中, 约定$ \xi _{Bl}^{ - 1} = \xi _{B0}^{ - 1} = 0 $, 从而$ X_{ - 1}^{\left( 1 \right)}\left( {{W_{Bl}}} \right) = 0 $.
定理2 对于受扰动的时间尺度上广义Birkhoff系统(方程(13)), 若存在规范函数$ {G^m}\left( {t,a_\varpi ^\sigma } \right) $使得Lie对称性生成元$ \xi _{B0}^m $, $ \xi _{Bl}^m $满足结构方程
$$\begin{split} &\left( {{R_l}a_l^\Delta - B} \right)\xi _{B0}^{m\Delta } + X_m^{\left( 1 \right)}\left( {{R_l}a_l^\Delta - B} \right) +\\ &\qquad \mu \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }}} \right)a_l^\Delta \xi _{B0}^{m\Delta } + {G^{m\Delta }} +\\ &\qquad {\varLambda _l}\left( {\xi _{Bl}^{m\sigma } - a_l^{\Delta \sigma }\xi _{B0}^{m\sigma }} \right) - {W_{Bl}}\left( {\xi _{Bl}^{\left( {m - 1} \right)\sigma } - a_l^\Delta \xi _{B0}^{\left( {m - 1} \right)\sigma }} \right) = 0 \end{split}$$ (19) 则该系统存在一个$ z $阶绝热不变量
$$ \begin{split} &{I_z} = \sum\limits_{m = 0}^z {{\varepsilon ^m}\left\{ {{R_l}\xi _{Bl}^m - \mu \left( t \right)\left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)\xi _{B0}^m - B\xi _{B0}^m} \right.} +\\ &\qquad {G^m} + \int_{{t_1}}^t \Bigg\{ {\left[ {B + \mu \left( \tau \right)\left( {\frac{{\partial {R_l}}}{{\partial \tau }}a_l^\Delta - \frac{{\partial B}}{{\partial \tau }}} \right) - {R_l}a_l^\Delta } \right]}^\Delta + \\ &\qquad R_l^\sigma {{\left( {a_l^\Delta } \right)}^\Delta } + \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }} + {\varLambda _l}} \right)a_l^\Delta + \\ &\qquad \left.\frac{{\partial {R_l}}}{{\partial \tau }}a_l^\Delta - \frac{{\partial B}}{{\partial \tau }} - {\varLambda _l}a_l^{\Delta \sigma } \Bigg\}\xi _{B0}^{m\sigma }\Delta \tau \right\} \end{split}$$ (20) 证明 根据定义2有
$$ \begin{split} &{{\Delta {I_z}} /{\Delta t}} = \sum\limits_{m = 0}^z {\varepsilon ^m}\left\{ {R_l^\Delta \xi _{Bl}^{m\sigma } + {R_l}\xi _{Bl}^{m\Delta } - \mu \left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)\xi _{B0}^{m\Delta }} \right. - \\ &\qquad {\left[ {\mu \left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)} \right]^\Delta }\xi _{B0}^{m\sigma } - {B^\Delta }\xi _{B0}^{m\sigma } - B\xi _{B0}^{m\Delta } + {G^{m\Delta }} +\\ &\qquad {B^\Delta }\xi _{B0}^{m\sigma } + {\left[ {\mu \left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)} \right]^\Delta }\xi _{B0}^{m\sigma } - {\left( {{R_l}a_l^\Delta } \right)^\Delta }\xi _{B0}^{m\sigma } +\\ &\qquad \left[ {R_l^\sigma {{\left( {a_l^\Delta } \right)}^\Delta } - {\varLambda _l}a_l^{\Delta \sigma }} + \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }} + {\varLambda _l}} \right)a_l^\Delta +\right.\\ &\qquad \left.\left.\left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right) \right]\xi _{B0}^{m\sigma } \right\} = \sum\limits_{m = 0}^z {\varepsilon ^m}\Bigg\{ R_l^\Delta \xi _{Bl}^{m\sigma } + {R_l}\xi _{Bl}^{m\Delta } +\\ &\qquad \left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)\xi _{B0}^m - B\xi _{B0}^{m\Delta } + {G^{m\Delta }} - \\ &\qquad \left[ {R_l^\Delta a_l^\Delta - \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }} + {\varLambda _l}} \right)a_l^\Delta + {\varLambda _l}a_l^{\Delta \sigma }} \right]\xi _{B0}^{m\sigma }\Bigg\}\end{split}$$ (21) 将式(19)代入式(21)且注意到
$$ \begin{split} &X_m^{\left( 1 \right)}\left( {{R_l}a_l^\Delta - B} \right) = X_m^{\left( 0 \right)}\left( {{R_l}a_l^\Delta - B} \right) + \\ &\qquad \left( {\xi _{Bl}^{m\Delta } - a_l^\Delta \xi _{B0}^{m\Delta }} \right)\left( {{R_l} + a_\nu ^\Delta \frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}\frac{{\partial a_l^\sigma }}{{\partial a_l^\Delta }} - \frac{{\partial B}}{{\partial a_l^\sigma }}\frac{{\partial a_l^\sigma }}{{\partial a_l^\Delta }}} \right) = \end{split} $$ $$ \begin{split} &\qquad \left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)\xi _{B0}^m + \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }}} \right)\xi _{Bl}^{m\sigma } +\\ &\qquad {R_l}\left( {\xi _{Bl}^{m\Delta }} \right. \left. { - a_l^\Delta \xi _{B0}^{m\Delta }} \right) - \mu a_l^\Delta \xi _{B0}^{m\Delta }\left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }}} \right)\end{split} $$ (22) 于是式(21)可写为
$$ \begin{split} &{{\Delta {I_z}} /{\Delta t}} = \sum\limits_{m = 0}^z {{\varepsilon ^m}\Bigg\{ {R_l^\Delta \xi _{Bl}^{m\sigma } + \varepsilon {W_{Bl}}a_l^\Delta \xi _{B0}^{m\sigma } - {\varLambda _l}\xi _{Bl}^{m\sigma }} } +\\ &\qquad { {W_{Bl}}\left[ {\xi _{Bl}^{\left( {m - 1} \right)\sigma } - a_l^\Delta \xi _{B0}^{\left( {m - 1} \right)\sigma }} \right] - \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }}} \right)\xi _{Bl}^{m\sigma }} \Bigg\} = \\ &\qquad \sum\limits_{m = 0}^z {\varepsilon ^m}\Bigg\{ {W_{Bl}}\left[ {\xi _{Bl}^{\left( {m - 1} \right)\sigma } - a_l^\Delta \xi _{B0}^{\left( {m - 1} \right)\sigma }} \right] -\\ &\qquad \varepsilon {W_{Bl}}\left( {\xi _{Bl}^{m\sigma } - a_l^\Delta \xi _{B0}^{m\sigma }} \right) \Bigg\}= - {\varepsilon ^{z + 1}}{W_{Bl}}\left( {\xi _{Bl}^{z\sigma } - a_l^\Delta \xi _{B0}^{z\sigma }} \right)\end{split} $$ 上式与$ {\varepsilon ^{z + 1}} $成比例, 本定理结论有效. 此外, 注意到若$ {W_{Bl}} = 0 $, 则$ {{\Delta {I_z}} \mathord{\left/ {\vphantom {{\Delta {I_z}} {\Delta t}}} \right. } {\Delta t}} = 0 $, 此时绝热不变量即为精确不变量.
注2 无扰动时的运动微分方程(5)和Lie对称性确定方程(7)、受扰动后的运动微分方程(13)和确定方程(16)与文献[23]一致, 但由于定理1和定理2采用的是Anerot等的方法, 故所得结果与文献[23]不同.
1.4 举例说明
例1 系统的广义Birkhoff函数$ B $, Birkhoff函数组$ {R_\mu } $和附加项$ {\varLambda _l} $分别为
$$\left. \begin{split} &B = \frac{1}{2}{\left({a}_{3}^{\sigma }\right)}^{2} + {a}_{2}^{\sigma }\text{, }\;\;{R}_{1} = {a}_{3}^{\sigma }\text{, }\;\;{R}_{2} = {a}_{4}^{\sigma }\text{, }\;\;{R}_{3} = {R}_{4} = 0\\ &{\varLambda }_{1} = {\varLambda }_{2} = {\varLambda }_{3} = 0\text{, }\;\;{\varLambda }_{4} = -{a}_{4}^{\sigma }\end{split}\right\} $$ (23) 基于时间尺度$ {\mathbb{T}} = \left\{ {{2^{n + 1}}:n \in \mathbb{N}} \right\} $, 我们讨论该系统Lie对称性的摄动.
首先, 根据文献[4], 令$ t = {2^{n + 1}} $易得前跳算子和步差函数分别为
$$ \left.\begin{split} &\sigma \left( t \right) = \inf \left\{ {s \in \mathbb{T}:s > {2^{n + 1}}} \right\} = {2^{n + 2}} = 2t \\ &\mu \left( t \right) = \sigma \left( t \right) - t = t\end{split}\right\} $$ (24) 由式(5)可得该系统的方程
$$ a_3^{\sigma \Delta } = 0 \text{, }\;\; a_4^{\sigma \Delta } + 1 = 0 \text{, }\;\; a_1^\Delta - a_3^\sigma = 0 \text{, }\;\; a_2^\Delta - a_4^\sigma = 0 $$ (25) 其次, 取满足Lie对称性确定方程(7)的生成元
$$ \xi _{B0}^0 = a_3^\sigma \text{, }\;\; \xi _{B1}^0 = \frac{1}{2}{a_2} \text{, }\;\; \xi _{B2}^0 = 0 \text{, }\;\; \xi _{B3}^0 = \frac{1}{2}{a_4} \text{, }\;\; \xi _{B4}^0 = {\text{0}} $$ (26) 结构方程(9)给出
$$ \begin{split} &\left( {a_1^\Delta a_3^\sigma + a_2^\Delta a_4^\sigma - B} \right)\xi _{B0}^{0\Delta } + X_0^{\left( 1 \right)}\left( {a_1^\Delta a_3^\sigma + a_2^\Delta a_4^\sigma - B} \right) +\\ &\qquad {G^{0\Delta }} - \mu \xi _{B0}^{0\Delta }\left[ {a_2^\Delta - \left( {a_1^\Delta - a_3^\sigma } \right)a_3^\Delta - a_2^\Delta a_4^\Delta } \right] -\\ &\qquad a_4^\sigma \left( {\xi _{B4}^{0\sigma } - a_4^{\Delta \sigma }\xi _{B0}^{0\sigma }} \right) = 0\end{split} $$ (27) 将生成元式(26)代入式(27)可得$ {G^0} = 0 $, 于是根据定理1可计算精确不变量
$$ \begin{split} &{I_0} = \frac{1}{2}{a_2}a_3^\sigma - a_2^\sigma a_3^\sigma - \frac{1}{2}{\left( {a_3^\sigma } \right)^3} + \\ &\qquad \int_{{t_1}}^t {\left\{ {{{\left[ {\frac{1}{2}{{\left( {a_3^\sigma } \right)}^2} + a_2^\sigma } \right]}^\Delta } + a_4^\sigma a_4^{\Delta \sigma }} \right\}a_3^\sigma \Delta\tau }\end{split} $$ (28) 此时, 以$ {a_1}\left( 1 \right) = {a_2}\left( 1 \right) = {a_3}\left( 2 \right) = 1 $, $ {a_4}\left( 2 \right) = 2 $作为初始值, 便可精确计算$ {a_2},{a_3},{a_4} $及$ {I_0} $的具体数值(如图1所示). 据此可知, 由式(28)所确定的$ {I_0} $始终不变, 确实是守恒量.
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图 1 例1中$ {a_2},{a_3},{a_4}和{I_0} $的值Figure 1. Simulations of $ {a_2},{a_3},{a_4}\;{\mathrm{and}}\;{I_0} $ in example 1下面考虑干扰力
$$ \varepsilon {W_{B1}} = \varepsilon {W_{B3}} = 0 \text{, }\;\; \varepsilon {W_{B2}} = \varepsilon a_4^\Delta \text{, }\;\; \varepsilon {W_{B4}} = - \varepsilon a_2^\Delta $$ (29) 此时方程变为
$$ a_3^{\sigma \Delta } = 0 \text{, }\;\; a_4^{\sigma \Delta } + 1 = \varepsilon a_4^\Delta \text{, }\;\; a_1^\Delta - a_3^\sigma = 0 \text{, }\;\; a_2^\Delta - a_4^\sigma = - \varepsilon a_2^\Delta $$ (30) 取满足Lie对称性确定方程(18)的生成元
$$ \xi _{B0}^1 = 0 \text{, }\;\; \xi _{B1}^1 = a_3^\sigma \text{, }\;\; \xi _{B2}^1 = \xi _{B3}^1 = \xi _{B4}^1 = 0 $$ (31) 此时, 结构方程(19)有
$$ \begin{split} &\left( {a_1^\Delta a_3^\sigma + a_2^\Delta a_4^\sigma - B} \right)\xi _{B0}^{1\Delta } + X_1^{\left( 1 \right)}\left( {a_1^\Delta a_3^\sigma + a_2^\Delta a_4^\sigma - B} \right) +\\ &\qquad {G^{1\Delta }} - \mu \xi _{B0}^{1\Delta }\left[ {a_2^\Delta - \left( {a_1^\Delta - a_3^\sigma } \right)a_3^\Delta - a_2^\Delta a_4^\Delta } \right] - \\ &\qquad a_4^\sigma \left( {\xi _{B4}^{1\sigma } - a_4^{\Delta \sigma }\xi _{B0}^{1\sigma }} \right) - a_4^\Delta \left( {\xi _{B2}^{0\sigma } - a_2^\Delta \xi _{B0}^{0\sigma }} \right) +\\ &\qquad a_2^\Delta \left( {\xi _{B4}^{0\sigma } - a_4^\Delta \xi _{B0}^{0\sigma }} \right) = 0\end{split} $$ (32) 生成元式(31)代入式(32)得$ {G^1} = 0 $, 因此根据定理2求得一阶绝热不变量
$$ \begin{split} &{I_1} = \frac{1}{2}{a_2}a_3^\sigma - a_2^\sigma a_3^\sigma - \frac{1}{2}{\left( {a_3^\sigma } \right)^3} + \\ &\qquad \int_{{t_1}}^t {\left\{ {{{\left[ {\frac{1}{2}{{\left( {a_3^\sigma } \right)}^2} + a_2^\sigma } \right]}^\Delta } + a_4^\sigma a_4^{\Delta \sigma }} \right\}a_3^\sigma \Delta \tau } + \varepsilon {\left( {a_3^\sigma } \right)^2}\end{split} $$ (33) 2. 时间尺度上受约束Birkhoff系统
2.1 建立方程
此时Birkhoff系统中的变量$ a_\varpi ^\sigma $不是彼此独立的, 而是受到一些约束
$$ {f}_{\beta }\left(t,{a}_{\varpi }^{\sigma }\right) = 0\text{, }\;\;\beta = 1,2,\cdots ,2k\text{, }\;\;k\in {\mathbb{N}} $$ (34) 从而无法导出Birkhoff方程. 上式对虚位移$ \delta a_l^\sigma $的限制表为
$$ \frac{\partial {f}_{\beta }}{\partial {a}_{l}^{\sigma }}\cdot \delta {a}_{l}^{\sigma } = 0\text{, }\;\;\beta = 1,2,\cdots ,2k\text{, }\;\;k\in \mathbb{N} $$ (35) 再利用Lagrange乘子法及Dubois-Reymond引理, 易得时间尺度上带乘子的方程[16]
$$ \frac{{\partial {R_\nu }\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }} \cdot a_\nu ^\Delta - \frac{{\partial B\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }} - R_l^\Delta = {\lambda _\beta }\frac{{\partial {f_\beta }\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }} $$ (36) 其中, $ {\lambda _\beta } = {\lambda _\beta }\left( {t,a_\varpi ^\sigma } \right) $称为约束乘子, $ t \in {\left[ {{t_1},{t_2}} \right]^\kappa } $, $ \nu ,l, \varpi = 1,2, \cdots ,2 n $, $ n \in \mathbb{N} $. 若通过方程(34)和方程(36)可以解出$ {\lambda _\beta } $, 方程(36)可表示为
$$ \frac{{\partial {R_\nu }\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }} \cdot a_\nu ^\Delta - \frac{{\partial B\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }} - R_l^\Delta = {Q_l}\left( {t,a_\varpi ^\sigma } \right) $$ (37) 其中, $ {Q_l} = {\lambda _\beta }\dfrac{{\partial {f_\beta }\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }} $, 称方程(37)为与约束Birkhoff系统(方程(34)和方程(36))相应的自由Birkhoff系统的运动方程. 受约束Birkhoff系统是一种特殊的广义Birkhoff系统, 对比广义Birkhoff系统方程(5)和自由Birkhoff系统方程(37)发现, 两者仅“附加项”不同, 因此在自由Birkhoff系统(方程(37))中, 相应的结论可自然得到.
2.2 Lie对称性和精确不变量
无限小变换、$ X_0^{\left( 0 \right)} $和$ X_0^{\left( 1 \right)} $仍采用1.2节中的式(6)和式(8), 于是在时间尺度上自由Birkhoff系统(方程(37))中, Lie对称性确定方程为
$$ \begin{split} &X_0^{\left( 1 \right)}\left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}} \right)a_\nu ^\Delta + \left( {\xi _{B\nu }^{0\Delta } - a_\nu ^\Delta \xi _{B0}^{0\Delta }} \right)\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }} -\\ &\qquad X_0^{\left( 1 \right)}\left( {\frac{{\partial B}}{{\partial a_l^\sigma }}} \right) - X_0^{\left( 1 \right)}\left( {R_l^\Delta } \right) - X_0^{\left( 1 \right)}\left( {{Q_l}} \right) = 0\end{split} $$ (38) 结构方程为
$$ \begin{split} &\left( {{R_l}a_l^\Delta - B} \right)\xi _{B0}^{0\Delta } + X_0^{\left( 1 \right)}\left( {{R_l}a_l^\Delta - B} \right) + G_B^{0\Delta } + \\ &\qquad \mu \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }} \cdot a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }}} \right)a_l^\Delta \xi _{B0}^{0\Delta } - {Q_l}\left( {\xi _{Bl}^{0\sigma } - a_l^{\Delta \sigma }\xi _{B0}^{0\sigma }} \right) = 0 \end{split}$$ (39) 精确不变量为
$$ \begin{split} &{I_0} = {R_l}\xi _{Bl}^0 - \mu \left( t \right)\left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)\xi _{B0}^0 - B\xi _{B0}^0 + G_B^0 +\\ &\qquad \int_{{t_1}}^t \Bigg\{\left[ B { + \mu \left( \tau \right)\left( {\frac{{\partial {R_l}}}{{\partial \tau }}a_l^\Delta - \frac{{\partial B}}{{\partial \tau }}} \right)} \right]^\Delta + \frac{{\partial {R_l}}}{{\partial \tau }}a_l^\Delta - \\ &\qquad \left.\frac{{\partial B}}{{\partial \tau }} + {Q_l}a_l^{\Delta \sigma } \right\}\xi _{B0}^{0\sigma }\Delta \tau \end{split}$$ (40) 于是自然得到如下定理.
定理3 对于时间尺度上自由Birkhoff系统(方程(37)), 若存在规范函数$ G_B^0 $使得Lie对称性的生成元$ \xi _{B0}^0 $和$ \xi _{Bl}^0 $满足结构方程(39), 则Lie对称性导致精确不变量式(40).
但对于时间尺度上约束Birkhoff系统(方程(34)和方程(36)), 还要考虑约束方程(34). 在无限小变换(6)下, 其不变性归于满足限制方程
$$ X_0^{\left( 1 \right)}\left[ {{f_\beta }\left( {t,a_\varpi ^\sigma } \right)} \right] = 0 $$ (41) 又注意到在导出方程(36)的过程中用到了方程(35), 因此也对$ \xi _{B0}^0 $和$ \xi _{Bl}^0 $的选取有所限制, 于是应用$ \delta {a_l} = \Delta {a_l} - a_l^\Delta \Delta t $, 方程(35)可归为附加限制方程
$$ \frac{\partial {f}_{\beta }}{\partial {a}_{l}^{\sigma }}\left({\xi }_{Bl}^{0\sigma }-{a}_{l}^{\Delta \sigma }{\xi }_{B0}^{0\sigma }\right) = 0\text{, }\;\;\beta = 1,2,\cdots ,2k\text{, }\;\;k\in {\mathbb{N}} $$ (42) 此时可得如下定义及定理.
定义3 若Lie对称性的生成元$ \xi _{B0}^0 $和$ \xi _{Bl}^0 $满足限制方程(41), 则相应对称性称为时间尺度上约束Birkhoff系统(方程(34)和方程(36))的弱Lie对称性.
定义4 若Lie对称性的生成元$ \xi _{B0}^0 $和$ \xi _{Bl}^0 $满足限制方程(41)以及附加限制方程(42), 则相应对称性称为时间尺度上约束Birkhoff系统(方程(34)和方程(36))的强Lie对称性.
定理4 对于时间尺度上约束Birkhoff系统(方程(34)和方程(36)), 若存在规范函数$ G_B^0 $使得弱(强) Lie对称性的生成元$ \xi _{B0}^0 $和$ \xi _{Bl}^0 $满足结构方程(39), 则该系统存在精确不变量式(40).
2.3 Lie对称性摄动和绝热不变量
本节仍从时间尺度上自由Birkhoff系统入手, 研究方法对照本文1.3节.
受$ {\varepsilon _B}{W_{Cl}}\left( {t,a_\varpi ^\sigma } \right) $作用的自由Birkhoff系统(方程(37))的运动微分方程为
$$ \frac{{\partial {R_\nu }\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }} \cdot a_\nu ^\Delta - \frac{{\partial B\left( {t,a_\varpi ^\sigma } \right)}}{{\partial a_l^\sigma }} - R_l^\Delta - {Q_l}\left( {t,a_\varpi ^\sigma } \right) = {\varepsilon _B}{W_{Cl}}\left( {t,a_\varpi ^\sigma } \right) $$ (43) 此时
$$ \left.\begin{split} &{\xi _{B0}} = \xi _{B0}^0 + {\varepsilon _B}\xi _{B0}^1 + \varepsilon _B^2\xi _{B0}^2 + \cdots = \varepsilon _B^m\xi _{B0}^m \\ &{\xi _{Bl}} = \xi _{Bl}^0 + {\varepsilon _B}\xi _{Bl}^1 + \varepsilon _B^2\xi _{Bl}^2 + \cdots = \varepsilon _B^m\xi _{Bl}^m \\ &{G_B} = G_B^0 + {\varepsilon _B}G_B^1 + \varepsilon _B^2G_B^2 + \cdots = \varepsilon _B^mG_B^m \end{split}\right\}$$ (44) 其中$ m = 0,1,2, \cdots $. 无限小变换、$ {X^{\left( 0 \right)}} $和$ {X^{\left( 1 \right)}} $仍采用1.3节中的式(15)和式(17), 于是在受扰的自由Birkhoff系统(方程(43))中, Lie对称性确定方程为
$$ \begin{split} &{X^{\left( 1 \right)}}\left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}} \right)a_\nu ^\Delta + \left( {\xi _{B\nu }^\Delta - a_\nu ^\Delta \xi _{B0}^\Delta } \right)\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }} - {X^{\left( 1 \right)}}\left( {\frac{{\partial B}}{{\partial a_l^\sigma }}} \right) - \\ &\qquad {X^{\left( 1 \right)}}\left( {R_l^\Delta } \right) - {X^{\left( 1 \right)}}\left( {{Q_l}} \right) = {\varepsilon _B}{X^{\left( 1 \right)}}\left( {{W_{Cl}}} \right) \end{split}$$ (45) 展开可得
$$ \begin{split} &X_m^{\left( 1 \right)}\left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}} \right)a_\nu ^\Delta + \left( {\xi _{B\nu }^{m\Delta } - a_\nu ^\Delta \xi _{B0}^{m\Delta }} \right)\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }} - X_m^{\left( 1 \right)}\left( {\frac{{\partial B}}{{\partial a_l^\sigma }}} \right) - \\ &\qquad X_m^{\left( 1 \right)}\left( {R_l^\Delta } \right) - X_m^{\left( 1 \right)}\left( {{Q_l}} \right) = X_{m - 1}^{\left( 1 \right)}\left( {{W_{Cl}}} \right)\end{split} $$ (46) 其中, 约定$ \xi _{Bl}^{ - 1} = \xi _{B0}^{ - 1} = 0 $, 从而$ X_{ - 1}^{\left( 1 \right)}\left( {{W_{Cl}}} \right) = 0 $.
结构方程为
$$ \begin{split} &\left( {{R_l}a_l^\Delta - B} \right)\xi _{B0}^{m\Delta } + X_m^{\left( 1 \right)}\left( {{R_l}a_l^\Delta - B} \right) + \\ &\qquad \mu \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }}} \right)a_l^\Delta \xi _{B0}^{m\Delta } + G_B^{m\Delta } - \\ &\qquad {Q_l}\left( {\xi _{Bl}^{m\sigma } - a_l^{\Delta \sigma }\xi _{B0}^{m\sigma }} \right) - {W_{Cl}}\left[ {\xi _{Bl}^{\left( {m - 1} \right)\sigma } - a_l^\Delta \xi _{B0}^{\left( {m - 1} \right)\sigma }} \right] = 0\end{split} $$ (47) $ z $阶绝热不变量为
$$ \begin{split} &{I_z} = \sum\limits_{m = 0}^z {\varepsilon _B^m\Bigg\{ {R_l}\xi _{Bl}^m - \mu \left( t \right)\left( {\frac{{\partial {R_l}}}{{\partial t}}a_l^\Delta - \frac{{\partial B}}{{\partial t}}} \right)\xi _{B0}^m - B\xi _{B0}^m} + \\ &\qquad G_B^m + \int_{{t_1}}^t \Bigg\{ {{\left[ {B + \mu \left( \tau \right)\left( {\frac{{\partial {R_l}}}{{\partial \tau }}a_l^\Delta - \frac{{\partial B}}{{\partial \tau }}} \right) - {R_l}a_l^\Delta } \right]}^\Delta } + \\ &\qquad R_l^\sigma {{\left( {a_l^\Delta } \right)}^\Delta } + \left( {\frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }}a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }} - {Q_l}} \right)a_l^\Delta + \\ &\qquad \frac{{\partial {R_l}}}{{\partial \tau }}a_l^\Delta - \frac{{\partial B}}{{\partial \tau }} + {Q_l}a_l^{\Delta \sigma } \Bigg\}\xi _{B0}^{m\sigma }\Delta \tau \Bigg\} \end{split}$$ (48) 于是有如下定理.
定理5 对于时间尺度上受$ {\varepsilon _B}{W_{Cl}} $作用的自由Birkhoff系统(方程(43)), 若存在$ G_B^m $使得Lie对称性的生成元$ \xi _{B0}^m $和$ \xi _{Bl}^m $满足结构方程(47), 则该系统存在一个$ z $阶绝热不变量式(48).
受$ {\varepsilon _B}{W_{Cl}} $作用的方程(36)变为
$$ \frac{{\partial {R_\nu }}}{{\partial a_l^\sigma }} \cdot a_\nu ^\Delta - \frac{{\partial B}}{{\partial a_l^\sigma }} - R_l^\Delta - {\lambda _\beta }\frac{{\partial {f_\beta }}}{{\partial a_l^\sigma }} = {\varepsilon _B}{W_{Cl}}\left( {t,a_\varpi ^\sigma } \right) $$ (49) 此时在时间尺度上受扰约束Birkhoff系统(方程(34)和方程(49))中, 限制方程(41)和附加限制方程(42)分别为
$$\qquad\qquad\quad X_m^{\left( 1 \right)}\left[ {{f_\beta }\left( {t,a_\varpi ^\sigma } \right)} \right] = 0 $$ (50) $$\qquad\qquad\quad \frac{{\partial {f_\beta }}}{{\partial a_l^\sigma }}\left( {\xi _{Bl}^{m\sigma } - a_l^{\Delta \sigma }\xi _{B0}^{m\sigma }} \right) = 0 $$ (51) 定义5 若Lie对称性的生成元$ \xi _{B0}^m $和$ \xi _{Bl}^m $满足限制方程(50), 则相应对称性称为时间尺度上受扰约束Birkhoff系统(方程(34)和方程(49))的弱Lie对称性.
定义6 若Lie对称性的生成元$ \xi _{B0}^m $和$ \xi _{Bl}^m $满足限制方程(50)以及附加限制方程(51), 则相应对称性称为时间尺度上受扰约束Birkhoff系统(方程(34)和方程(49))的强Lie对称性.
定理6 对于时间尺度上受$ {\varepsilon _B}{W_{Cl}} $作用的约束Birkhoff系统(方程(34)和方程(49)), 若存在$ G_B^m $使得弱(强)Lie对称性的生成元$ \xi _{B0}^m $和$ \xi _{Bl}^m $满足结构方程(47), 则该系统存在一个$ z $阶绝热不变量式(48).
2.4 举例说明
例2 对于Birkhoff函数, Birkhoff函数组
$$ \left.\begin{split} &B = \frac{1}{2}{\left( {a_3^\sigma } \right)^2} + {\left( {a_4^\sigma } \right)^2} - ca_1^\sigma \\ &{R}_{1} = {a}_{3}^{\sigma }\text{, }\;\;{R}_{2} = {a}_{4}^{\sigma }\text{, }\;\;{R}_{3} = {R}_{4} = 0 \end{split}\right\}$$ (52) 和约束方程
$$ {f_1} = a_1^\sigma - a_2^\sigma = 0 \text{, }\;\; {f_2} = a_3^\sigma - 2a_4^\sigma = 0 $$ (53) 其中, $ c $是常数, 试在时间尺度$ {\mathbb{T}} = \left\{ {{2^{n + 1}}:n \in \mathbb{N}} \right\} $上讨论该约束Birkhoff系统Lie对称性的摄动.
首先, 本例时间尺度与例1一样, 故$ \sigma \left( t \right) $和$ \mu \left( t \right) $仍为式(24). 由式(36)可得
$$\left. \begin{split} &c - a_3^{\sigma \Delta } = {\lambda _1} \text{, }\;\; - a_4^{\sigma \Delta } = - {\lambda _1} \\ &a_1^\Delta - a_3^\sigma = {\lambda _2} \text{, }\;\; a_2^\Delta - 2a_4^\sigma = - 2{\lambda _2} \end{split}\right\}$$ (54) 根据式(54)和约束(53)可求得约束乘子
$$ {\lambda _1} = \frac{1}{3}c \text{, }\;\; {\lambda _2} = 0 $$ (55) 因此
$$ {Q_1} = - {Q_2} = \frac{1}{3}c \text{, }\;\; {Q_3} = {Q_4} = 0 $$ (56) 易得自由Birkhoff方程为
$$ a_3^{\sigma \Delta } = \frac{2}{3}c \text{, }\;\; a_4^{\sigma \Delta } = \frac{1}{3}c \text{, }\;\; a_1^\Delta - a_3^\sigma = 0 \text{, }\;\; a_2^\Delta - 2a_4^\sigma = 0 $$ (57) 取
$$ \xi _{B0}^0 = 0 \text{, }\;\; \xi _{B1}^0 = - \frac{1}{2} \text{, }\;\; \xi _{B2}^0 = 1 \text{, }\;\; \xi _{B3}^0 = \xi _{B4}^0 = 0 $$ (58) $$ \bar \xi _{B0}^0 = 0 \text{, }\;\; \bar \xi _{B1}^0 = \bar \xi _{B2}^0 = t \text{, }\;\; \bar \xi _{B3}^0 = 1 \text{, }\;\; \bar \xi _{B4}^0 = \frac{1}{2} $$ (59) 显然生成元式(58)和式(59)满足Lie对称性的确定方程(38). 再考虑限制方程(41)
$$ X_0^{\left( 1 \right)}\left( {a_1^\sigma - a_2^\sigma } \right) = 0 \text{, }\;\; X_0^{\left( 1 \right)}\left( {a_3^\sigma - 2a_4^\sigma } \right) = 0 $$ (60) 和附加限制方程(42)
$$ \left.\begin{split} &\xi _{B1}^{0\sigma } - a_1^{\Delta \sigma }\xi _{B0}^{0\sigma } - \xi _{B2}^{0\sigma } + a_2^{\Delta \sigma }\xi _{B0}^{0\sigma } = 0 \\ &\xi _{B3}^{0\sigma } - a_3^{\Delta \sigma }\xi _{B0}^{0\sigma } - 2\xi _{B4}^{0\sigma } + 2a_4^{\Delta \sigma }\xi _{B0}^{0\sigma } = 0 \end{split}\right\}$$ (61) 生成元(59)满足方程(60)和方程(61), 而生成元式(58)不满足, 因此生成元式(59)对应约束Birkhoff系统(方程(53)和式(54))的强Lie对称性, 生成元式(58)对应自由Birkhoff系统(方程(57))的Lie对称性.
结构方程(39)给出
$$ \begin{split} &\left( {a_3^\sigma a_1^\Delta + a_4^\sigma a_2^\Delta - B} \right)\xi _{B0}^{0\Delta } + X_0^{\left( 1 \right)}\left( {a_3^\sigma a_1^\Delta + a_4^\sigma a_2^\Delta - B} \right) + \\ &\qquad \mu \xi _{B0}^{0\Delta }\left[ {ca_1^\Delta + \left( {a_1^\Delta - a_3^\sigma } \right)a_3^\Delta + \left( {a_2^\Delta - 2a_4^\sigma } \right)a_4^\Delta } \right] +\\ &\qquad G_B^{0\Delta } - \frac{1}{3}c\left( {\xi _{B1}^{0\sigma } - a_1^{\Delta \sigma }\xi _{B0}^{0\sigma }} \right) + \frac{1}{3}c\left( {\xi _{B2}^{0\sigma } - a_2^{\Delta \sigma }\xi _{B0}^{0\sigma }} \right) = 0 \end{split}$$ (62) 对生成元式(58)和式(59), 式(62)分别给出
$$ G_B^0 = 0 \text{, }\;\; \bar G_B^0 = - {a_1} - \frac{1}{2}{a_2} - \frac{2}{3}c{t^2} $$ (63) 因此根据定理3和定理4可分别得到精确不变量
$$ {I_0} = - \frac{1}{2}a_3^\sigma + a_4^\sigma \text{, }\;\; {\bar I_0} = ta_3^\sigma + ta_4^\sigma - {a_1} - \frac{1}{2}{a_2} - \frac{2}{3}c{t^2} $$ (64) 此时, 令$ c = 3 $, 以$ {a_1}\left( 1 \right) = {a_4}\left( 2 \right) = 1 $作为初始值, 可计算$ {a_2},{a_3},{a_4} $及$ {I_0},{\bar I_0} $的精确数值, 显示为图2. 由此可见, 由式(64)所确定的$ {I_0}和{\bar I_0} $确实是守恒量.
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图 2 例2中$ {a_2},{a_3},{a_4},{I_0}和{\bar I_0} $的值Figure 2. Simulations of $ {a_2},{a_3},{a_4},{I_0}\;{\mathrm{and}}\;{\bar I_0} $in example 2假设干扰力
$$ {\varepsilon _B}{W_{C1}} = {\varepsilon _B}{W_{C2}} = 0 \text{, }\;\; {\varepsilon _B}{W_{C3}} = {\varepsilon _B}a_3^\Delta \text{, }\;\; {\varepsilon _B}{W_{C4}} = {\varepsilon _B}a_4^\Delta $$ (65) 此时受扰自由Birkhoff方程为
$$ \left.\begin{split} &a_3^{\sigma \Delta } = \frac{2}{3}c \text{, }\;\; a_4^{\sigma \Delta } = \frac{1}{3}c \\ &a_1^\Delta - a_3^\sigma = {\varepsilon _B}a_3^\Delta \text{, }\;\; a_2^\Delta - 2a_4^\sigma = {\varepsilon _B}a_4^\Delta\end{split}\right\} $$ (66) 取满足Lie对称性确定方程(46)的生成元
$$ \xi _{B0}^1 = 0 \text{, }\;\; \xi _{B1}^1 = 1 \text{, }\;\; \xi _{B2}^1 = t \text{, }\;\; \xi _{B3}^1 = 0 \text{, }\;\; \xi _{B4}^1 = \frac{1}{2} $$ (67) $$ \bar \xi _{B0}^1 = 0 \text{, }\;\; \bar \xi _{B1}^1 = \bar \xi _{B2}^1 = 1 \text{, }\;\; \bar \xi _{B3}^1 = \bar \xi _{B4}^1 = 0 $$ (68) 受干扰力式(65)作用的方程(54)为
$$\left.\begin{split} &c - a_3^{\sigma \Delta } - {\lambda _1} = 0 \text{, }\;\; - a_4^{\sigma \Delta } + {\lambda _1} = 0 \\ &a_1^\Delta - a_3^\sigma - {\lambda _2} = {\varepsilon _B}a_3^\Delta \text{, }\;\; a_2^\Delta - 2a_4^\sigma + 2{\lambda _2} = {\varepsilon _B}a_4^\Delta \end{split}\right\}$$ (69) 此时限制方程(50)为
$$ X_1^{\left( 1 \right)}\left( {a_1^\sigma - a_2^\sigma } \right) = 0 \text{, }\;\; X_1^{\left( 1 \right)}\left( {a_3^\sigma - 2a_4^\sigma } \right) = 0 $$ (70) 附加限制方程(51)为
$$ \left.\begin{split} &\xi _{B1}^{1\sigma } - a_1^{\Delta \sigma }\xi _{B0}^{1\sigma } - \xi _{B2}^{1\sigma } + a_2^{\Delta \sigma }\xi _{B0}^{1\sigma } = 0 \\ &\xi _{B3}^{1\sigma } - a_3^{\Delta \sigma }\xi _{B0}^{1\sigma } - 2\xi _{B4}^{1\sigma } + 2a_4^{\Delta \sigma }\xi _{B0}^{1\sigma } = 0\end{split}\right\} $$ (71) 生成元式(68)满足方程(70)和方程(71), 而生成元(67)不满足, 因此生成元式(68)对应受扰约束Birkhoff系统(方程(53)和方程(69))的强Lie对称性, 生成元式(67)对应受扰自由Birkhoff系统(方程(66))的Lie对称性.
结构方程(47)给出
$$ \begin{split} &\left( {a_3^\sigma a_1^\Delta + a_4^\sigma a_2^\Delta - B} \right)\xi _{B0}^{1\Delta } + X_1^{\left( 1 \right)}\left( {a_3^\sigma a_1^\Delta + a_4^\sigma a_2^\Delta - B} \right) +\\ &\qquad G_B^{1\Delta } + \mu \xi _{B0}^{1\Delta }\left[ {ca_1^\Delta + \left( {a_1^\Delta - a_3^\sigma } \right)a_3^\Delta + \left( {a_2^\Delta - 2a_4^\sigma } \right)a_4^\Delta } \right] - \\ &\qquad \frac{1}{3}c\left( {\xi _{B1}^{1\sigma }} \right. \left. { - a_1^{\Delta \sigma }\xi _{B0}^{1\sigma }} \right) + \frac{1}{3}c\left( {\xi _{B2}^{1\sigma } - a_2^{\Delta \sigma }\xi _{B0}^{1\sigma }} \right) + \\ &\qquad \frac{1}{2}{W_{C1}} - {W_{C2}} = 0\\[-1pt]\end{split} $$ (72) 对生成元式(67)和式(68), 式(72)分别给出
$$ G_B^1 = - \frac{1}{2}{a_2} - \frac{2}{9}c{t^2} - \frac{2}{3}ct \text{, }\;\; \bar G_B^1 = - ct $$ (73) 因此根据定理5和定理6可分别得到一阶绝热不变量
$$ \left.\begin{split} &{I_1} = - \frac{1}{2}a_3^\sigma + a_4^\sigma + {\varepsilon _B}\left( {a_3^\sigma + ta_4^\sigma } \right) \\ &{\bar I_1} = ta_3^\sigma + ta_4^\sigma - {a_1} - \frac{1}{2}{a_2} - \frac{2}{3}c{t^2} + {\varepsilon _B}\left( {a_3^\sigma + a_4^\sigma } \right)\end{split}\right\} $$ (74) 类似可求得该系统高阶的绝热不变量, 以及其他时间尺度下该系统的绝热不变量.
3. 结 论
Birkhoff力学是最一般可能的力学, 本文在时间尺度上讨论了广义Birkhoff系统和受约束Birkhoff系统Lie对称性的摄动和绝热不变量. 主要有以下两个新成果: 首先, 得到时间尺度上广义Birkhoff系统的精确不变量和绝热不变量; 其次, 给出约束Birkhoff系统的Lie对称性导致的精确不变量及其对称性摄动导致绝热不变量的条件. 此外, 当$ {\varLambda _l} = 0 $时, 相应结论退化到时间尺度上Birkhoff系统. 当$ \mathbb{T} = \mathbb{R} $时, 相应结论退化到经典情况.
值得注意的是, Birkhoff系统的Lie对称性在一定条件下可导致Hojman型和Noether型两种守恒量, 本文所得的守恒量均为Noether型的, Hojman型守恒量也是一个值得研究的课题. 时间尺度微积分包括delta积分和nabla积分, 本文讨论了前者, 后者的研究也可纳入参考. 约束力学系统分为迁移和非迁移, 本文讨论了迁移情况, 非迁移情况也可开展相关研究. 除此之外, 时间尺度与分数阶结合的研究开始兴起[32-33], 有关对称性摄动和绝热不变量的研究也可在此领域内开展. 综上, 时间尺度对称性摄动和绝热不变量相关课题是未来研究的重要方向之一, 具有广阔的发展潜力与未来前景.
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图 1 例1中$ {a_2},{a_3},{a_4}和{I_0} $的值
Figure 1. Simulations of $ {a_2},{a_3},{a_4}\;{\mathrm{and}}\;{I_0} $ in example 1
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