NOETHER-TYPE ADIABATIC INVARIANTS FOR CONSTRAINED MECHANICS SYSTEMS ON TIME SCALES
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摘要: 时间尺度微积分是近年来的研究热点之一, 其不仅统一连续分析与离散分析, 还能协助完成更复杂动力学系统的建模. Noether对称性方法是一种近代的积分方法, 揭示了力学系统守恒量与其内在的动力学对称性之间的潜在关系, Noether对称性的摄动与绝热不变量和系统的可积性之间也有着密切的联系. 时间尺度上约束力学系统的Noether对称性问题虽然已有学者研究, 但是由于时间尺度微积分理论的不成熟, 研究成果的深度及正确性均有待探究. 文章的重点是探讨时间尺度上约束力学系统Noether对称性的摄动与绝热不变量, 这其中包含了Lagrange系统、Hamilton系统以及Birkhoff系统. 首先, 探讨了3个受小扰动作用的系统, 其Noether对称性的变化, 并给出了相应的绝热不变量; 然后提供了在无扰动条件下, 3个系统的精确不变量. Lagrange系统得到的精确不变量与原有结果吻合, Hamilton系统和Birkhoff系统得到的精确不变量是新的. 其次, 时间尺度上的导数分为delta导数和nabla导数, 由这两种导数所组成的系统, 互为对偶系统. 基于文章delta导数下所得到的结果, 采用对偶原理的方法, 给出了3个系统对偶空间的绝热不变量和精确不变量. 最后, 文末分别讨论了时间尺度上Kepler问题和Hojman-Urrutia问题的Noether型绝热不变量, 从而借助例题对3个系统中所得到的结果和所采用的方法进行说明.Abstract: Time scale calculus is one of the research hotspots in recent years. It not only unifies continuous analysis and discrete analysis, but also assists in modeling more complex dynamics systems. The Noether symmetry method is a modern integration method that reveals the potential relationship between the conserved quantity of a mechanics system and its inherent dynamics symmetry. Perturbation to Noether symmetry, as well as adiabatic invariants, are also closely related to the integrability of the system. Although the problems of symmetry for constrained mechanics systems on time scales have been studied by scholars, the depth and accuracy of research results need to be explored due to the immaturity of time scale calculus theory. The focus of this article is to explore the perturbation to Noether symmetry, and the adiabatic invariants for the constrained mechanics systems on time scales, including Lagrangian system, Hamiltonian system and Birkhoffian system. Firstly, for the three perturbed systems, we discuss the changes of the Noether symmetry, and present the corresponding adiabatic invariants. Then, we provide the exact invariants of the three systems under disturbance free conditions. The exact invariant obtained of the Lagrangian system is consistent with the original result, while the exact invariants obtained of the Hamiltonian and Birkhoffian systems are new. Secondly, there are two derivatives on time scales, namely, the delta derivative and the nabla one, and the systems composed of the two derivatives are dual. Based on the results obtained under the delta derivative in this article, the adiabatic invariants and exact invariants of the three dual spaces are given using the method of dual principle. Thirdly, at the end of the article, the Noether type adiabatic invariants of the Kepler problem and the Hojman-Urrutia problem on time scales were discussed respectively, to illustrate the results and methods presented in the three systems of this article by examples.
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Keywords:
- time scale /
- perturbation to symmetry /
- exact invariant /
- dual system /
- dual principle
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引 言
力学是微分方程的起源之一, 大多力学问题都用微分方程来描述, 进而研究其各种性质. 而微分方程也可以力学化为Lagrange方程、Hamilton方程或Birkhoff方程, 从而利用力学的方法进行求解. 力学和数学的交缘, 特别是分析力学与微分方程的交缘, 使两个学科在相互促进中得到了重要发展. 本文的研究是在分析力学框架下开展的.
Mei等[1]指出, 动力学中最重要的工作是寻找运动微分方程的解, 而一个守恒量便是一个解, 因此, 人们致力于研究系统的守恒量. 然而, 当系统受到小干扰力的作用时, 原有的守恒量也可能会产生改变. 1917年, Burgers[2]提出了关于绝热不变量的理论. 动力学系统的可积性与对称性的摄动和绝热不变量存在紧密联系, 并且在此领域, 研究人员已经取得了许多研究成果[3-12].
在科学家们尝试将微分方程、差分方程、连续分析和离散分析融为一体的情况下, 德国学者Hilger[13]提出了时间尺度分析理论. 时间尺度微积分不仅能够协助我们明确连续和离散系统的差异及相似性, 并且能够使我们对连续系统、离散系统和其他复杂动力学系统有更精确和更明了的认识. 近年来, 时间尺度上的动力学理论已经在诸如经济学、生物学、动力学与控制以及物理学等多个领域得到了广泛运用[14-19].
时间尺度上变分问题的研究始于2004年[20-21], Noether对称性的研究始于2008年[22]. 随后, 变分问题与Noether对称性的研究被拓展到非保守非完整系统[23-24]、Hamilton系统[25-26]和Birkhoff系统[27]等, 这时研究Noether对称性采用的是Bartosiewicz等[22]的方法. 特别地, 2020年, Anerot等[28]提出了一种其认为更合理的方法, 重新给出了时间尺度上的Noether定理. 此后, 学者们对时间尺度上的Noether定理及其相关性质进行研究时, 多采用Anerot的方法. 近年来, 张毅团队分别对时间尺度上的Lie对称性[29-30]、Mei对称性[31-33]、分数阶Noether对称性[34]及Lagrange框架下非迁移系统的Noether对称性[35]进行了研究.
另外, 时间尺度上的导数主要有两种类型, 一种是Hilger[13]提出的delta导数, 另一种是Atici等[36]提出的nabla导数. 对于这两种导数相关性质的介绍及其使用可参考文献[14-15]. 实际上, 由这两种导数组成的系统互为对偶系统, 它们之间的转化可以采用一种名为对偶原理的方法[37]. 自从2010年Caputo[37]提出时间尺度上的对偶原理并在Lagrange系统中进行验证后, 该方法相继被应用于Hamilton系统[38]和Birkhoff系统[39]. 本工作基于时间尺度微积分, 采用Anerot的方法, 进一步探讨对称性的摄动对约束力学系统可积性的影响, 并采用对偶原理的方法, 给出对偶约束力学系统相应的结果, 以期为时间尺度上分析力学中积分方法的完善提供支撑.
时间尺度用符号$ \mathbb{T} $表示. 特别地, 实数集$ \mathbb{R} $和整数集$ \mathbb{Z} $均为其特例. 时间尺度的定义及时间尺度微积分的相关性质可参考文献[14-15], 这里不再做详细介绍.
1. 对称性摄动
一个量$ C $称为精确不变量, 当且仅当$ {{\Delta C} \mathord{\left/ {\vphantom {{\Delta C} {\Delta t}}} \right. } {\Delta t}} = 0 $(时间尺度上对$ C $求delta导数等于0)(或$ {{\nabla C} \mathord{\left/ {\vphantom {{\nabla C} {\nabla t}}} \right. } {\nabla t}} = 0 $(时间尺度上对$ C $求nabla导数等于0)). 一个量$ C $称为$ z $阶绝热不变量, 当且仅当$ C $中含有小参数, 其最高次幂为$ z $, 且$ {{\Delta C} \mathord{\left/ {\vphantom {{\Delta C} {\Delta t}}} \right. } {\Delta t}} $(或$ {{\nabla C} \mathord{\left/ {\vphantom {{\nabla C} {\nabla t}}} \right. } {\nabla t}} $)与小参数的$ z + 1 $次方成比例.
1.1 Lagrange系统绝热不变量
受小扰动作用的Lagrange方程表示为
$$ {\partial _j}L\left( {t,q_i^\sigma ,q_i^\Delta } \right) - {\left( {{\partial _{n + j}}L} \right)^\Delta } = {\varepsilon _L}{W_{Lj}}\left( {t,q_i^\sigma ,q_i^\Delta } \right) $$ (1) 其中$ \sigma $称为向前跳跃算子, $ \Delta $表示delta导数, $ q_i^\sigma = {q_i} \circ \sigma $, $ q_i^\Delta = {{\Delta {q_i}} \mathord{\left/ {\vphantom {{\Delta {q_i}} {\Delta t}}} \right. } {\Delta t}} $, $ t \in {[{t_1},{t_2}]^\kappa } $, $ i,j = 1,2, \cdots ,n $, $ L:\mathbb{T} \times {\mathbb{R}^n} \times {\mathbb{R}^n} \to \mathbb{R} $为Lagrange函数, $ {\partial _j}L = {{\partial L} \mathord{\left/ {\vphantom {{\partial L} {\partial q_j^\sigma }}} \right. } {\partial q_j^\sigma }} $, $ {\partial _{n + j}}L = {{\partial L} \mathord{\left/ {\vphantom {{\partial L} {\partial q_j^\Delta }}} \right. } {\partial q_j^\Delta }} $, $ {\varepsilon _L} $是无限小参数, $ {W_{Lj}}\left( {t,q_i^\sigma ,q_i^\Delta } \right) $表示扰动力. 除此之外, 记$ {\partial _0}L = {{\partial L} \mathord{\left/ {\vphantom {{\partial L} {\partial t}}} \right. } {\partial t}} $.
对于受小扰动作用的Lagrange系统, 其无限小变换表示为如下形式
$$ \bar t = t + {\delta _L}{\xi _{L0}}\left( {t,{q_i}} \right) + o\left( {{\delta _L}} \right) \text{, } {\bar q_j} = {q_j} + {\delta _L}{\xi _{Lj}}\left( {t,{q_i}} \right) + o\left( {{\delta _L}} \right) $$ (2) 其中, $ {\xi _{L0}} = \xi _{L0}^0 + {\gamma _L}\xi _{L0}^1 + \gamma _L^2\xi _{L0}^2 + \cdots = \gamma _L^m\xi _{L0}^m $, $ {\xi _{Lj}} = \xi _{Lj}^0 + {\gamma _L}\xi _{Lj}^1 + \gamma _L^2\xi _{Lj}^2 + \cdots = \gamma _L^m\xi _{Lj}^m $, $ m = 0,1,2, \cdots $, $ {\delta _L} $ 和 $ {\gamma _L} $是无限小参数, $ {\xi _{L0}} $, $ \xi _{L0}^m $, $ {\xi _{Lj}} $和$ \xi _{Lj}^m $称为无限小生成元.
定理1 对于受小扰动作用的Lagrange系统, 如果存在函数$ G_L^m\left( {t,q_i^\sigma ,q_i^\Delta } \right) $使得$ \xi _{L0}^m $和$ \xi _{Lj}^m $满足
$$\begin{split} &{\partial _0}L \cdot \xi _{L0}^m + {\partial _j}L \cdot \xi _{Lj}^{m\sigma } + {\partial _{n + j}}L \cdot \xi _{Lj}^{m\Delta } + L\xi _{L0}^{m\Delta } + G_L^{m\Delta } -\\ &\qquad {\partial _{n + j}}L \cdot q_j^\Delta \xi _{L0}^{m\Delta } - {W_{Lj}}\left( {\xi _{Lj}^{\left( {m - 1} \right)\sigma } - q_j^\Delta \xi _{L0}^{\left( {m - 1} \right)\sigma }} \right) = 0\end{split} $$ (3) 则该系统存在一个Noether型绝热不变量
$$ \begin{split} & {C_{Lz}} = \sum\limits_{m = 0}^z {\varepsilon _L^m} \left\{ {\left( {L - {\partial _{n + j}}L \cdot q_j^\Delta } \right)\xi _{L0}^m + {\partial _{n + j}}L \cdot \xi _{Lj}^m - \mu \left( t \right){\partial _0}L} \right.\cdot\\ &\qquad \xi _{L0}^m + G_L^m + \int_{{t_1}}^t {\left[ {{{\left( {\mu \left( \tau \right){\partial _0}L - L} \right)}^\Delta } + {\partial _j}L \cdot q_j^\Delta + {{\left( {{\partial _{n + j}}L} \right)}^\sigma }} \right.} \cdot \\ &\qquad \left. {\left. { q_j^{\Delta \Delta } + {\partial _0}L} \right]\xi _{L0}^{m\sigma }{\mathrm{d}} \tau } \right\} \text{, } t \in {\left[ {{t_1},{t_2}} \right]^\kappa }\end{split} $$ (4) 其中, $ G_L^m $满足$ {G_L} = G_L^0 + {\gamma _L}G_L^1 + \gamma _L^2 G_L^2 + \cdots = \gamma _L^mG_L^m $, $ {G_L} = {G_L}\left( {t,q_i^\sigma ,q_i^\Delta } \right) $, 且当$ m = 0 $时, $ \xi _{Lj}^{m - 1} = \xi _{L0}^{m - 1} = 0 $.
证明 由方程(1)和式(3)可得
$$ \begin{split} &\frac{\Delta }{{\Delta t}}{C_{Lz}} = \sum\limits_{m = 0}^z {\varepsilon _L^m} \left[ {\left( {L - {\partial _{n + j}}L \cdot q_j^\Delta } \right)\xi _{L0}^{m\Delta } + {{\left( {L - {\partial _{n + j}}L \cdot q_j^\Delta } \right)}^\Delta }} \right.\cdot \\ &\qquad \xi _{L0}^{m\sigma } + {\partial _{n + j}}L \cdot \xi _{Lj}^{m\Delta } + {\left( {{\partial _{n + j}}L} \right)^\Delta } \cdot \xi _{Lj}^{m\sigma } -\\ &\qquad \mu \left( t \right){\partial _0}L \cdot \xi _{L0}^{m\Delta } + G_L^{m\Delta }- {\left( {\mu \left( t \right){\partial _0}L} \right)^\Delta }\xi _{L0}^{m\sigma } + \\ &\qquad {\left( {\mu \left( t \right){\partial _0}L} \right)^\Delta }\xi _{L0}^{m\sigma } - {L^\Delta }\xi _{L0}^{m\sigma } + q_j^\Delta \xi _{L0}^{m\sigma } \cdot\\ &\qquad \left. { {\partial _j}L + q_j^{\Delta \Delta } \cdot \left( {{\partial _{n + j}}L + \mu \left( t \right) \cdot {{\left( {{\partial _{n + j}}L} \right)}^\Delta }} \right)\xi _{L0}^{m\sigma } + {\partial _0}L \cdot \xi _{L0}^{m\sigma }} \right]= \end{split}$$ $$\begin{split} &\qquad \sum\limits_{m = 0}^z {\varepsilon _L^m} \left[ - {\partial _j}L \cdot \xi _{Lj}^{m\sigma } + {W_{Lj}}\left( {\xi _{Lj}^{\left( {m - 1} \right)\sigma } - q_j^\Delta \xi _{L0}^{\left( {m - 1} \right)\sigma }} \right) -\right.\\ &\qquad {\partial _{n + j}}L \cdot q_j^{\Delta \Delta } \cdot \xi _{L0}^{m\sigma } - {\left( {{\partial _{n + j}}L} \right)^\Delta } \cdot q_j^{\Delta \sigma } \cdot \xi _{L0}^{m\sigma } +\\ &\qquad {\left( {{\partial _{n + j}}L} \right)^\Delta }\xi _{Lj}^{m\sigma } + {\partial _j}L\cdot q_j^\Delta \xi _{L0}^{m\sigma } + q_j^{\Delta \Delta } \cdot {\partial _{n + j}}L \cdot \xi _{L0}^{m\sigma } +\\ &\qquad \left.{{\left( {{\partial _{n + j}}L} \right)}^\Delta } \cdot \xi _{L0}^{m\sigma }\left( {q_j^{\Delta \sigma } - q_j^\Delta } \right) \right] = \sum\limits_{m = 0}^z \varepsilon _L^m\Bigg[ {{\left( {{\partial _{n + j}}L} \right)}^\Delta }\\ &\qquad \left( {\xi _{Lj}^{m\sigma } - q_j^\Delta \xi _{L0}^{m\sigma }} \right) - {\partial _j}L \cdot \left( {\xi _{Lj}^{m\sigma } - q_j^\Delta \xi _{L0}^{m\sigma }} \right) +\\ &\qquad { {W_{Lj}}\left( {\xi _{Lj}^{\left( {m - 1} \right)\sigma } - q_j^\Delta \xi _{L0}^{\left( {m - 1} \right)\sigma }} \right)} \Bigg] =\\ &\qquad \sum\limits_{m = 0}^z {\varepsilon _L^m} \left[ - \varepsilon {W_{Lj}}\left( {\xi _{Lj}^{m\sigma } - q_j^\Delta \xi _{L0}^{m\sigma }} \right) +\right.\\ &\qquad \left.{W_{Lj}}\left( {\xi _{Lj}^{\left( {m - 1} \right)\sigma } - q_j^\Delta \xi _{L0}^{\left( {m - 1} \right)\sigma }} \right) \right]= - \varepsilon _L^{z + 1}{W_{Lj}}\left( {\xi _{Lj}^{z\sigma } - q_j^\Delta \xi _{L0}^{z\sigma }} \right) \end{split}$$ 1.2 Hamilton系统绝热不变量
引入广义动量和Hamilton函数
$$ {p_j} = {\partial _{n + j}}L\left( {t,q_i^\sigma ,q_i^\Delta } \right) \text{, }\quad H\left( {t,q_i^\sigma ,{p_i}} \right) = {p_j}q_j^\Delta - L $$ (5) 则受小扰动作用的Hamilton方程可表示为
$$ q_j^\Delta = {\partial _{n + j}}H \text{, } p_j^\Delta = - {\partial _j}H - {\varepsilon _H}{W_{Hj}}\left( {t,q_i^\sigma ,{p_i}} \right) $$ (6) 其中, $ q_i^\sigma = {q_i} \circ \sigma $, $ q_i^\Delta = {{\Delta {q_i}} \mathord{\left/ {\vphantom {{\Delta {q_i}} {\Delta t}}} \right. } {\Delta t}} $, $ t \in {[{t_1},{t_2}]^\kappa } $, $ {\partial _{n + j}}H = {\partial H} / {\partial {p_j}} $, $ {\partial _j}H = {{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial q_j^\sigma }}} \right. } {\partial q_j^\sigma }} $, $ {\varepsilon _H} $是无限小参数, $ {W_{Hj}}\left( {t,q_i^\sigma ,{p_i}} \right) $表示扰动力, $ i,j = 1,2, \cdots ,n $. 除此之外, 记$ {\partial _0}H = {{\partial H} \mathord{\left/ {\vphantom {{\partial H} {\partial t}}} \right. } {\partial t}} $.
当Hamilton系统受到小扰动时, 该系统的无限小变换可表示为
$$\left. \begin{split} &\bar t = t + {\delta _H}{\xi _{H0}}\left( {t,{q_i},{p_i}} \right) \text{, } {\bar q_j} = {q_j} + {\delta _H}{\xi _{Hj}}\left( {t,{q_i},{p_i}} \right) \\ &{\bar p_j} = {p_j} + {\delta _H}{\eta _{Hj}}\left( {t,{q_i},{p_i}} \right) \end{split}\right\}$$ (7) 其中, $ {\xi _{H0}} = \xi _{H0}^0 + {\gamma _H}\xi _{H0}^1 + \gamma _H^2\xi _{H0}^2 + \cdots = \gamma _H^m\xi _{H0}^m $, $ {\xi _{Hj}} = \xi _{Hj}^0 + {\gamma _H}\xi _{Hj}^1 + \gamma _H^2\xi _{Hj}^2 + \cdots = \gamma _H^m\xi _{Hj}^m $, $ {\eta _{Hj}} = \eta _{Hj}^0 + {\gamma _H}\eta _{Hj}^1 + \gamma _H^2\eta _{Hj}^2 + \cdots = \gamma _H^m\eta _{Hj}^m $, $ m = 0,1,2, \cdots $, $ {\delta _H} $和$ {\gamma _H} $是无限小参数, $ {\xi _{H0}} $, $ \xi _{H0}^m $, $ {\xi _{Hj}} $, $ \xi _{Hj}^m $, $ {\eta _{Hj}} $和$ \eta _{Hj}^m $称为无限小生成元.
定理2 对于受到小扰动作用的Hamilton系统, 如果存在函数$ G_H^m\left( {t,q_i^\sigma ,{p_i}} \right) $使得无限小生成元$ \xi _{H0}^m $, $ \xi _{Hj}^m $和$ \eta _{Hj}^m $满足
$$ \begin{split} &{p_j}\xi _{Hj}^{m\Delta } - {\partial _0}H \cdot \xi _{H0}^m - {\partial _j}H \cdot \xi _{Hj}^{m\sigma } - H\xi _{H0}^{m\Delta } -\\ &\qquad {W_{Hj}}\left( {\xi _{Hj}^{\left( {m - 1} \right)\sigma } - q_j^\Delta \xi _{H0}^{\left( {m - 1} \right)\sigma }} \right) + G_H^{m\Delta } = 0 \end{split}$$ (8) 则该系统存在如下Noether型绝热不变量
$$ \begin{split} &{C_{Hz}} = \sum\limits_{m = 0}^z {\varepsilon _H^m} \left\{ {{p_j}\xi _{Hj}^m + \left( {\mu \left( t \right){\partial _0}H - H} \right)\xi _{H0}^m + } \right.\int_{{t_1}}^t {\left[ {\Big( { - \mu \left( \tau \right)} } \right.}\cdot \\ &\qquad { { {\partial _0}H - {p_j} \cdot {\partial _{n + j}}H + H} \Big)^\Delta } - {\partial _j}H \cdot {\partial _{n + j}}H + {\left( {{\partial _{n + j}}H} \right)^\Delta }p_j^\sigma\\ &\qquad \left. {\left. { - {\partial _0}H} \right]\xi _{H0}^{m\sigma }\mathrm{d} \tau + G_H^m} \right\} \text{, }\quad t \in {\left[ {{t_1},{t_2}} \right]^\kappa }\end{split} $$ (9) 其中, $ G_H^m $满足$ {G_H} = G_H^0 + {\gamma _H}G_H^1 + \gamma _H^2 G_H^2 + \cdots = \gamma _H^mG_H^m $, $ {G_H} = {G_H}\left( {t,q_i^\sigma ,{p_i}} \right) $, 且当$ m = 0 $时, $ \xi _{Hj}^{m - 1} = \xi _{H0}^{m - 1} = 0 $.
证明 由方程(6)和式(8)可得
$$ \begin{split} &\frac{\Delta }{{\Delta t}}{C_{Hz}} = \sum\limits_{m = 0}^z {\varepsilon _H^m} \Big[ {{p_j}\xi _{Hj}^{m\Delta } + p_j^\Delta \xi _{Hj}^{m\sigma } - H\xi _{H0}^{m\Delta } + \mu \left( t \right)\xi _{H0}^{m\Delta }{\partial _0}H} + \\ &\qquad {\left( {\mu \left( t \right){\partial _0}H} \right)^\Delta }\xi _{H0}^{m\sigma } - {\left( {\mu \left( t \right){\partial _0}H} \right)^\Delta }\xi _{H0}^{m\sigma } - {H^\Delta }\xi _{H0}^{m\sigma }-\\ &\qquad {\left( {{p_j}{\partial _{n + j}}H} \right)^\Delta }\xi _{H0}^{m\sigma } + {H^\Delta }\xi _{H0}^{m\sigma } - \frac{{\partial H}}{{\partial q_j^\sigma }} \cdot \frac{{\partial H}}{{\partial {p_j}}} \cdot \xi _{H0}^{m\sigma }+\\ &\qquad { {{\left( {{\partial _{n + j}}H} \right)}^\Delta }p_j^\sigma \xi _{H0}^{m\sigma } - {\partial _0}H \cdot \xi _{H0}^{m\sigma } + G_H^{m\Delta }} \Big]=\\ &\qquad \sum\limits_{m = 0}^z \varepsilon _H^m\left[ {\partial _j}H \cdot \xi _{Hj}^{m\sigma } + {W_{Hj}}\left( {\xi _{Hj}^{\left( {m - 1} \right)\sigma } - q_j^\Delta \xi _{H0}^{\left( {m - 1} \right)\sigma }} \right) +\right. \\ &\qquad p_j^\Delta \xi _{Hj}^{m\sigma } - \left. { p_j^\Delta q_j^\Delta \cdot \xi _{H0}^{m\sigma } - {\partial _j}H \cdot q_j^\Delta \xi _{H0}^{m\sigma }} \right]=\\ &\qquad \sum\limits_{m = 0}^z {\varepsilon _H^m\left[ {{\partial _j}H \cdot \left( {\xi _{Hj}^{m\sigma } - q_j^\Delta \xi _{H0}^{m\sigma }} \right) + p_j^\Delta \left( {\xi _{Hj}^{m\sigma } - q_j^\Delta \xi _{H0}^{m\sigma }} \right)} \right.}+\\ &\qquad \left. { {W_{Hj}}\left( {\xi _{Hj}^{\left( {m - 1} \right)\sigma } - q_j^\Delta \xi _{H0}^{\left( {m - 1} \right)\sigma }} \right)} \right]=\\ &\qquad \sum\limits_{m = 0}^z {\varepsilon _H^m{W_{Hj}}} \left[ - {\varepsilon _H}\left( {\xi _{Hj}^{m\sigma } - q_j^\Delta \xi _{H0}^{m\sigma }} \right) + \left( \xi _{Hj}^{\left( {m - 1} \right)\sigma } -\right.\right.\\ &\qquad \left.\left. q_j^\Delta \xi _{H0}^{\left( {m - 1} \right)\sigma } \right) \right]= - \varepsilon _H^{z + 1}{W_{Hj}}\left( {\xi _{Hj}^{z\sigma } - q_j^\Delta \xi _{H0}^{z\sigma }} \right) \end{split}$$ 1.3 Birkhoff系统绝热不变量
受小扰动作用的Birkhoff方程可表示为
$$ {\partial _l}{R_k}\left( {t,a_\varpi ^\sigma } \right) \cdot a_k^\Delta - {\partial _l}B\left( {t,a_\varpi ^\sigma } \right) - R_l^\Delta = {\varepsilon _B}{W_{Bl}}\left( {t,a_\varpi ^\sigma } \right) $$ (10) 其中, $ a_\varpi ^\sigma = {a_\varpi } \circ \sigma $, $ a_l^\Delta = {{\Delta {a_l}} \mathord{\left/ {\vphantom {{\Delta {a_l}} {\Delta t}}} \right. } {\Delta t}} $, $ {\varepsilon _B} $是无限小参数, $ {W_{Bl}} $为扰动力, $ t \in {[{t_1},{t_2}]^\kappa } $, $ l,k,\varpi = 1,2, \cdots ,2 n $, $ {R_l}: \mathbb{T} \times {\mathbb{R}^{2 n}} \to \mathbb{R} $称为Birkhoff函数组, $ B:\mathbb{T} \times {\mathbb{R}^{2 n}} \to \mathbb{R} $称为Birkhoff函数, $ {\partial _l}{R_k} = {{\partial {R_k}} \mathord{\left/ {\vphantom {{\partial {R_k}} {\partial a_l^\sigma }}} \right. } {\partial a_l^\sigma }} $, $ {\partial _l}B = {{\partial B} \mathord{\left/ {\vphantom {{\partial B} {\partial a_l^\sigma }}} \right. } {\partial a_l^\sigma }} $. 除此之外, 记$ {\partial _0}{R_k} = {{\partial {R_k}} \mathord{\left/ {\vphantom {{\partial {R_k}} {\partial t}}} \right. } {\partial t}} $, $ {\partial _0}B = {{\partial B} \mathord{\left/ {\vphantom {{\partial B} {\partial t}}} \right. } {\partial t}} $.
对于受到小扰动作用的Birkhoff系统, 其无限小变换可表示为
$$ \bar t = t + {\delta _B}{\xi _{B0}}\left( {t,{a_\varpi }} \right) \text{, } {\bar a_k} = {a_k} + {\delta _B}{\xi _{Bk}}\left( {t,{a_\varpi }} \right) $$ (11) 其中, $ {\xi _{B0}} = \xi _{B0}^0 + {\gamma _B}\xi _{B0}^1 + \gamma _B^2\xi _{B0}^2 + \cdots = \gamma _B^m\xi _{B0}^m $, $ {\xi _{Bk}} = \xi _{Bk}^0 + {\gamma _B}\xi _{Bk}^1 + \gamma _B^2\xi _{Bk}^2 + \cdots = \gamma _B^m\xi _{Bk}^m $, $ m = 0,1,2, \cdots $, $ {\delta _B} $和$ {\gamma _B} $是无限小参数, $ {\xi _{B0}} $, $ \xi _{B0}^m $, $ {\xi _{Bk}} $和$ \xi _{Bk}^m $称为无限小生成元.
定理3 对于受到小扰动作用的Birkhoff系统, 如果存在函数$ G_B^m $使得$ \xi _{B0}^m $和$ \xi _{Bk}^m $满足
$$\begin{split} &\left( {{\partial _0}{R_k} \cdot a_k^\Delta - {\partial _0}B} \right)\xi _{B0}^m + \left( {{\partial _l}{R_k} \cdot a_k^\Delta - {\partial _l}B} \right)\xi _{Bl}^{m\sigma } + {R_k}\xi _{Bk}^{m\Delta }-\\ &\qquad B\xi _{B0}^{m\Delta } + G_B^{m\Delta } - {W_{Bl}}\left( {\xi _{Bl}^{\left( {m - 1} \right)\sigma } - a_l^\Delta \xi _{B0}^{\left( {m - 1} \right)\sigma }} \right) = 0\end{split} $$ (12) 则该系统存在如下Noether型绝热不变量
$$\begin{split} &{C_{Bz}} = \sum\limits_{m = 0}^z {\varepsilon _B^m\Bigg\{ {{R_l}\xi _{Bl}^m - \mu \left( t \right)\left( {{\partial _0}{R_l} \cdot a_l^\Delta - {\partial _0}B} \right)\xi _{B0}^m - B\xi _{B0}^m} }+\\ &\qquad G_B^m + \int_{{t_1}}^t {\Bigg\{ {{{\left[ {\mu \left( \tau \right)\left( {{\partial _0}{R_l} \cdot a_l^\Delta - {\partial _0}B} \right) + B - {R_l}a_l^\Delta } \right]}^\Delta } - {\partial _0}B} }+\\ &\qquad {{ \left( {{\partial _l}{R_k} \cdot a_k^\Delta - {\partial _l}B} \right)a_l^\Delta + a_l^{\Delta \Delta }R_l^\sigma + {\partial _0}{R_l} \cdot a_l^\Delta } \Bigg\}\xi _{B0}^{m\sigma }\mathrm{d} \tau } \Bigg\}\end{split} $$ (13) 其中, $ G_B^m $满足$ {G_B} = G_B^0 + {\gamma _B}G_B^1 + \gamma _B^2 G_B^2 + \cdots = \gamma _B^mG_B^m $, $ {G_B} = {G_B}\left( {t,a_\varpi ^\sigma } \right) $, 且当$ m = 0 $时, $ \xi _{Bl}^{m - 1} = \xi _{B0}^{m - 1} = 0 $.
证明 由方程(10)和式(12)可得
$$ \begin{split} &\frac{\Delta }{{\Delta t}}{C_{Bz}} = \sum\limits_{m = 0}^z {\varepsilon _B^m\left\{ {{R_l}\xi _{Bl}^{m\Delta } + R_l^\Delta \xi _{Bl}^{m\sigma } - \mu \left( t \right)\left( {{\partial _0}{R_l} \cdot a_l^\Delta - {\partial _0}B} \right)} \right.}\cdot \\ &\qquad \xi _{B0}^{m\Delta } - {\left[ {\mu \left( t \right)\left( {{\partial _0}{R_l} \cdot a_l^\Delta - {\partial _0}B} \right)} \right]^\Delta }\xi _{B0}^{m\sigma } - B\xi _{B0}^{m\Delta } - {B^\Delta }\xi _{B0}^{m\sigma }+\\ &\qquad {B^\Delta }\xi _{B0}^{m\sigma } + {\left[ {\mu \left( t \right)\left( {{\partial _0}{R_l} \cdot a_l^\Delta - {\partial _0}B} \right)} \right]^\Delta }\xi _{B0}^{m\sigma } - {\left( {{R_l}a_l^\Delta } \right)^\Delta }\xi _{B0}^{m\sigma } +\\ &\qquad \left( {{\partial _l}{R_k} \cdot a_k^\Delta - {\partial _l}B} \right)a_l^\Delta \xi _{B0}^{m\sigma } + a_l^{\Delta \Delta }R_l^\sigma \xi _{B0}^{m\sigma } + {\partial _0}{R_l} \cdot a_l^\Delta \cdot \xi _{B0}^{m\sigma }- \\ &\qquad \left. { {\partial _0}B \cdot \xi _{B0}^{m\sigma } + G_B^{m\Delta }} \right\}=\sum\limits_{m = 0}^z {\varepsilon _B^m}\left[- \left( {a_k^\Delta {\partial _l}{R_k} - {\partial _l}B} \right)\xi _{Bl}^{m\sigma } +\right. \\ &\qquad \left. {W_{Bl}}\left( {\xi _{Bl}^{\left( {m - 1} \right)\sigma } - a_l^\Delta \xi _{B0}^{\left( {m - 1} \right)\sigma }} \right) \right.- \\ &\qquad R_l^\Delta a_l^\Delta \xi _{B0}^{m\sigma } - R_l^\sigma a_l^{\Delta \Delta }\xi _{B0}^{m\sigma } + \left( {{\partial _l}{R_k} \cdot a_k^\Delta - {\partial _l}B} \right)a_l^\Delta \xi _{B0}^{m\sigma }+\\ &\qquad \left. { R_l^\Delta \xi _{Bl}^{m\sigma } + a_l^{\Delta \Delta }R_l^\sigma \xi _{B0}^{m\sigma }} \right]=\sum\limits_{m = 0}^z \varepsilon _B^m\Bigg[ \Bigg( {\partial _l}B - \\ &\qquad {\partial _l}{R_k} \cdot a_k^\Delta + R_l^\Delta \Bigg)\left( {\xi _{Bl}^{m\sigma } - a_l^\Delta \xi _{B0}^{m\sigma }} \right) +{W_{Bl}}\left( \xi _{Bl}^{\left( {m - 1} \right)\sigma } - \right.\\ &\qquad \left.a_l^\Delta \xi _{B0}^{\left( {m - 1} \right)\sigma } \right) \Bigg]= \sum\limits_{m = 0}^z {\varepsilon _B^m{W_{Bl}}}\left[ - {\varepsilon _B}\left( {\xi _{Bl}^{m\sigma } - a_l^\Delta \xi _{B0}^{m\sigma }} \right) +\right. \\ &\qquad \left. \left( {\xi _{Bl}^{\left( {m - 1} \right)\sigma } - a_l^\Delta \xi _{B0}^{\left( {m - 1} \right)\sigma }} \right) \right]= - \varepsilon _B^{z + 1}{W_{Bl}}\left( {\xi _{Bl}^{z\sigma } - a_l^\Delta \xi _{B0}^{z\sigma }} \right)\end{split}$$ 1.4 特例
当系统不受小扰动作用时, 即$ m = 0 $, 此时可得如下结论.
定理4 对于Lagrange系统
$$ {\partial _j}L - {\left( {{\partial _{n + j}}L} \right)^\Delta } = 0 \text{, } t \in {\left[ {{t_1},{t_2}} \right]^\kappa } $$ (14) 如果存在函数$ G_L^0\left( {t,q_i^\sigma ,q_i^\Delta } \right) $使得无限小生成元$ \xi _{L0}^0 $和$ \xi _{Lj}^0 $满足
$$ \begin{split} &{\partial _0}L \cdot \xi _{L0}^0 + {\partial _j}L \cdot \xi _{Lj}^{0\sigma } + {\partial _{n + j}}L \cdot \xi _{Lj}^{0\Delta } + L\xi _{L0}^{0\Delta }-\\ &\qquad {\partial _{n + j}}L \cdot q_j^\Delta \xi _{L0}^{0\Delta } + G_L^{0\Delta } = 0 \end{split}$$ (15) 则该系统存在一个精确不变量
$$ \begin{split} &{C_{L0}} = \left( {L - {\partial _{n + j}}L \cdot q_j^\Delta } \right)\xi _{L0}^0 + {\partial _{n + j}}L \cdot \xi _{Lj}^0 - \mu \left( t \right){\partial _0}L \cdot \xi _{L0}^0+\\ &\qquad G_L^0 + \int_{{t_1}}^t {\Big[ {{{\left( {\mu \left( \tau \right){\partial _0}L - L} \right)}^\Delta } + {\partial _j}L \cdot q_j^\Delta + q_j^{\Delta \Delta } \cdot {{\left( {{\partial _{n + j}}L} \right)}^\sigma }} }+\\ &\qquad { {\partial _0}L} \Big]\xi _{L0}^{0\sigma }{\mathrm{d}} \tau \text{, } t \in {\left[ {{t_1},{t_2}} \right]^\kappa } \text{, } i,j = 1,2, \cdots ,n \end{split} $$ (16) 定理5 对于Hamilton系统
$$ q_j^\Delta = {\partial _{n + j}}H \text{, } p_j^\Delta = - {\partial _j}H \text{, } t \in {\left[ {{t_1},{t_2}} \right]^\kappa } $$ (17) 如果存在函数$ G_H^0\left( {t,q_i^\sigma ,{p_i}} \right) $使得无限小生成元$ \xi _{H0}^0 $, $ \xi _{Hj}^0 $和$ \eta _{Hj}^0 $满足
$$ {p_j}\xi _{Hj}^{0\Delta } - {\partial _0}H \cdot \xi _{H0}^0 - {\partial _j}H \cdot \xi _{Hj}^{0\sigma } - H\xi _{H0}^{0\Delta } + G_H^{0\Delta } = 0 $$ (18) 则该系统存在如下精确不变量
$$\begin{split} &{C_{H0}} = {p_j}\xi _{Hj}^0 + \left( {\mu \left( t \right){\partial _0}H - H} \right)\xi _{H0}^0 + G_H^0 + \int_{{t_1}}^t {\Bigg[ {{{\left( {{\partial _{n + j}}H} \right)}^\Delta }} }\cdot\\ &\qquad p_j^\sigma - {\left( {\mu \left( \tau \right){\partial _0}H + {p_j}{\partial _{n + j}}H - H} \right)^\Delta } - {\partial _j}H \cdot {\partial _{n + j}}H- \\ &\qquad { {\partial _0}H} \Bigg]\xi _{H0}^{0\sigma }{\mathrm{d}} \tau \text{, } t \in {\left[ {{t_1},{t_2}} \right]^\kappa } \text{, } i,j = 1,2, \cdots ,n \\[-1pt]\end{split}$$ (19) 证明 由方程(17)和式(18)可得$ {{\Delta {C_{H0}}} \mathord{\left/ {\vphantom {{\Delta {C_{H0}}} {\Delta t}}} \right. } {\Delta t}} = 0 $.
定理6 对于Birkhoff系统
$$\left.\begin{split} &{\partial _l}{R_k}\left( {t,a_\varpi ^\sigma } \right) \cdot a_k^\Delta - {\partial _l}B\left( {t,a_\varpi ^\sigma } \right) - R_l^\Delta = 0 \text{, } t \in {\left[ {{t_1},{t_2}} \right]^\kappa } \\ &\qquad l,k,\varpi = 1,2, \cdots ,2n\end{split} \right\}$$ (20) 如果存在函数$ G_B^0\left( {t,a_\varpi ^\sigma } \right) $使得无限小生成元$ \xi _{B0}^0 $和$ \xi _{Bk}^0 $满足
$$ \begin{split} &\left( {{\partial _0}{R_k} \cdot a_k^\Delta - {\partial _0}B} \right)\xi _{B0}^0 + \left( {{\partial _l}{R_k} \cdot a_k^\Delta - {\partial _l}B} \right)\xi _{Bl}^{0\sigma } + {R_k}\xi _{Bk}^{0\Delta }-\\ &\qquad B\xi _{B0}^{0\Delta } + G_B^{0\Delta } = 0 \text{, } t \in {\left[ {{t_1},{t_2}} \right]^\kappa } \end{split}$$ (21) 则该系统存在如下精确不变量
$$ \begin{split} & {C_{B0}} = {R_l}\xi _{Bl}^0 - \mu \left( t \right)\left( {{\partial _0}{R_l} \cdot a_l^\Delta - {\partial _0}B} \right)\xi _{B0}^0 - B\xi _{B0}^0 + G_B^0 +\\ &\qquad \int_{{t_1}}^t {\Bigg\{ {{{\left[ {\mu \left( \tau \right)\left( {{\partial _0}{R_l} \cdot a_l^\Delta - {\partial _0}B} \right) + B - {R_l}a_l^\Delta } \right]}^\Delta } + {\partial _0}{R_l} \cdot a_l^\Delta } }+\\ &\qquad { \left( {{\partial _l}{R_k} \cdot a_k^\Delta - {\partial _l}B} \right)a_l^\Delta + a_l^{\Delta \Delta }R_l^\sigma - {\partial _0}B} \Bigg\}\xi _{B0}^{0\sigma }{\mathrm{d}} \tau \\[-1pt]\end{split} $$ (22) 证明 由方程(20)和式(21)可得$ {{\Delta {C_{B0}}} \mathord{\left/ {\vphantom {{\Delta {C_{B0}}} {\Delta t}}} \right. } {\Delta t}} = 0 $.
定理5和定理6分别与文献[26]和文献[27]所得结果不一致. 文献[26]和文献[27]采用了Bartosiewicz等[22]的方法, 定理5和定理6是定理2和定理3的特殊情况, 而定理2和定理3借鉴了Anerot等[28]的方法.
给定一个时间尺度$ \mathbb{T} $, 则可定义一个对偶时间尺度$ {\mathbb{T}^ * }: = \left\{ {\left. {s \in \mathbb{R}} \right| - s \in \mathbb{T}} \right\} $. 若在时间尺度$ \mathbb{T} $中分别用$ \sigma $, $ \rho $, $ \mu $和$ \nu $表示向前跳跃算子、向后跳跃算子、向前步差函数和向后步差函数, 则在对偶时间尺度$ {\mathbb{T}^ * } $中分别用$ \hat \sigma $, $ \hat \rho $, $ \hat \mu $和$ \hat \nu $来表示. 对偶时间尺度$ {\mathbb{T}^ * } $中讨论的约束力学系统, 本文称其为对偶约束力学系统. 下面将讨论对偶Lagrange系统、对偶Hamilton系统和对偶Birkhoff系统的绝热不变量问题.
2. 对偶约束力学系统绝热不变量
2.1 对偶Lagrange系统绝热不变量
定义1[37] 给定一个Lagrange函数$ L:\mathbb{T} \times {\mathbb{R}^n} \times {\mathbb{R}^n} \to \mathbb{R} $, 定义其对偶Lagrange函数$ {L^*}:{\mathbb{T}^*} \times {\mathbb{R}^n} \times {\mathbb{R}^n} \to \mathbb{R} $为$ {L^*}\left( {s,x,y} \right) = L\left( { - s,x, - y} \right) $, $ \left( {s,x,y} \right) \in {\mathbb{T}^*} \times {\mathbb{R}^n} \times {\mathbb{R}^n} $.
这里的$ s $, $ x $和$ y $没有特别意义, 仅用来显示函数在Lagrange系统及其对偶系统里各分量的变化情况. 后面讨论Hamilton系统和Birkhoff系统时, 采用了类似的记法.
除此之外, 还需要定义如下关系式
$$\left.\begin{split} &{\partial _0}{L^ * }\left( {s,x,y} \right) = - {\partial _0}L\left( { - s,x, - y} \right) \\ &\xi _{L0}^{m * }\left( {s,x} \right) = \xi _{L0}^m\left( { - s,x} \right) \text{, } {\partial _j}{L^ * }\left( {s,x,y} \right) = {\partial _j}L\left( { - s,x, - y} \right) \\ &{\partial _{n + j}}{L^ * }\left( {s,x,y} \right) = - {\partial _{n + j}}L\left( { - s,x, - y} \right)\\ &\xi _{Lj}^{m * }\left( {s,x} \right) = \xi _{Lj}^m\left( { - s,x} \right) \text{, } W_{Lj}^ * \left( {s,x,y} \right) = {W_{Lj}}\left( { - s,x, - y} \right)\\ &G_L^{m * }\left( {s,x,y} \right) = - G_L^m\left( { - s,x, - y} \right) \end{split}\right\} $$ (23) 利用定理1中给出的受扰Lagrange系统运动微分方程(方程(1))、满足的条件(方程(3))和绝热不变量(方程(4))及定义1和方程(23)给出的Lagrange系统与其对偶系统之间的关系式可推出对偶Lagrange系统中相应的结果.
首先, 利用对偶原理给出对偶Lagrange系统中受扰的Lagrange方程. 令$ s \in {\left[ { - {t_2}, - {t_1}} \right]^\kappa } $, 则有
$$\begin{split} &{\partial _j}{L^ * }\left[ {s,{{\left( {q_i^ * } \right)}^{\hat \sigma }}\left( s \right),{{\left( {q_i^ * } \right)}^{\hat \Delta }}\left( s \right)} \right] - \Bigg\{ {{\partial _{n + j}}{L^ * }\Bigg[ {s,{{\left( {q_i^ * } \right)}^{\hat \sigma }}\left( s \right),} } \\ &\qquad { { {{{\left( {q_i^ * } \right)}^{\hat \Delta }}\left( s \right)} \Bigg]} \Bigg\}^{\hat \Delta }} = {\varepsilon _L}W_{Lj}^ * \left[ {s,{{\left( {q_i^ * } \right)}^{\hat \sigma }}\left( s \right),{{\left( {q_i^ * } \right)}^{\hat \Delta }}\left( s \right)} \right]\end{split} $$ (24) 即
$$\begin{split} &{\partial _j}L\left[ { - s,q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right)} \right] - \left\{ {{\partial _{n + j}}L\left[ { - s,q_i^\rho \left( { - s} \right),} \right.} \right.\\ &\qquad {\left. {\left. {q_i^\nabla \left( { - s} \right)} \right]} \right\}^\nabla } = {\varepsilon _L}{W_{Lj}}\left[ { - s,q_i^\rho \left( { - s} \right),q_i^\nabla \left( s \right)} \right] \end{split}$$ (25) 再令$ t = - s \in {\left[ {{t_1},{t_2}} \right]_\kappa } $, 则可得
$${\partial _j}L\left( {t,q_i^\rho ,q_i^\nabla } \right) - {\left[ {{\partial _{n + j}}L\left( {t,q_i^\rho ,q_i^\nabla } \right)} \right]^\nabla } = {\varepsilon _L}{W_{Lj}}\left( {t,q_i^\rho ,q_i^\nabla } \right) $$ (26) 其中$ \rho $称为向后跳跃算子, $ \nabla $表示nabla导数, $ q_i^\rho = {q_i} \circ \rho ,$ $ q_i^\nabla = {{\nabla {q_i}} \mathord{\left/ {\vphantom {{\nabla {q_i}} {\nabla t}}} \right. } {\nabla t}} ,$ $ t \in {[{t_1},{t_2}]_\kappa } ,$ $ i,j = 1,2, \cdots ,n ,$ $ L\left( {t,q_i^\rho ,q_i^\nabla } \right) $为Lagrange函数, $ {\partial _j}L = {{\partial L} \mathord{\left/ {\vphantom {{\partial L} {\partial q_j^\rho }}} \right. } {\partial q_j^\rho }} $, $ {\partial _{n + j}}L = {{\partial L} \mathord{\left/ {\vphantom {{\partial L} {\partial q_j^\nabla }}} \right. } {\partial q_j^\nabla }} $, $ {\varepsilon _L} $是小参数, $ {W_{Lj}}\left( {t,q_i^\rho ,q_i^\nabla } \right) $是扰动力. 除此之外, 记$ {\partial _0}L = {{\partial L} \mathord{\left/ {\vphantom {{\partial L} {\partial t}}} \right. } {\partial t}} $.
其次, 利用对偶原理推出对偶Lagrange系统中需要满足的条件. 令$ s \in {\left[ { - {t_2}, - {t_1}} \right]^\kappa } $, 则有
$$ \begin{split} & - {\partial _0}{L^ * }\left[ {s,{{\left( {q_i^ * } \right)}^{\hat \sigma }}\left( s \right),{{\left( {q_i^ * } \right)}^{\hat \Delta }}\left( s \right)} \right] \cdot \xi _{L0}^{m * }\left[ {s,q_i^ * \left( s \right)} \right]+\\ &\qquad {\partial _j}{L^ * }\left[ {s,{{\left( {q_i^ * } \right)}^{\hat \sigma }}\left( s \right),{{\left( {q_i^ * } \right)}^{\hat \Delta }}\left( s \right)} \right]\xi_{Lj}^{m * \hat \sigma }\left[ {s,q_i^ * \left( s \right)} \right]+\\ &\qquad {\partial _{n + j}}{L^ * }\left[ {s,{{\left( {q_i^ * } \right)}^{\hat \sigma }}\left( s \right),{{\left( {q_i^ * } \right)}^{\hat \Delta }}\left( s \right)} \right] \cdot \left\{ {\xi _{Lj}^{m * }\left[ {s,q_i^ * \left( s \right)} \right]} \right\}^{\hat \Delta }-\\ &\qquad L\left[ {s,{{\left( {q_i^ * } \right)}^{\hat \sigma }}\left( s \right),{{\left( {q_i^ * } \right)}^{\hat \Delta }}\left( s \right)} \right] \cdot \left\{ {\xi _{L0}^{m * }\left[ {s,q_i^ * \left( s \right)} \right]} \right\}^{\hat \Delta }+\\ &\qquad {\partial _{n + j}}{L^ * }\left[ {s,{{\left( {q_i^ * } \right)}^{\hat \sigma }}\left( s \right),{{\left( {q_i^ * } \right)}^{\hat \Delta }}\left( s \right)} \right]\left\{ {\xi _{L0}^{m * }\left[ {s,q_i^ * \left( s \right)} \right]} \right\}^{\hat \Delta }+ \\ &\qquad \left\{ G_L^{m * }\left[ {s,{{\left( {q_i^ * } \right)}^{\hat \sigma }}\left( s \right),{{\left( {q_i^ * } \right)}^{\hat \Delta }}\left( s \right)} \right] \right\}^{\hat \Delta }- \\ &\qquad \left\{ \xi_{Lj}^{\left( {m - 1} \right) * \hat \sigma }\left[ s,q_i^ * \left( s \right) \right] - q_j^{* \hat \Delta }\left( s \right) \cdot \xi_{L0}^{\left( {m - 1} \right) * \hat \sigma }\left[ {s,q_i^ * \left( s \right)} \right] \right\}\cdot \\ &\qquad {W_{Lj}}\left[ s,\left( {q_i^ * } \right)^{\hat \sigma }\left( s \right),\left( {q_i^ * } \right)^{\hat \Delta }\left( s \right) \right]=\\ &\qquad {\partial _0}L\left[ - s,q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right) \right] \cdot \xi _{L0}^m\left[ - s,{q_i}\left( { - s} \right)\right]+\\ &\qquad {\partial _{n + j}}L\left[ { - s,q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right)} \right] \cdot \xi _{Lj}^{m\nabla }\left[ - s,{q_i}\left( { - s} \right) \right]+\\ & \qquad {\partial _{n + j}}L\left[ - s,q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right) \right] \cdot \xi _{Lj}^{m\nabla }\left[ - s,{q_i}\left( { - s} \right) \right]+\\ &\qquad L\left[ { - s,q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right)} \right] \cdot \xi _{L0}^{m\nabla }\left[ - s,{q_i}\left( { - s} \right)\right]-\\ &\qquad {\partial _{n + j}}L\left[ - s,q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right) \right] \cdot q_j^\nabla \left( { - s} \right)\xi _{L0}^{m\nabla }\left[ - s,{q_i}\left( { - s} \right) \right]+ \\ &\qquad G_L^{m\nabla }\left[ - s,q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right) \right]-\\ &\qquad \left\{ \xi _{Lj}^{\left( {m - 1} \right)\rho }\left[ - s,{q_i}\left( { - s} \right) \right] - q_j^\nabla \left( { - s} \right) \cdot \xi _{L0}^{\left( {m - 1} \right)\rho }\left[ { - s,{q_i}\left( { - s} \right)} \right] \right\}\cdot \\ &\qquad {W_{Lj}}\left[ - s,q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right) \right]=0 \end{split}$$ 再令$ t = - s \in {\left[ {{t_1},{t_2}} \right]_\kappa } $, 可得
$$ \begin{split} &{\partial _0}L\left( {t,q_i^\rho ,q_i^\nabla } \right)\xi _{L0}^m\left( {t,{q_i}} \right) + {\partial _j}L \cdot \xi _{Lj}^{m\rho }\left( {t,{q_i}} \right) + {\partial _{n + j}}L \cdot \xi _{Lj}^{m\nabla }+\\ &\qquad L \cdot \xi _{L0}^{m\nabla } - {\partial _{n + j}}L \cdot q_j^\nabla \cdot \xi _{L0}^{m\nabla } + G_L^{m\nabla }\left( {t,q_i^\rho ,q_i^\nabla } \right)-\\ &\qquad {W_{Lj}}\left( {t,q_i^\rho ,q_i^\nabla } \right) \cdot \left( {\xi _{Lj}^{\left( {m - 1} \right)\rho } - q_j^\nabla \cdot \xi _{L0}^{\left( {m - 1} \right)\rho }} \right) = 0 \end{split}$$ (27) 最后, 利用对偶原理找出对偶Lagrange系统中的Noether型绝热不变量. 令$ s \in {\left[ { - {t_2}, - {t_1}} \right]^\kappa } $, 则有
$$\begin{split} &C_{Lz}^ * \left( s \right) = \sum\limits_{m = 0}^z {\varepsilon _L^m} \left\{ { - \left\{ {L\left[ { - s,q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right)} \right] - {\partial _{n + j}}L\Big[ { - s,} } \right.} \right.\\ &\quad \left. { {q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right)} \Big] \cdot q_j^\nabla \left( { - s} \right)} \right\}\xi _{L0}^m\left[ { - s,{q_i}\left( { - s} \right)} \right] - \xi _{Lj}^m\left[ { - s,} \right. \\ &\quad \left. {{q_i}\left( { - s} \right)} \right] \cdot {\partial _{n + j}}L\left[ { - s,q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right)} \right] - \nu \left( { - s} \right){\partial _0}L\Big[ { - s,} \\ &\quad {q_i^\rho \left( { - s} \right),q_i^\nabla \left( { - s} \right)} \Big]\xi _{L0}^m\left[ { - s,{q_i}\left( { - s} \right)} \right] - G_L^m\left[ { - s,q_i^\rho \left( { - s} \right),} \right.\\ &\quad \left. {q_i^\nabla \left( { - s} \right)} \right] - \int_{ - {t_2}}^s {\left\{ { - \left\{ { - \nu \left( { - \omega } \right){\partial _0}L\left[ { - \omega ,q_i^\rho \left( { - \omega } \right),q_i^\nabla \left( { - \omega } \right)} \right]} \right.} \right.}- \\ &\quad {\left. { L\left[ { - \omega ,q_i^\rho \left( { - \omega } \right),q_i^\nabla \left( { - \omega } \right)} \right]} \right\}^\nabla } + {\partial _j}L\left[ { - \omega ,q_i^\rho \left( { - \omega } \right),} \right.\\ &\quad \left. {q_i^\nabla \left( { - \omega } \right)} \right] \cdot \left[ { - q_j^\nabla \left( { - \omega } \right)} \right] - q_j^{\nabla \nabla }\left( { - \omega } \right) \cdot {\left( {{\partial _{n + j}}L} \right)^\rho }\Big[ { - \omega ,} \\ &\quad \left. { {q_i^\rho \left( { - \omega } \right),q_i^\nabla \left( { - \omega } \right)} \Big] - {\partial _0}L\left[ { - \omega ,q_i^\rho \left( { - \omega } \right),q_i^\nabla \left( { - \omega } \right)} \right]} \right\}\cdot \\ &\quad \left. { \xi _{L0}^{m\rho }\left[ { - \omega ,{q_i}\left( { - \omega } \right)} \right]\hat {\mathrm{d}} \omega } \right\}\end{split}$$ 再令$ t = - s \in {\left[ {{t_1},{t_2}} \right]_\kappa } $, 可得
$$ \begin{split} &{C_{Lz}} = \sum\limits_{m = 0}^z {\varepsilon _L^m} \left\{ { - \left[ {L\left( {t,q_i^\rho ,q_i^\nabla } \right) - {\partial _{n + j}}L \cdot q_j^\nabla } \right]\xi _{L0}^m\left( {t,{q_i}} \right)} \right.-\\ &\quad {\partial _{n + j}}L \cdot \xi _{Lj}^m - \xi _{L0}^m \cdot \nu \left( t \right){\partial _0}L - \int_t^{{t_2}} {\left\{ {{{\left[ {\nu \left( \tau \right) \cdot {\partial _0}L + L} \right]}^\nabla } - {\partial _j}L} \right.}\cdot \\ &\quad \left. {\left. { q_j^\nabla - q_j^{\nabla \nabla } \cdot {{\left( {{\partial _{n + j}}L} \right)}^\rho } - {\partial _0}L} \right\}\xi _{L0}^{m\rho }{\mathrm{d}} \tau - G_L^m\left[ {t,q_i^\rho ,q_i^\nabla } \right]} \right\} \end{split} $$ (28) 综上, 利用方程(26) ~ 式(28)可将对偶Lagrange系统中得到的结果概括为如下定理.
定理7 对于对偶的受扰Lagrange系统(方程(26))
$$ {\partial _j}L - {\left( {{\partial _{n + j}}L} \right)^\nabla } = {\varepsilon _L}{W_{Lj}} $$ 如果存在一个函数$ G_L^m $使得无限小生成元$ \xi _{L0}^m $和$ \xi _{Lj}^m $满足方程(27)
$$ \begin{split} &{\partial _0}L \cdot \xi _{L0}^m + {\partial _j}L \cdot \xi _{Lj}^{m\rho } + {\partial _{n + j}}L \cdot \xi _{Lj}^{m\nabla } + L \cdot \xi _{L0}^{m\nabla } - {\partial _{n + j}}L\cdot\\ &\qquad q_j^\nabla \cdot \xi _{L0}^{m\nabla } + G_L^{m\nabla } - {W_{Lj}} \cdot \left( {\xi _{Lj}^{\left( {m - 1} \right)\rho } - q_j^\nabla \cdot \xi _{L0}^{\left( {m - 1} \right)\rho }} \right) = 0\end{split} $$ 则该系统存在如下Noether型绝热不变量(方程(28))
$$ \begin{split} &{C_{Lz}} = \sum\limits_{m = 0}^z {\varepsilon _L^m} \left\{ { - \left( {L - {\partial _{n + j}}L \cdot q_j^\nabla } \right)\xi _{L0}^m - {\partial _{n + j}}L \cdot \xi _{Lj}^m} \right.-\\ &\qquad \nu \left( t \right) \cdot {\partial _0}L \cdot \xi _{L0}^m - \int_t^{{t_2}} {\left[ {{{\left( {\nu \left( \tau \right) \cdot {\partial _0}L + L} \right)}^\nabla } - {\partial _j}L \cdot q_j^\nabla } \right.}- \\ &\qquad \left. {\left. { q_j^{\nabla \nabla } \cdot {{\left( {{\partial _{n + j}}L} \right)}^\rho } - {\partial _0}L} \right]\xi _{L0}^{m\rho }{\mathrm{d}} \tau - G_L^m} \right\}\end{split} $$ 其中, 当$ m = 0 $时, $ \xi _{Lj}^{m - 1} = \xi _{L0}^{m - 1} = 0 $.
证明 由方程(26)和式(27)可得
$$\begin{split} &\frac{\nabla }{{\nabla t}}{C_{Lz}} = - \sum\limits_{m = 0}^z {\varepsilon _L^m} \Big[ {\left( {L - {\partial _{n + j}}L \cdot q_j^\nabla } \right)\xi _{L0}^{m\nabla } + {{\left( {L - {\partial _{n + j}}L \cdot q_j^\nabla } \right)}^\nabla }} \cdot \\ &\qquad \xi _{L0}^{m\rho } + {\partial _{n + j}}L \cdot \xi _{Lj}^{m\nabla } + {\left( {{\partial _{n + j}}L} \right)^\nabla }\xi _{Lj}^{m\rho } + \nu \left( t \right){\partial _0}L \cdot \xi _{L0}^{m\nabla }+\\ &\qquad {\left( {\nu \left( t \right){\partial _0}L} \right)^\nabla }\xi _{L0}^{m\rho } + G_L^{m\nabla } - {\left( {\nu \left( t \right) \cdot {\partial _0}L + L} \right)^\nabla }\xi _{L0}^{m\rho }+\\ &\qquad { {\partial _j}L \cdot q_j^\nabla \xi _{L0}^{m\rho } + q_j^{\nabla \nabla }{{\left( {{\partial _{n + j}}L} \right)}^\rho }\xi _{L0}^{m\rho } + {\partial _0}L \cdot \xi _{L0}^{m\rho }} \Big]=\\ &\qquad - \sum\limits_{m = 0}^z \varepsilon _L^m\Big[ { - {\partial _j}L \cdot \xi _{Lj}^{m\rho } + {W_{Lj}}\left( {\xi _{Lj}^{\left( {m - 1} \right)\rho } - q_j^\nabla \cdot \xi _{L0}^{\left( {m - 1} \right)\rho }} \right)} -\\ &\qquad {{\left( {{\partial _{n + j}}L} \right)}^\nabla } \cdot q_j^\nabla \cdot \xi _{L0}^{m\rho } + {{\left( {{\partial _{n + j}}L} \right)}^\nabla } \cdot \xi _{Lj}^{m\rho } + {\partial _j}L \cdot q_j^\nabla \cdot \xi _{L0}^{m\rho } \Big]= \\ &\qquad - \sum\limits_{m = 0}^z \varepsilon _L^m\Big\{ {\left( {\xi _{Lj}^{m\rho } - q_j^\nabla \xi _{L0}^{m\rho }} \right)\left[ { - {\partial _j}L + {{\left( {{\partial _{n + j}}L} \right)}^\nabla }} \right]} +\\ &\qquad {W_{Lj}}\left( {\xi _{Lj}^{\left( {m - 1} \right)\rho } - q_j^\nabla \cdot \xi _{L0}^{\left( {m - 1} \right)\rho }} \right) \Big\}=\\ &\qquad - \sum\limits_{m = 0}^z {\varepsilon _L^m{W_{Lj}}} \left[ - {\varepsilon _L}\left( {\xi _{Lj}^{m\rho } - q_j^\nabla \xi _{L0}^{m\rho }} \right) \right.+\\ &\qquad \left.\left( {\xi _{Lj}^{\left( {m - 1} \right)\rho } - q_j^\nabla \xi _{L0}^{\left( {m - 1} \right)\rho }} \right) \right]= \varepsilon _L^{z + 1}{W_{Lj}}\left( {\xi _{Lj}^{z\rho } - q_j^\nabla \xi _{L0}^{z\rho }} \right) \end{split}$$ 定理8 对于对偶Lagrange系统
$$ {\partial _j}L - {\left( {{\partial _{n + j}}L} \right)^\nabla } = 0 $$ (29) 如果存在一个函数$ G_L^0 $使得无限小生成元$ \xi _{L0}^0 $和$ \xi _{Lj}^0 $满足
$$\begin{split} &{\partial _0}L \cdot \xi _{L0}^0 + {\partial _j}L \cdot \xi _{Lj}^{0\rho } + {\partial _{n + j}}L \cdot \xi _{Lj}^{0\nabla } + L \cdot \xi _{L0}^{0\nabla }- \\ &\qquad {\partial _{n + j}}L\cdot q_j^\nabla \cdot \xi _{L0}^{0\nabla } + G_L^{0\nabla } = 0\end{split} $$ (30) 则该系统存在一个精确不变量
$$ \begin{split} &{C_{L0}} = - \left( {L - {\partial _{n + j}}L \cdot q_j^\nabla } \right)\xi _{L0}^0 - {\partial _{n + j}}L \cdot \xi _{Lj}^0 - \nu \left( t \right) \cdot {\partial _0}L \cdot \xi _{L0}^0-\\ &\quad \int_t^{{t_2}} {\left\{ {{{\left[ {\nu \left( \tau \right) \cdot {\partial _0}L + L} \right]}^\nabla } - {\partial _j}L \cdot q_j^\nabla - q_j^{\nabla \nabla }{{\left( {{\partial _{n + j}}L} \right)}^\rho } - {\partial _0}L} \right\}}\cdot\\ &\quad { \xi _{L0}^{0\rho }{\mathrm{d}} \tau - G_L^0} \end{split} $$ (31) 证明 由方程(29)和式(30)可得$ {{\nabla {C_{L0}}} \mathord{\left/ {\vphantom {{\nabla {C_{L0}}} {\nabla t}}} \right. } {\nabla t}} = 0 $.
2.2 对偶Hamilton系统绝热不变量
定义2[38] 给定一个Hamilton函数$ H:\mathbb{T} \times {\mathbb{R}^n} \times {\mathbb{R}^n} \to \mathbb{R} $和广义动量$ p:\mathbb{T} \to \mathbb{R} $, 对偶Hamilton函数$ {H^*}:{\mathbb{T}^*} \times {\mathbb{R}^n} \times {\mathbb{R}^n} \to \mathbb{R} $和对偶广义动量$ {p^*}:{\mathbb{T}^*} \to \mathbb{R} $定义为$ {H^ * }\left( {s,x,y} \right) = H\left( { - s,x, - y} \right) $, $ {p^ * }\left( s \right) = - p\left( { - s} \right) $, 且满足
$$\left.\begin{split} &{\partial _0}{H^ * }\left( {s,x,y} \right) = - {\partial _0}H\left( { - s,x, - y} \right) \\ &{\partial _j}{H^ * }\left( {s,x,y} \right) = {\partial _j}H\left( { - s,x, - y} \right) \\ &{\partial _{n + j}}{H^ * }\left( {s,x,y} \right) = - {\partial _{n + j}}H\left( { - s,x, - y} \right) \end{split}\right. $$ 除此之外, 还需要定义如下关系式
$$\left. \begin{split} &\xi _{H0}^ * \left( {s,x,y} \right) = \xi _{H0}^m\left( { - s,x, - y} \right)\\ &\xi _{Hj}^{m * }\left( {s,x,y} \right) = \xi _{Hj}^m\left( { - s,x, - y} \right)\\ &W_{Hj}^ * \left( {s,x,y} \right) = {W_{Hj}}\left( { - s,x, - y} \right)\\ &G_H^{m * }\left( {s,x,y} \right) = - G_H^m\left( { - s,x, - y} \right) \end{split}\right\} $$ (32) 类似地, 利用对偶原理, 可推出对偶Hamilton系统中受小扰动作用时的运动方程、需要满足的条件及绝热不变量, 并将所得结果进行综合, 得如下定理.
定理9 对于对偶系统中受小扰动作用的Hamilton系统
$$ q_j^\nabla = {\partial _{n + j}}H\left( {t,q_i^\rho ,{p_i}} \right) \text{, } p_j^\nabla = - {\partial _j}H - {\varepsilon _H}{W_{Hj}}\left( {t,q_i^\rho ,{p_i}} \right) $$ (33) 如果存在一个函数$ G_H^m\left( {t,q_i^\rho ,{p_i}} \right) $使得$ \xi _{H0}^m\left( {t,{q_i},{p_i}} \right) $和$ \xi _{Hj}^m\left( {t,{q_i},{p_i}} \right) $满足
$$ \begin{split} &{p_j} \cdot \xi _{Hj}^{m\nabla } - {\partial _0}H \cdot \xi _{H0}^m - {\partial _j}H \cdot \xi _{Hj}^{m\rho } - H \cdot \xi _{H0}^{m\nabla }-\\ &\qquad {W_{Hj}}\left[ {\xi _{Hj}^{\left( {m - 1} \right)\rho } - q_i^\nabla \left( t \right) \cdot \xi _{H0}^{\left( {m - 1} \right)\rho }} \right] + G_H^{m\nabla } = 0 \end{split}$$ (34) 则该系统存在如下Noether型绝热不变量
$$ \begin{split} &{C_{Hz}}\left( t \right) = \sum\limits_{m = 0}^z {\varepsilon _H^m} \left\{ { - {p_j} \cdot \xi _{Hj}^m + \left[ {\nu \left( t \right) \cdot {\partial _0}H + H} \right]\xi _{H0}^m} \right.- \\ &\qquad \int_t^{{t_2}} {\Big\{ {{{\left[ { - \nu \left( \tau \right) \cdot {\partial _0}H + {p_j} \cdot {\partial _{n + j}}H - H} \right]}^\nabla } + {\partial _j}H} }\cdot \\ &\qquad \left. { { {\partial _{n + j}}H - {\partial _{n + j}}{H^\nabla } \cdot p_j^\rho + {\partial _0}H} \Big\}\xi _{H0}^{m\rho }{\mathrm{d}} \tau - G_H^m} \right\}\end{split} $$ (35) 其中, 当$ m = 0 $时, $ \xi _{Hj}^{m - 1} = \xi _{H0}^{m - 1} = 0 $.
证明 由方程(33)和式(34)可得
$$ \begin{split} &\frac{\nabla }{{\nabla t}}{C_{Hz}} = - \sum\limits_{m = 0}^z {\varepsilon _H^m} \left\{ {{p_j}\xi _{Hj}^{m\nabla } + p_j^\nabla \xi _{Hj}^{m\rho } - \left[ {\nu \left( t \right) \cdot {\partial _0}H + H} \right]\xi _{H0}^{m\nabla }} \right.- \\ &\qquad {\left[ {\nu \left( t \right) \cdot {\partial _0}H + H} \right]^\nabla }\xi _{H0}^{m\rho } + \xi _{H0}^{m\rho }{\left[ {\nu \left( t \right){\partial _0}H + H} \right]^\nabla }- \\ &\qquad {\left( {{p_j} \cdot {\partial _{n + j}}H} \right)^\nabla }\xi _{H0}^{m\rho } - {\partial _j}H \cdot {\partial _{n + j}}H \cdot \xi _{H0}^{m\rho } + {\left( {{\partial _{n + j}}H} \right)^\nabla }\cdot \\ &\qquad \left. { p_j^\rho \xi _{H0}^{m\rho } - {\partial _0}H \cdot \xi _{H0}^{m\rho } + G_H^{m\nabla }} \right\}=\\ &\qquad - \sum\limits_{m = 0}^z {\varepsilon _H^m} \Big[ {\partial _j}H \cdot \xi _{Hj}^{m\rho } + {W_{Hj}}\left( {\xi _{Hj}^{\left( {m - 1} \right)\rho } - q_i^\nabla \xi _{H0}^{\left( {m - 1} \right)\rho }} \right) + \\ &\qquad p_j^\nabla \cdot \xi _{Hj}^{m\rho } - p_j^\nabla \cdot {\partial _{n + j}}H \cdot \xi _{H0}^{m\rho } - p_j^\rho {\left( {{\partial _{n + j}}H} \right)^\nabla }\xi _{H0}^{m\rho } -\\ &\qquad {\partial _j}H \cdot {\partial _{n + j}}H \cdot \xi _{H0}^{m\rho } + { {{\left( {{\partial _{n + j}}H} \right)}^\nabla } \cdot p_j^\rho \xi _{H0}^{m\rho }} \Big]=\\ &\qquad \varepsilon _H^{z + 1}{W_{Hj}}\left( {\xi _{Hj}^{z\rho } - q_j^\nabla \xi _{H0}^{z\rho }} \right) \end{split}$$ 定理10 对于对偶Hamilton系统
$$ q_j^\nabla = {\partial _{n + j}}H \text{, } p_j^\nabla = - {\partial _j}H $$ (36) 如果存在一个函数$ G_H^0 $使得$ \xi _{H0}^0 $, $ \xi _{Hj}^0 $和$ \eta _{Hj}^0 $满足
$$ {p_j} \cdot \xi _{Hj}^{m\nabla } - {\partial _0}H \cdot \xi _{H0}^m - {\partial _j}H \cdot \xi _{Hj}^{m\rho } - H \cdot \xi _{H0}^{m\nabla } + G_H^{m\nabla } = 0 $$ (37) 则该系统存在一个精确不变量
$$ \begin{split} & {C_{H0}} = - {p_j} \cdot \xi _{Hj}^0 + \left( {\nu \left( t \right) \cdot {\partial _0}H + H} \right)\xi _{H0}^0 - \int_t^{{t_2}} {\Big[ {{\partial _j}H \cdot {\partial _{n + j}}H} }-\\ &\quad { {{\left( {\nu \left( \tau \right) \cdot {\partial _0}H - {p_j} \cdot {\partial _{n + j}}H + H} \right)}^\nabla } - {{\left( {{\partial _{n + j}}H} \right)}^\nabla }p_j^\rho + {\partial _0}H} \Big] \cdot\\ &\quad \xi _{H0}^{0\rho }{\mathrm{d}} \tau - G_H^0 \end{split} $$ (38) 证明 由方程(36)和式(37)可得$ {{\nabla {C_{H0}}} \mathord{\left/ {\vphantom {{\nabla {C_{H0}}} {\nabla t}}} \right. } {\nabla t}} = 0 $.
2.3 对偶Birkhoff系统绝热不变量
定义3[39] 给定Birkhoff函数组$ {R_k}:\mathbb{T} \times {\mathbb{R}^{2 n}} \to \mathbb{R} $, $ k = 1,2, \cdots ,2 n $和Birkhoff函数$ B:\mathbb{T} \times {\mathbb{R}^{2 n}} \to \mathbb{R} $, 定义对偶Birkhoff函数组$ R_k^*:{\mathbb{T}^*} \times {\mathbb{R}^{2 n}} \to \mathbb{R} $和对偶Birkhoff函数$ {B^*}:{\mathbb{T}^*} \times {\mathbb{R}^{2 n}} \to \mathbb{R} $为$ R_k^ * \left( {s,x} \right) = - {R_k}\left( { - s,x} \right) $, $ {B^ * }\left( {s,x} \right) = B\left( { - s,x} \right) $, 且满足
$$\left.\begin{split} &{\partial _0}R_k^ * \left( {s,x} \right) = {\partial _0}{R_k}\left( { - s,x} \right) \text{, } {\partial _0}{B^ * }\left( {s,x} \right) = - {\partial _0}B\left( { - s,x} \right)\\ &{\partial _l}R_k^ * \left( {s,x} \right) = - {\partial _l}{R_k}\left( { - s,x} \right) \text{, } {\partial _l}{B^ * }\left( {s,x} \right) = {\partial _l}B\left( { - s,x} \right)\end{split}\right. $$ 除此之外, 还需要定义如下关系式
$$\left.\begin{split} &\xi _{B0}^{m * }\left( {s,x} \right) = \xi _{B0}^m\left( { - s,x} \right) \text{, } \xi _{Bl}^{m * }\left( {s,x} \right) = \xi _{Bl}^m\left( { - s,x} \right)\\ &G_B^{m * }\left( {s,x} \right) = - G_B^m\left( { - s,x} \right) \text{, } W_{Bl}^ * \left( {s,x} \right) = {W_{Bl}}\left( { - s,x} \right)\end{split}\right\} $$ (39) 利用对偶原理, 可推导出对偶Birkhoff系统中受小扰动作用的运动方程、需要满足的条件及绝热不变量. 再将所得结果进行综合, 可得如下定理.
定理11 对于对偶系统中受小扰动作用的Birkhoff系统
$$ {\partial _l}{R_k}\left( {t,a_\varpi ^\rho } \right) \cdot a_k^\nabla - {\partial _l}B\left( {t,a_\varpi ^\rho } \right) - R_l^\nabla = {\varepsilon _B}{W_{Bl}}\left( {t,a_\varpi ^\rho } \right) $$ (40) 如果存在一个函数$ G_B^m\left( {t,a_\varpi ^\rho } \right) $使得无限小生成元$ \xi _{B0}^m\left( {t,{a_\varpi }} \right) $和$ \xi _{Bk}^m\left( {t,{a_\varpi }} \right) $满足
$$ \begin{split} &\left( {{\partial _0}{R_k} \cdot a_k^\nabla - {\partial _0}B} \right) \cdot \xi _{B0}^m + \left( {{\partial _l}{R_k} \cdot a_k^\nabla - {\partial _l}B} \right) \cdot \xi _{Bl}^{m\rho } + \\ &\qquad {R_k} \cdot \xi _{Bk}^{m\nabla } - B \cdot \xi _{B0}^{m\nabla } + G_B^{m\nabla } - {W_{Bl}} \cdot \left( \xi _{Bl}^{\left( {m - 1} \right)\rho } -\right. \\ &\qquad \left.a_l^\nabla \cdot \xi _{B0}^{\left( {m - 1} \right)\rho } \right) = 0\end{split} $$ (41) 则该系统存在如下Noether型绝热不变量
$$\begin{split} & {C_{B{\text{z}}}}\left( t \right) = - \sum\limits_{m = 0}^z {\varepsilon _B^m} \Big\{ {{R_l}\xi _{Bl}^m + \nu \left( t \right)\left( {{\partial _0}{R_l} \cdot a_l^\nabla - {\partial _0}B} \right)\xi _{B0}^m - \xi _{B0}^m} \cdot\\ &\quad B - \int_t^{{t_2}} {\Big\{ {\left( {{\partial _l}{R_k} \cdot a_k^\nabla - {\partial _l}B} \right)a_l^\nabla + a_l^{\nabla \nabla }R_l^\rho + {\partial _0}{R_l} \cdot a_l^\nabla - {\partial _0}B} }+\\ &\quad { { {{\left[ {B - \nu \left( \tau \right)\left( {{\partial _0}{R_l} \cdot a_l^\nabla - {\partial _0}B} \right) - {R_l}a_l^\nabla } \right]}^\nabla }} \Big\}\xi _{B0}^{m\rho }{\mathrm{d}} \tau + G_B^m} \Big\}\end{split} $$ (42) 其中, 当$ m = 0 $时, $ \xi _{Bl}^{m - 1} = \xi _{B0}^{m - 1} = 0 $.
证明 由方程(40)和式(41)可得
$$ \begin{split} &\frac{\nabla }{{\nabla t}}{C_{B{\text{z}}}} = - \sum\limits_{m = 0}^z {\varepsilon _B^m} \left\{ {{R_l}\xi _{Bl}^{m\nabla } + R_l^\nabla \xi _{Bl}^{m\rho } + \left( {{\partial _0}{R_l} \cdot a_l^\nabla - {\partial _0}B} \right)} \right. \cdot \\ &\qquad \xi _{B0}^{m\nabla } \cdot \nu \left( t \right) + {\left[ {\left( {{\partial _0}{R_l} \cdot a_l^\nabla - {\partial _0}B} \right)\nu \left( t \right)} \right]^\nabla }\xi _{B0}^{m\rho } - B\xi _{B0}^{m\nabla }-\\ &\qquad {B^\nabla }\xi _{B0}^{m\rho } + {B^\nabla }\xi _{B0}^{m\rho } - {\left[ {\nu \left( t \right)\left( {{\partial _0}{R_l} \cdot a_l^\nabla } \right.\left. { - {\partial _0}B} \right)} \right]^{^\nabla }}\xi _{B0}^{m\rho }-\\ & \qquad {\left( {{R_l}a_l^\nabla } \right)^\nabla }\xi _{B0}^{m\rho } + \left( {{\partial _l}{R_k} \cdot a_k^\nabla - {\partial _l}B} \right)a_l^\nabla \xi _{B0}^{m\rho } + G_B^{m\nabla }+\\ &\qquad \left. { a_l^{\nabla \nabla }R_l^\rho \xi _{B0}^{m\rho } + {\partial _0}{R_l} \cdot a_l^\nabla \xi _{B0}^{m\rho } - {\partial _0}B \cdot \xi _{B0}^{m\rho }} \right\}=\\ &\qquad - \sum\limits_{m = 0}^z {\varepsilon _B^m\left[ { - \left( {{\partial _l}{R_k} \cdot a_k^\nabla - {\partial _l}B} \right)\left( {\xi _{Bl}^{m\rho } - a_l^\nabla \xi _{B0}^{m\rho }} \right)} \right.}+\\ &\qquad \left. { R_l^\nabla \left( {\xi _{Bl}^{m\rho } - a_l^\nabla \xi _{B0}^{m\rho }} \right) + {W_{Bl}}\left( {\xi _{Bl}^{\left( {m - 1} \right)\rho } - a_l^\nabla \xi _{B0}^{\left( {m - 1} \right)\rho }} \right)} \right]=\\ &\qquad \varepsilon _B^{z + 1}{W_{Bl}}\left( {\xi _{Bl}^{z\rho } - a_l^\nabla \xi _{B0}^{z\rho }} \right) \end{split}$$ 定理12 对于对偶Birkhoff系统
$$ {\partial _l}{R_k} \cdot a_k^\nabla - {\partial _l}B - R_l^\nabla = 0 $$ (43) 如果存在一个函数$ G_B^0 $使得无限小生成元$ \xi _{B0}^0 $和$ \xi _{Bk}^0 $满足
$$ \begin{split} &\left( {{\partial _0}{R_k} \cdot a_k^\nabla - {\partial _0}B} \right)\xi _{B0}^0 + \left( {{\partial _l}{R_k} \cdot a_k^\nabla - {\partial _l}B} \right)\xi _{Bl}^{0\rho } +\\ &\qquad {R_k}\xi _{Bk}^{0\nabla } - B\xi _{B0}^{0\nabla } + G_B^{0\nabla } = 0 \end{split}$$ (44) 则该系统存在一个精确不变量
$$ \begin{split} & {C_{B0}} = \nu \left( t \right)\left[ {{\partial _0}{R_l} \cdot a_l^\nabla \left( t \right) - {\partial _0}B} \right]\xi _{B0}^0 + G_B^0 + {R_l}\xi _{Bl}^0 - B\xi _{B0}^0-\\ &\quad \int_t^{{t_2}} {\Big\{ {{{\left[ {B - \nu \left( \tau \right)\left( {{\partial _0}{R_l} \cdot a_l^\nabla - {\partial _0}B} \right) - {R_l}a_l^\nabla } \right]}^\nabla } + {\partial _0}{R_l} \cdot a_l^\nabla } }+ \\ &\quad { \left( {{\partial _l}{R_k} \cdot a_k^\nabla - {\partial _l}B} \right) \cdot a_l^\nabla + a_l^{\nabla \nabla } \cdot R_l^\rho - {\partial _0}B} \Big\} \cdot \xi _{B0}^{0\rho }{\mathrm{d}} \tau \end{split} $$ (45) 证明 由方程(43)和式(44)可得$ {{\nabla {C_{B0}}} \mathord{\left/ {\vphantom {{\nabla {C_{B0}}} {\nabla t}}} \right. } {\nabla t}} = 0 $.
定理8、定理10和定理12的精确不变量(方程(31)、方程(38)和方程(45))分别与文献[40]、文献[38]及文献[39]中所得结果不同, 因为文献[38-40]采用了Bartosiewicz等[22]的方法, 而定理8、定理10和定理12借鉴了Anerot等[28]的方法.
3. 应用举例
例1 基于时间尺度$ \mathbb{T} = \left\{{2}^{n}:n\in \mathbb{N}\cup \left\{0\right\}\right\} $, 研究Lagrange系统Noether型绝热不变量, 其中$ {L_1}\left( {t,{q^\sigma },{q^\Delta }} \right) = {{{{\left( {{q^\sigma }} \right)}^2}} \mathord{\left/ {\vphantom {{{{\left( {{q^\sigma }} \right)}^2}} t}} \right. } t} + t{\left( {{q^\Delta }} \right)^2} $
首先, 由方程(14)可得Lagrange方程为
$$ {{{q^\sigma }} \mathord{\left/ {\vphantom {{{q^\sigma }} t}} \right. } t} - {\left( {t{q^\Delta }} \right)^\Delta } = 0 $$ (46) 再由方程(15)可得
$$ \begin{split} &\left[ { - \frac{1}{{{t^2}}}{{\left( {{q^\sigma }} \right)}^2} + {{\left( {{q^\Delta }} \right)}^2}} \right]\xi _{L0}^0 + \frac{1}{t} \cdot 2{q^\sigma }\xi _L^{0\sigma } + 2t{q^\Delta }\xi _L^{0\Delta }+\\ &\qquad \left[ {\frac{1}{t}{{\left( {{q^\sigma }} \right)}^2} - t{{\left( {{q^\Delta }} \right)}^2}} \right]\xi _{L0}^{0\Delta } + G_L^{0\Delta } = 0\end{split} $$ (47) 经计算可得
$$ G_L^0 = 0 \text{, } \xi _{L0}^0 = t \text{, } \xi _L^0 = 0 $$ (48) 为方程(47)的一组解, 那么由定理4可得该系统的一个精确不变量为
$$\begin{split} &{C_{L0}} = 2\left[ {{{\left( {{q^\sigma }} \right)}^2} - {t^2}{{\left( {{q^\Delta }} \right)}^2}} \right] + 2\int_{{t_1}}^t {\Bigg\{ {\left[ { - \frac{1}{{{\tau ^2}}}{{\left( {{q^\sigma }} \right)}^2} + {{\left( {{q^\Delta }} \right)}^2}} \right]} } +\\ &\qquad { 2{{\left[ { - \frac{1}{\tau }{{\left( {{q^\sigma }} \right)}^2} + \tau {{\left( {{q^\Delta }} \right)}^2}} \right]}^\Delta }} \Bigg\}\tau {\mathrm{d}} \tau \end{split}$$ (49) 其次, 当Lagrange系统受到扰动力$ {W_L} = \left( {\dfrac{{{q^\sigma }}}{t} - \dfrac{{{q^\Delta }}}{2}} \right) $的小扰动时, 可表示为
$$ \frac{{{q^\sigma }}}{t} - {\left( {t{q^\Delta }} \right)^\Delta } = {\varepsilon _L} \cdot \left( {\frac{{{q^\sigma }}}{t} - \frac{{{q^\Delta }}}{2}} \right) $$ (50) 由方程(3)可得
$$\begin{split} &\left[ { - \frac{1}{{{t^2}}}{{\left( {{q^\sigma }} \right)}^2} + {{\left( {{q^\Delta }} \right)}^2}} \right]\xi _{L0}^1 + \frac{1}{t} \cdot 2{q^\sigma }\xi _L^{1\sigma } + 2t{q^\Delta }\xi _L^{1\Delta }+\\ &\qquad \left[ {\frac{1}{t}{{\left( {{q^\sigma }} \right)}^2} - t{{\left( {{q^\Delta }} \right)}^2}} \right]\xi _{L0}^{1\Delta } + {W_L} \cdot 2t{q^\Delta } + G_L^{1\Delta } = 0\end{split} $$ (51) 此时可找到
$$ G_L^1 = - {\left( {{q^\sigma } - t{q^\Delta }} \right)^2} \text{, } \xi _{L0}^1 = t \text{, } \xi _L^1 = 0 $$ (52) 为方程(51)的一组解, 从而由定理1可得该扰动系统的Noether型绝热不变量为
$$ \begin{split} &{C_{L1}} = {C_{L0}} + \varepsilon \Bigg\{ {{{\left( {{q^\sigma }} \right)}^2} - 3{t^2}{{\left( {{q^\Delta }} \right)}^2} + 2t{q^\sigma }{q^\Delta }} + \\ &\qquad 2\int_{{t_1}}^t {\left\{ { - \frac{1}{{{\tau ^2}}}{{\left( {{q^\sigma }} \right)}^2} + {{\left( {{q^\Delta }} \right)}^2} + \frac{{2{q^\sigma }}}{\tau } \cdot {q^\Delta } + {q^{\Delta \Delta }}{{\left( {2t{q^\Delta }} \right)}^\sigma }} \right.}+\\ &\qquad {\left. { 2\frac{\Delta }{{\Delta \tau }}\left[ { - \frac{1}{\tau }{{\left( {{q^\sigma }} \right)}^2} + \tau {{\left( {{q^\Delta }} \right)}^2}} \right]} \right\}\tau {\mathrm{d}} \tau } \Bigg\} \end{split} $$ (53) 例2 研究时间尺度上Kepler问题的Noether型绝热不变量, 其中
$$ L = \dfrac{1}{2}\left[ {{{\left( {q_1^\Delta } \right)}^2} + {{\left( {q_2^\Delta } \right)}^2}} \right] - \left[ {{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2} \right]^{ - \frac{1}{2}}$$ 首先, 由方程(5)可得
$${p_1} = q_1^\Delta \text{, } {p_2} = q_2^\Delta \text{, } H = \frac{1}{2}\left( {{p_1}^2 + {p_2}^2} \right) + {\left[ {{{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2}} \right]^{ - \frac{1}{2}}} $$ (54) 由方程(17)可得该系统方程为
$$\left.\begin{split} &q_1^\Delta = {p_1} \text{, } q_2^\Delta = {p_2} \text{, } p_1^\Delta = - q_1^\sigma {\left[ {{{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2}} \right]^{ - \frac{3}{2}}} \\ &p_2^\Delta = - q_2^\sigma {\left[ {{{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2}} \right]^{ - \frac{3}{2}}} \end{split}\right\}$$ (55) 由方程(18)可得
$$ \begin{split} &{p_1}\xi _{H1}^{0\Delta } + {p_2}\xi _{H2}^{0\Delta } - q_1^\sigma {\left[ {{{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2}} \right]^{ - \frac{3}{2}}}\xi _{H1}^{0\sigma }- \\ &\qquad q_2^\sigma {\left[ {{{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2}} \right]^{ - \frac{3}{2}}}\xi _{H2}^{0\sigma } + G_H^{0\Delta }-\\ &\qquad \left\{ {\frac{1}{2}\left( {{p_1}^2 + {p_2}^2} \right) + {{\left[ {{{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2}} \right]}^{ - \frac{1}{2}}}} \right\}\xi _{H0}^{0\Delta } = 0 \end{split}$$ (56) 经计算, 可找到
$$ G_H^0 = 0 \text{, } \xi _{H0}^0 = - 1 \text{, } \xi _{H1}^0 = \xi _{H2}^0 = 0 $$ (57) 为方程(56)的一组解, 因此, 由定理5可得该系统的一个精确不变量为
$$ {C_{H0}} = H + \int_{{t_1}}^t {{H^\Delta }} \cdot \left( { - 1} \right){\mathrm{d}} \tau $$ (58) 受小扰动作用的Hamilton系统可表示为
$$\left.\begin{split} & q_1^\Delta = {p_1} \text{, } q_2^\Delta = {p_2} \text{, } p_1^\Delta = - q_1^\sigma {\left[ {{{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2}} \right]^{ - \frac{3}{2}}} - {\varepsilon _H}{W_{H1}} \\ &p_2^\Delta = - q_2^\sigma {\left[ {{{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2}} \right]^{ - \frac{3}{2}}} - {\varepsilon _H}{W_{H2}}\end{split}\right\} $$ (59) 其中, $ {W_{H1}} = q_2^\sigma - \mu \left( t \right){p_2} $, $ {W_{H2}} = q_1^\sigma $. 由方程(8)可得
$$ \begin{split} &- q_1^\sigma {\left[ {{{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2}} \right]^{ - \frac{3}{2}}}\xi _{H1}^{1\sigma } - q_2^\sigma {\left[ {{{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2}} \right]^{ - \frac{3}{2}}}\xi _{H2}^{1\sigma }+ \\ &\qquad {p_1}\xi _{H1}^{1\Delta } + {p_2}\xi _{H2}^{1\Delta } + G_H^{1\Delta } - {W_{H1}}q_1^\Delta - {W_{H2}}q_2^\Delta-\\ &\qquad \left\{ {\frac{1}{2}\left( {{p_1}^2 + {p_2}^2} \right) + {{\left[ {{{\left( {q_1^\sigma } \right)}^2} + {{\left( {q_2^\sigma } \right)}^2}} \right]}^{ - \frac{1}{2}}}} \right\}\xi _{H0}^{1\Delta } = 0\end{split} $$ (60) 而
$$\left. \begin{split} &G_H^1 = \left( {q_1^\sigma - \mu \left( t \right){p_1}} \right)\left( {q_2^\sigma - \mu \left( t \right){p_2}} \right) \text{, } \xi _{H0}^1 = 0 \\ &\xi _{H1}^1 = - {q_2} \text{, } \xi _{H2}^1 = {q_1}\end{split}\right\} $$ (61) 为方程(60)的一组解. 因此, 定理2给出该扰动系统的一个Noether型绝热不变量为
$$ {C_{H1}} = {C_{H0}} + {\varepsilon _H}\left( {{q_1}{p_2} - {p_1}{q_2} + {q_1}{q_2}} \right) $$ (62) 例3 研究时间尺度上Hojman-Urrutia问题的Noether型绝热不变量, 其中$ \mathbb{T} = \left\{{2}^{n}:n\in \mathbb{N}\cup \left\{0\right\}\right\} $, $ B = \dfrac{1}{2}\left[ {{{\left( {a_3^\sigma } \right)}^2} + 2 a_2^\sigma a_3^\sigma - {{\left( {a_4^\sigma } \right)}^2}} \right] $, $ {R_1} = a_2^\sigma + a_3^\sigma $, $ {R_2} = 0 $, $ {R_3} = a_4^\sigma $, $ {R_4} = 0 $.
首先, 方程(20)给出Birkhoff方程为
$$\left.\begin{split} &{\left( {a_2^\sigma + a_3^\sigma } \right)^\Delta } = 0 \text{, } a_1^\Delta - a_3^\sigma = 0 \text{, }a_1^\Delta - a_3^\sigma - a_2^\sigma - {\left( {a_4^\sigma } \right)^\Delta } = 0 \\ &a_3^\Delta + a_4^\sigma = 0 \end{split}\right\}$$ (63) 再由方程(21)可得
$$\begin{split} &\left( {a_1^\Delta - a_3^\sigma } \right)\xi _{B2}^{0\sigma } + \left( {a_1^\Delta - a_3^\sigma - a_2^\sigma } \right)\xi _{B3}^{0\sigma } + \left( {a_3^\Delta + a_4^\sigma } \right)\xi _{B4}^{0\sigma }- \\ &\qquad \frac{1}{2}\left[ {{{\left( {a_3^\sigma } \right)}^2} + 2a_2^\sigma a_3^\sigma - {{\left( {a_4^\sigma } \right)}^2}} \right]\xi _{B0}^{0\Delta } + \left( {a_2^\sigma + a_3^\sigma } \right)\xi _{B1}^{0\Delta }+\\ &\qquad a_4^\sigma \xi _{B3}^{0\Delta } + G_B^{0\Delta } = 0\end{split} $$ (64) 经计算, 可找到
$$ G_B^0 = 0 \text{, } \xi _{B0}^0 = \xi _{B2}^0 = \xi _{B3}^0 = \xi _{B4}^0 = 0 \text{, } \xi _{B1}^0 = 1 $$ (65) 满足方程(64), 定理6给出该系统一个精确不变量为
$$ {C_{B0}} = a_2^\sigma + a_3^\sigma $$ (66) 设受小扰动作用的Birkhoff系统为
$$ \left.\begin{split} & - {\left( {a_2^\sigma + a_3^\sigma } \right)^\Delta } = {\varepsilon _B}{W_{B1}} \text{, } a_1^\Delta - a_3^\sigma = {\varepsilon _B}{W_{B2}} \\ &a_1^\Delta - a_3^\sigma - a_2^\sigma - {\left( {a_4^\sigma } \right)^\Delta } = {\varepsilon _B}{W_{B3}} \text{, } a_3^\Delta + a_4^\sigma = {\varepsilon _B}{W_{B4}}\end{split} \right\}$$ (67) 其中, $ {W_{B1}} = 3 t $, $ {W_{B2}} $, $ {W_{B3}} $和$ {W_{B4}} $可任意取, 则方程(12)给出
$$ \begin{split} &\left( {a_1^\Delta - a_3^\sigma } \right)\xi _{B2}^{1\sigma } + \left( {a_1^\Delta - a_3^\sigma - a_2^\sigma } \right)\xi _{B3}^{1\sigma } + \left( {a_3^\Delta + a_4^\sigma } \right)\xi _{B4}^{1\sigma }-\\ &\qquad \frac{1}{2}\left[ {{{\left( {a_3^\sigma } \right)}^2} + 2a_2^\sigma a_3^\sigma - {{\left( {a_4^\sigma } \right)}^2}} \right]\xi _{B0}^{1\Delta } + \left( {a_2^\sigma + a_3^\sigma } \right)\xi _{B1}^{1\Delta }+\\ &\qquad a_4^\sigma \xi _{B3}^{1\Delta } + G_B^{1\Delta } - {W_{B1}} = 0\end{split} $$ (68) 由于
$$G_B^1 = - {a_1} + {t^2} \text{, } \xi _{B0}^1 = 0 \text{, } \xi _{B1}^1 = t \text{, } \xi _{B2}^1 = 0 \text{, } \xi _{B3}^1 = 1 \text{, } \xi _{B4}^1 = 0 $$ (69) 为方程(68)的一组解, 则由定理3可得该扰动系统的一阶Noether型绝热不变量为
$$ {C_{B1}} = {C_{B0}} + {\varepsilon _B}\left[ {\left( {a_2^\sigma + a_3^\sigma + t} \right)t + a_4^\sigma - {a_1}} \right] $$ (70) 例1 ~ 例3分别给出了delta导数下Lagrange系统、Hamilton系统和Birkhoff系统的Noether型绝热不变量, 其中的求解过程用到了定理1 ~ 定理6. 而对于nabla导数下3个系统的Noether型绝热不变量同样可以举出例子, 然后利用定理7 ~ 定理12进行求解, 这里不再详细阐述.
4. 结论
本文研究了时间尺度上Lagrange系统、Hamilton系统和Birkhoff系统及它们对偶系统的Noether对称性摄动与绝热不变量问题, 所得定理1 ~ 定理3、定理7、定理9和定理11均是新成果. 特别地, 本文将所得结果进行特殊化处理, 可得时间尺度上Lagrange系统、Hamilton系统和Birkhoff系统及它们对偶系统的Noether定理, 即定理4 ~ 定理6、定理8、定理10和定理12. 除了定理4与已知结论一致外, 其余结果均为新成果.
时间尺度约束力学系统分为迁移和非迁移两种类型, 本文是在迁移系统中开展的研究. 虽然迁移系统很重要, 但是Bourdin[41]指出, 在研究保结构算法时, 非迁移系统更关键. 因此, 可尝试将本文研究内容推广到非迁移系统. 另外, 分数阶时间尺度微积分也是近年来研究的一个热点, 因此基于分数阶时间尺度研究约束力学系统对称性及其摄动问题也有待尝试.
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[1] Mei FX, Wu HB, Zhang YF. Symmetries and conserved quantities of constrained mechanical systems. International Journal of Dynamics and Control, 2014, 2: 285-303 doi: 10.1007/s40435-013-0043-8
[2] Burgers JM. Die adiabatischen invarianten bedingt periodischer systems. Annalen der Physik, 1917, 357(2): 195-202 doi: 10.1002/andp.19173570206
[3] 赵跃宇. 力学系统对称性的摄动与绝热不变量. 湖南大学学报(自然科学版), 1996, 23(1): 45-50 (Zhao Yueyu. Perturbation to the symmetry of dynamical systems and adiabatic invariants. Journal of Hunan University (Natural Sciences), 1996, 23(1): 45-50 (in Chinese) Zhao Yueyu. Perturbation to the symmetry of dynamical systems and adiabatic invariants. Journal of Hunan University (Natural Sciences), 1996, 23(1): 45-50 (in Chinese)
[4] Chen XW, Wang MQ, Wang XM. Perturbation to symmetries and adiabatic invariants of nonholonomic dynamical system of relative motion. Communications in Theoretical Physics, 2005, 43(4): 577-581 doi: 10.1088/0253-6102/43/4/001
[5] 张毅. 广义Birkhoff系统Noether对称性的摄动与绝热不变量. 科技通报, 2010, 26(4): 477-481 (Zhang Yi. Perturbation to Noether symmetries and adiabatic invariants for generalized Birkhoffian systems. Bulletin of Science and Technology, 2010, 26(4): 477-481 (in Chinese) doi: 10.3969/j.issn.1001-7119.2010.04.001 Zhang Yi. Perturbation to Noether symmetries and adiabatic invariants for generalized Birkhoffian systems. Bulletin of Science and Technology, 2010, 26(4): 477-481 (in Chinese) doi: 10.3969/j.issn.1001-7119.2010.04.001
[6] Wang P. Perturbation to symmetry and adiabatic invariants of discrete nonholonomic nonconservative mechanical system. Nonlinear Dynamics, 2012, 68(1-2): 53-62 doi: 10.1007/s11071-011-0203-3
[7] Chen XW, Li YM. Exact invariants and adiabatic invariants of holonomic system in terms of quasi-coordinates. Chinese Physics, 2005, 14(4): 663-668 doi: 10.1088/1009-1963/14/4/005
[8] 傅景礼, 陈立群, 谢凤萍. 相对论性Birkhoff系统的对称性摄动及其逆问题. 物理学报, 2003, 52(11): 2664-2670 (Fu Jingli, Chen Liqun, Xie Fengping. Perturbation to the symmetries of relativistic Birkhoffian systems and the inverse problems. Acta Physica Sinica, 2003, 52(11): 2664-2670 (in Chinese) doi: 10.3321/j.issn:1000-3290.2003.11.004 Fu Jingli, Chen Liqun, Xie Fengping. Perturbation to the symmetries of relativistic Birkhoffian systems and the inverse problems. Acta Physica Sinica, 2003, 52(11): 2664-2670 (in Chinese) doi: 10.3321/j.issn:1000-3290.2003.11.004
[9] Luo SK, Chen XW, Guo YX. Lie symmetrical perturbation and adiabatic invariants of generalized Hojman type for Lagrange systems. Chinese Physics, 2007, 16(11): 3176-3181 doi: 10.1088/1009-1963/16/11/005
[10] Jiang WA, Li L, Li ZJ, et al. Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems. Nonlinear Dynamics, 2012, 67(2): 1075-1081 doi: 10.1007/s11071-011-0051-1
[11] Xia LL, Li YC. Perturbation to symmetries and Hojman adiabatic invariant for nonholonomic controllable mechanical systems with non-Chetaev type constraints. Chinese Physics, 2007, 16(6): 1516-1520 doi: 10.1088/1009-1963/16/6/004
[12] Zhang KJ, Fang JH, Li Y. Perturbation to Mei symmetry and Mei adiabatic invariants for discrete generalized Birkhoffian system. Chinese Physics B, 2011, 20(5): 054501 doi: 10.1088/1674-1056/20/5/054501
[13] Hilger S. Ein maßkettenkalkiil mit anwendung auf zentrumsmannigfaltigkeiten. [PhD Thesis]. Würzburg: Universität Würzburg, 1988
[14] Bohner M, Peterson A. Dynamic Equations on Time Scales—An Introduction with Applications. Boston: Birkhauser, 2001
[15] Bohner M, Peterson A. Advances in Dynamic Equations on Time Scales. Boston: Birkhauser, 2003
[16] Agarwal R, Bohner M, O’Regan D, et al. Dynamic equations on time scales: A survey. Journal of Computational & Applied Mathematics, 2004, 141(1-2): 1-26
[17] Atici FM, Biles DC, Ledebinsky A. An application of time scales to economics. Mathematical & Computer Modelling, 2006, 43(7-8): 718-726
[18] Bohner M, Fan M, Zhang J. Periodicity of scalar dynamic equations and applications to population model. Journal of Mathematical Analysis and Applications, 2007, 330(1): 1-9 doi: 10.1016/j.jmaa.2006.04.084
[19] Bohner M, Hudson T. Euler-type boundary value problems in quantum calculus. International Journal of Applied Mathematics & Statistics, 2007, 9(7): 19-23
[20] Bohner M. Calculus of variations on time scales. Dynamic Systems & Applications, 2004, 13(12): 339-349
[21] Hilscher R, Zeidan V. Calculus of variations on time scales: weaklocal piecewise $ {{C}}_{{\text{rd}}}^1 $ solutions with variable endpoints. Journal of Mathematical Analysis & Applications, 2004, 289(1): 143-166
[22] Bartosiewicz Z, Torres DFM. Noether’s theorem on time scales. Journal of Mathematical Analysis and Applications, 2008, 342(2): 1220-1226 doi: 10.1016/j.jmaa.2008.01.018
[23] Cai PP, Fu JL, Guo YX. Noether symmetries of the nonconservative and nonholonomic systems on time scales. Science China - Physics, Mechanics & Astronomy, 2013, 56(5): 1017-1028
[24] Zu QH, Zhu JQ. Noether theorem for nonholonomic nonconservative mechanical systems in phase space on time scales. Journal of Mathematical Physics, 2016, 57(8): 082701 doi: 10.1063/1.4960471
[25] Peng KK, Luo YP. Dynamics symmetries of Hamiltonian system on time scales. Journal of Mathematical Physics, 2014, 55(4): 042702 doi: 10.1063/1.4871545
[26] 张毅. 时间尺度上Hamilton系统的Noether理论. 力学季刊, 2016, 37(2): 214-224 (Zhang Yi. Noether theory for Hamiltonian system on time scales. Chinese Quarterly of Mechanics, 2016, 37(2): 214-224 (in Chinese) Zhang Yi. Noether theory for Hamiltonian system on time scales. Chinese Quarterly of Mechanics, 2016, 37(2): 214-224 (in Chinese)
[27] Song CJ, Zhang Y. Noether theorem for Birkhoffian systems on time scales. Journal of Mathematical Physics, 2015, 56: 102701 doi: 10.1063/1.4932607
[28] Anerot B, Cresson J, Belgacem KH, et al. Noether’s-type theorems on time scales. Journal of Mathematical Physics, 2020, 61: 113502 doi: 10.1063/1.5140201
[29] Zhai XH, Zhang Y. Lie symmetry analysis on time scales and its application on mechanical systems. Journal of Vibration and Control, 2019, 25(3): 581-592 doi: 10.1177/1077546318790864
[30] 张毅, 田雪, 翟相华等. 时间尺度上Lagrange系统的Hojman守恒量. 力学学报, 2021, 53(10): 2814-2822 (Zhang Yi, Tian Xue, Zhai Xianghua, et al. Hojman conserved quantity for time scales Lagrange systems. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(10): 2814-2822 (in Chinese) doi: 10.6052/0459-1879-21-413 Zhang Yi, Tian Xue, Zhai Xianghua, et al. Hojman conserved quantity for time scales Lagrange systems. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(10): 2814-2822 (in Chinese) doi: 10.6052/0459-1879-21-413
[31] Zhai XH, Zhang Y. Mei symmetry of time-scales Euler-Lagrange equations and its relation to Noether symmetry. Acta Physica Polonica A, 2019, 136(3): 439-443 doi: 10.12693/APhysPolA.136.439
[32] 张毅. 时间尺度上非迁移Birkhoff系统的Mei对称性定理. 物理学报, 2021, 70(24): 244501 (Zhang Y. Mei’s symmetry theorems for non-migrated Birkhoffian systems on a time scale. Acta Physica Sinica, 2021, 70(24): 244501 (in Chinese) doi: 10.7498/aps.70.20210372 Zhang Y. Mei’s symmetry theorems for non-migrated Birkhoffian systems on a time scale. Acta Physica Sinica, 2021, 70(24): 244501 (in Chinese) doi: 10.7498/aps.70.20210372
[33] Zhang Y. Mei’s symmetry theorem for time scales nonshifted mechanical systems. Theoretical and Applied Mechanics Letters, 2021, 11(5): 100286 doi: 10.1016/j.taml.2021.100286
[34] 田雪, 张毅. Caputo型分数阶时间尺度Noether定理. 力学学报, 2021, 53(7): 2010-2022 (Tian Xue, Zhang Yi. Caputo-type fractional time-scales Noether theorem. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(7): 2010-2022 (in Chinese) doi: 10.6052/0459-1879-21-108 Tian Xue, Zhang Yi. Caputo-type fractional time-scales Noether theorem. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(7): 2010-2022 (in Chinese) doi: 10.6052/0459-1879-21-108
[35] Zhang Y. Nonshifted dynamics of constrained systems on time scales under Lagrange framework and its Noether’s theorem. Communications in Nonlinear Science and Numerical Simulation, 2022, 108: 106214 doi: 10.1016/j.cnsns.2021.106214
[36] Atici FM, Guseinov GSh. On Green’s functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics, 2002, 141: 75-99 doi: 10.1016/S0377-0427(01)00437-X
[37] Caputo MC. Time scales: From nabla calculus to delta calculus and vice versa via duality. International Journal of Difference Equations, 2010, 5(1): 25-40
[38] Song CJ, Zhang Y. Conserved quantities for Hamiltonian systems on time scales. Applied Mathematics and Computation, 2017, 313: 24-36 doi: 10.1016/j.amc.2017.05.074
[39] Song CJ, Zhang Y. Noether theory for Birkhoffian systems with nabla derivatives. Journal of Nonlinear Sciences and Applications, 2017, 10: 2268-2282 doi: 10.22436/jnsa.010.04.76
[40] Martins N, Torres DFM. Noether’s symmetry theorem for nabla problems of the calculus of variations. Applied Mathematics Letters, 2010, 23: 1432-1438 doi: 10.1016/j.aml.2010.07.013
[41] Bourdin L. Nonshifted calculus of variations on time scales with $ \nabla $-differentiable $ \sigma $. Journal of Mathematical Analysis and Applications, 2014, 411: 543-554 doi: 10.1016/j.jmaa.2013.10.013
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