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研究并证明时间尺度上非迁移Birkhoff系统的Mei对称性定理. 首先, 建立任意时间尺度上Pfaff-Birkhoff原理和广义Pfaff-Birkhoff原理, 由此导出时间尺度上非迁移Birkhoff系统(包括自由Birkhoff系统、广义Birkhoff系统和约束Birkhoff系统)的动力学方程. 其次, 基于非迁移Birkhoff方程中的动力学函数经历变换后仍满足原方程的不变性, 给出了时间尺度上Mei对称性的定义, 导出了相应的判据方程. 再次, 建立并证明了时间尺度上非迁移Birkhoff系统的Mei对称性定理, 得到了时间尺度上Birkhoff系统的Mei守恒量. 并通过3个算例说明了结果的应用.
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关键词:
- Birkhoff系统 /
- Mei对称性定理 /
- 时间尺度 /
- 非迁移变分学
The Mei symmetry and its corresponding conserved quantities for non-migrated Birkhoffian systems on a time scale are proposed and studied. Firstly, the dynamic equations of non-migrated Birkhoffian systems (including free Birkhoffian systems, generalized Birkhoffian systems and constrained Birkhoffian systems) on a time scale are derived based on the time-scale Pfaff-Birkhoff principle and time-scale generalized Birkhoff principle. Secondly, based on the fact that the dynamical functions in the non-migrated Birkhoff’s equations still satisfy the original equations after they have been transformed, the definitions of Mei symmetry on an arbitrary time scale are given, and the corresponding criterion equations are derived. Thirdly, Mei’s symmetry theorems for non-migrated Birkhoffian systems on a time scales are established and proved, and Mei conserved quantities of Birkhoffian systems on a time scale are obtained. The results are illustrated by three examples.1. 引 言
Birkhoff力学起源于Birkhoff[1]的著作《动力系统》. Santilli[2]首次提出Birkhoff力学一词, 并详细地讨论了Birkhoff方程的构造、变换理论及其对强子物理的应用. 梅凤翔等[3]和Galiullin等[4]从各自角度分别独立地研究了Birkhoff系统动力学, 他们的研究各具特色且更侧重于分析力学. 文献[5]构建了广义Birkhoff系统动力学. 梅凤翔先生[6]指出Birkhoff力学是分析力学发展的第4个阶段. 近年来, Birkhoff力学在对称性理论[7-13]、几何动力学[14,15]、全局分析与稳定性[16,17]、数值计算[18-22]等研究方向上都取得了重要进展.
时间尺度, 即实数集的任意非空闭子集, 最早是由Hilger博士[23]引进的. 由于实数集和整数集本身就是一类特殊的时间尺度, 因而在时间尺度上不仅可以统一地处理连续系统和离散系统, 而且可以处理既有连续又有离散的复杂动力学过程. 近20年来, 时间尺度分析理论不仅在理论上不断完善[24-26], 其应用领域也在不断拓广[27-34]. 文献[35]最早提出并研究了时间尺度上基于delta导数的自由Birkhoff系统动力学及其Noether对称性. 文献[36]利用对偶原理将文献[35]的结果拓展到nabla导数情形. 文献[37]给出了时间尺度上非迁移Birkhoff系统的Noether定理. 但是, 这些研究尚限于: 1)自由Birkhoff系统; 2) Noether对称性; 3)守恒量是Noether型的. 文献[38, 39]初步研究了时间尺度上Birkhoff系统的Lie对称性和Mei对称性, 但是其守恒量的证明基于第二Euler-Lagrange方程, 而数值计算表明该方程并不成立[34]. 此外, 根据Bourdin[33]的研究, 在离散层面非迁移情形的结果是保变分结构及其相关性质的, 尽管迄今时间尺度上非迁移变分问题研究还很少. 本文研究时间尺度上非迁移Birkhoff系统的Mei对称性, 包括自由Birkhoff系统、广义Birkhoff系统和约束Birkhoff系统, 建立并证明上述3类Birkhoff系统的Mei对称性定理, 给出时间尺度上新型守恒量, 称之为Mei守恒量.
2. 时间尺度上非迁移Birkhoff方程
关于时间尺度上微积分及其基本性质, 读者可参阅文献[24, 25].
2.1 Pfaff-Birkhoff原理及其推广
在时间尺度上, 非迁移Pfaff作用量为
$$ A = \int_{{t_1}}^{{t_2}} {\left[ {{R_\beta }\left( {t,{a_\gamma }\left( t \right)} \right)a_\beta ^\Delta - B\left( {t,{a_\gamma }\left( t \right)} \right)} \right]} \Delta t , $$ (1) 其中
$ {R_\beta }:{\mathbb{T}} \times {\mathbb{R}^{2 n}} \to \mathbb{R} $ 是时间尺度上Birkhoff函数组,$ B:{\mathbb{T}} \times {\mathbb{R}^{2 n}} \to \mathbb{R} $ 是时间尺度上Birkhoff函数,$ a_\beta ^\Delta $ 是Birkhoff变量$ {a_\beta } $ 对时间的delta导数. 设所有函数都是$ C_{{\text{rd}}}^{{\text{1, }}\Delta }\left( {\mathbb{T}} \right) $ 函数.$ \beta , \gamma = 1, 2, \cdots , 2 n $ . 非迁移是指作用量(1)中的变量$ {a_\gamma } $ 没有经过前跳算子$\sigma $ 或后跳算子$\rho $ 的作用而发生跃迁[33].等时变分原理
$$ {\text{δ}}A = 0 , $$ (2) 且满足端点条件
$$ {\left. {{\text{δ}}{a_\beta }} \right|_{t = {t_1}}} = {\left. {{\text{δ}}{a_\beta }} \right|_{t = {t_2}}} = 0 , $$ (3) 以及互易关系
$$ {\text{δ}}a_\beta ^\Delta = {\left( {{\text{δ}}{a_\beta }} \right)^\Delta } . $$ (4) 原理(2)称为时间尺度上非迁移Pfaff-Birkhoff原理.
等时变分原理(2) 可推广为
$$ \int_{{t_1}}^{{t_2}} {\left[ {\text{δ}\left({{R_\beta }a_\beta ^\Delta - B} \right) + {\varPhi _\beta }{\text{δ}}{a_\beta }} \right]} \Delta t = 0 , $$ (5) 式中
${\varPhi _\beta } = {\varPhi _\beta }\left( {t, {a_\gamma }} \right)$ 表示附加项[5]. 原理(5)式可称为时间尺度上非迁移广义Pfaff-Birkhoff原理.2.2 自由Birkhoff系统
由原理(2), 容易导出
$$ \int_{{t_1}}^{{t_2}} {\left[ {{R_\beta } + \int_{{t_1}}^{\sigma \left( t \right)} {\left( {\frac{{\partial B}}{{\partial {a_\beta }}} - \frac{{\partial {R_\gamma }}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right)\Delta \tau } } \right]{\text{δ}}a_\beta ^\Delta \Delta t} = 0 , $$ (6) 其中
$ \sigma \left( t \right) $ 是前跳算子. 考虑到${\text{δ}}a_\beta ^\Delta$ 的独立性, 由时间尺度上Dubois-Reymond引理[24], 得到$$ {R_\beta } + \int_{{t_1}}^{\sigma \left( t \right)} {\left( {\frac{{\partial B}}{{\partial {a_\beta }}} - \frac{{\partial {R_\gamma }}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right)\Delta \tau } = {C_\beta } , $$ (7) 其中
$ {C_\beta } $ 为常数. 因此有$$\begin{split} & \frac{\nabla }{{\nabla t}}{R_\beta } + {\sigma ^\nabla }\left( {\frac{{\partial B}}{{\partial {a_\beta }}} - \frac{{\partial {R_\gamma }}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right) = 0 \\ & \qquad \left( {\beta = 1,2, \cdots ,2n} \right) . \end{split} $$ (8) 方程(8)为时间尺度上非迁移Birkhoff方程.
2.3 广义Birkhoff系统
由原理(5), 可导出
$$\begin{split} &\int_{{t_1}}^{{t_2}} \left[ {{R_\beta } + \int_{{t_1}}^{\sigma \left( t \right)} {\left( {\frac{{\partial B}}{{\partial {a_\beta }}} - \frac{{\partial {R_\gamma }}}{{\partial {a_\beta }}}a_\gamma ^\Delta - {\varPhi _\beta }} \right)\Delta \tau } } \right]\\&\times{\text{δ}}a_\beta ^\Delta \Delta t = 0 . \end{split}$$ (9) 类似于方程(8), 有
$$\begin{split} &\frac{\nabla }{{\nabla t}}{R_\beta } + {\sigma ^\nabla }\left( {\frac{{\partial B}}{{\partial {a_\beta }}} - \frac{{\partial {R_\gamma }}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right) \\=\;& {\sigma ^\nabla }{\varPhi _\beta }\quad\left( {\beta = 1,2, \cdots ,2n} \right) . \end{split}$$ (10) 方程(10)可称为时间尺度上非迁移广义Birkhoff方程.
2.4 约束Birkhoff系统
约束方程为
$$ {f_j}\left( {t,{a_\beta }} \right) = 0~~\left( {j = 1,2, \cdots ,2g} \right) , $$ (11) 将(11)式取变分, 得
$$ \frac{{\partial {f_j}}}{{\partial {a_\beta }}}{\text{δ}}{a_\beta } = 0 . $$ (12) $$ \frac{\nabla }{{\nabla t}}{R_\beta } + {\sigma ^\nabla }\left( {\frac{{\partial B}}{{\partial {a_\beta }}} - \frac{{\partial {R_\gamma }}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right) = {\sigma ^\nabla }{\lambda _j}\frac{{\partial {f_j}}}{{\partial {a_\beta }}} , $$ (13) 其中
$ {\lambda _j} = {\lambda _j}\left( {t, {a_\beta }} \right) $ 为约束乘子. 假设约束(11)式相互独立, 则由(11)式和(13)式可解出$ {\lambda _j} $ . 于是方程(13)可写成$$ \frac{\nabla }{{\nabla t}}{R_\beta } + {\sigma ^\nabla }\left( {\frac{{\partial B}}{{\partial {a_\beta }}} - \frac{{\partial {R_\gamma }}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right) = {\sigma ^\nabla }{P_\beta } , $$ (14) 其中
${P_\beta } = {\lambda _j}\dfrac{{\partial {f_j}}}{{\partial {a_\beta }}}$ . 方程(14)可视作与约束Birkhoff系统(13)和(11)相应的自由Birkhoff系统. 只要初始条件满足约束方程(11), 那么方程(14)的解就给出约束Birkhoff系统的运动.3. Mei对称性
3.1 自由Birkhoff系统
引进无限小变换
$$\begin{split}&\; \bar t = t + \upsilon {\xi _0}\left( {t,{a_\gamma }} \right),\\&\;{\bar a_\beta }\left( {\bar t} \right) = {a_\beta }\left( t \right) + \upsilon {\xi _\beta }\left( {t,{a_\gamma }} \right) \\&\left( {\beta ,\gamma = 1,2, \cdots ,2n} \right) , \end{split}$$ (15) 其中映射
$ t \mapsto \vartheta \left( t \right) = t + \upsilon {\xi _0} + o\left( \upsilon \right) $ 是1个严格递增$C_{{\text{rd}}}^{1, \Delta }$ 函数,$\upsilon \in {\mathbb{R}}$ 是无限小参数,$ \vartheta \left( t \right) $ 是一个新的时间尺度${\bar {\mathbb{T}}}$ , 前跳算子为${\bar \sigma}$ , delta导数为${\bar \Delta}$ .在变换(15)下, 动力学函数
$B$ 和${R_\beta }$ 变换为$\bar B$ 和${\bar R_\beta }$ , 有$$\begin{split} &\bar B = B\left( {\bar t,{{\bar a}_\gamma }\left( {\bar t} \right)} \right) = B\left( {\vartheta \left( t \right),\left( {{{\bar a}_\gamma } \circ \vartheta } \right)\left( t \right)} \right),\\ &{\bar R_\beta } = {R_\beta }\left( {\bar t,{{\bar a}_\gamma }\left( {\bar t} \right)} \right) = {R_\beta }\left( {\vartheta \left( t \right),\left( {{{\bar a}_\gamma } \circ \vartheta } \right)\left( t \right)} \right) . \end{split}$$ (16) 将(16)式在
$\upsilon = 0$ 处Taylor级数展开, 得到$$\begin{split} &\bar B = B\left( {t,{a_\gamma }} \right) + \upsilon {Y^{\left( 0 \right)}}\left( B \right) + O\left( {{\upsilon ^2}} \right) ,\\&{\bar R_\beta } = {R_\beta }\left( {t,{a_\gamma }} \right) + \upsilon {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right) + O\left( {{\upsilon ^2}} \right) , \end{split}$$ (17) 其中
${Y^{\left( 0 \right)}} = {\xi _0}{\partial \mathord{\left/ {\vphantom {\partial {\partial t}}} \right. } {\partial t}} + {\xi _\beta }{\partial \mathord{\left/ {\vphantom {\partial {\partial {a_\beta }}}} \right. } {\partial {a_\beta }}}$ .定义1 对于时间尺度上非迁移Birkhoff系统(8), 如果
$$ \frac{\nabla }{{\nabla t}}{\bar R_\beta } + {\sigma ^\nabla }\left( {\frac{{\partial \bar B}}{{\partial {a_\beta }}} - \frac{{\partial {{\bar R}_\gamma }}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right) = 0 $$ (18) 成立, 则变换(15)称为Mei对称性的.
判据1 如果变换(15)满足如下判据方程:
$$ \frac{\nabla }{{\nabla t}}{Y^{\left( 0 \right)}}\left( {{R_\beta }} \right) + {\sigma ^\nabla }\left[ {\frac{{\partial {Y^{\left( 0 \right)}}\left( B \right)}}{{\partial {a_\beta }}} - \frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\gamma }} \right)}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right] = 0 , $$ (19) 则变换相应于时间尺度上非迁移Birkhoff系统(8)的Mei对称性.
3.2 广义Birkhoff系统
设时间尺度上动力学函数
$B$ ,${R_\beta }$ 和${\varPhi _\beta }$ 经历变换(15)后, 成为$\bar B$ ,${\bar R_\beta }$ 和${\bar \varPhi _\beta }$ , 有$$\begin{split}& \bar B = B\left( {t,{a_\gamma }} \right) + \upsilon {Y^{\left( 0 \right)}}\left( B \right) + O\left( {{\upsilon ^2}} \right) ,\\&{\bar R_\beta } = {R_\beta }\left( {t,{a_\gamma }} \right) + \upsilon {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right) + O\left( {{\upsilon ^2}} \right) ,\\&{\bar \varPhi _\beta } = {\varPhi _\beta }\left( {t,{a_\gamma }} \right) + \upsilon {Y^{\left( 0 \right)}}\left( {{\varPhi _\beta }} \right) + O\left( {{\upsilon ^2}} \right) . \end{split}$$ (20) 于是有下述定义2和判据2.
定义2 对于时间尺度上非迁移广义Birkhoff系统(10), 如果
$$ \frac{\nabla }{{\nabla t}}{\bar R_\beta } + {\sigma ^\nabla }\left( {\frac{{\partial \bar B}}{{\partial {a_\beta }}} - \frac{{\partial {{\bar R}_\gamma }}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right) = {\sigma ^\nabla }{\bar \varPhi _\beta } $$ (21) 成立, 则变换(15)称为Mei对称性的.
判据2 如果变换(15)满足如下判据方程:
$$\begin{split} &\frac{\nabla }{{\nabla t}}{Y^{\left( 0 \right)}}\left( {{R_\beta }} \right) + {\sigma ^\nabla }\left[ {\frac{{\partial {Y^{\left( 0 \right)}}\left( B \right)}}{{\partial {a_\beta }}} - \frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\gamma }} \right)}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right] \\=\;& {\sigma ^\nabla }{Y^{\left( 0 \right)}}\left( {{\varPhi _\beta }} \right) , \\[-10pt]\end{split}$$ (22) 则变换相应于时间尺度上非迁移广义Birkhoff系统(10)的Mei对称性.
3.3 约束Birkhoff系统
设时间尺度上动力学函数
$B$ ,${R_\beta }$ 和${P_\beta }$ , 以及约束${f_j}$ 经历变换(15)后, 成为$\bar B$ ,${\bar R_\beta }$ ,${\bar P_\beta }$ 和${\bar f_j}$ , 有$$\begin{split}& \bar B = B\left( {t,{a_\gamma }} \right) + \upsilon {Y^{\left( 0 \right)}}\left( B \right) + O\left( {{\upsilon ^2}} \right) ,\\&{\bar R_\beta } = {R_\beta }\left( {t,{a_\gamma }} \right) + \upsilon {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right) + O\left( {{\upsilon ^2}} \right) ,\\ &{\bar P_\beta } = {P_\beta }\left( {t,{a_\gamma }} \right) + \upsilon {Y^{\left( 0 \right)}}\left( {{P_\beta }} \right) + O\left( {{\upsilon ^2}} \right) ,\\ &{\bar f_j} = {f_j}\left( {t,{a_\gamma }} \right) + \upsilon {Y^{\left( 0 \right)}}\left( {{f_j}} \right) + O\left( {{\upsilon ^2}} \right) , \end{split}$$ (23) 于是有下述定义3和判据3.
定义3 对于时间尺度上与约束Birkhoff系统(13)和(11)相应的自由Birkhoff系统(14), 如果
$$ \frac{\nabla }{{\nabla t}}{\bar R_\beta } + {\sigma ^\nabla }\left( {\frac{{\partial \bar B}}{{\partial {a_\beta }}} - \frac{{\partial {{\bar R}_\gamma }}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right) = {\sigma ^\nabla }{\bar P_\beta } $$ (24) 成立, 则变换(15)称为Mei对称性的.
判据3 如果变换(15)满足如下判据方程:
$$\begin{split} &\frac{\nabla }{{\nabla t}}{Y^{\left( 0 \right)}}\left( {{R_\beta }} \right) + {\sigma ^\nabla }\left[ {\frac{{\partial {Y^{\left( 0 \right)}}\left( B \right)}}{{\partial {a_\beta }}} - \frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\gamma }} \right)}}{{\partial {a_\beta }}}a_\gamma ^\Delta } \right] \\=\;& {\sigma ^\nabla }{Y^{\left( 0 \right)}}\left( {{P_\beta }} \right) , \\[-10pt]\end{split}$$ (25) 则变换相应于时间尺度上相应自由Birkhoff系统(14)的Mei对称性.
定义4 对于时间尺度上约束Birkhoff系统(13)和(11), 如果方程(24)以及如下方程
$$ {\bar f_j} = {f_j}\left( {\bar t,{{\bar a}_\gamma }\left( {\bar t} \right)} \right) = 0\quad\left( {j = 1,2, \cdots ,g} \right) $$ (26) 成立, 则变换(15)称为Mei对称性的.
判据4 如果变换(15)满足判据方程(25)和如下限制方程:
$$ {Y^{\left( 0 \right)}}\left( {{f_j}} \right) = 0 , $$ (27) 则变换相应于时间尺度上约束Birkhoff系统(13)和(11)的Mei对称性.
4. Mei对称性定理
4.1 自由Birkhoff系统
定理1 假设变换(15)满足判据方程(19), 则时间尺度上非迁移Birkhoff系统(8)存在新型守恒量
$$\begin{split} {I_{\text{M}}} =\;& {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)\xi _\beta ^\sigma - {Y^{\left( 0 \right)}}\left( B \right)\xi _0^\sigma + G_{\text{M}}^\sigma + \int_{{t_1}}^t {\xi _0}\Bigg\{ {\sigma ^\nabla }a_\beta ^\Delta \frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)}}{{\partial t}} + \frac{\nabla }{{\nabla t}}{Y^{\left( 0 \right)}}\left( B \right) - {\sigma ^\nabla }\frac{{\partial {Y^{\left( 0 \right)}}\left( B \right)}}{{\partial t}} \Bigg\}\nabla t , \\[-15pt]\end{split}$$ (28) 其中
${G_{\text{M}}}$ 是规范函数, 满足$$\begin{split} {Y^{\left( 0 \right)}}\big[ {{Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)} \big]a_\beta ^\Delta - {Y^{\left( 0 \right)}}\big[ {{Y^{\left( 0 \right)}}\left( B \right)} \big] + {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)\xi _\beta ^\Delta - {Y^{\left( 0 \right)}}\left( B \right)\xi _0^\Delta + G_{\text{M}}^\Delta = 0 . \end{split}$$ (29) 证明
$$\begin{split} \frac{\nabla }{{\nabla t}}{I_{\text{M}}} =\;& {\xi _\beta }\frac{\nabla }{{\nabla t}}{Y^{\left( 0 \right)}}\left( {{R_\beta }} \right) + {\sigma ^\nabla }\xi _\beta ^\Delta {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right) - {\xi _0}\frac{\nabla }{{\nabla t}}{Y^{\left( 0 \right)}}\left( B \right) - {\sigma ^\nabla }\xi _0^\Delta {Y^{\left( 0 \right)}}\left( B \right) + \frac{\nabla }{{\nabla t}}G_{\text{M}}^\sigma \\& + {\xi _0}{\sigma ^\nabla }a_\beta ^\Delta \frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)}}{{\partial t}} + {\xi _0}\frac{\nabla }{{\nabla t}}{Y^{\left( 0 \right)}}\left( B \right) - {\xi _0}{\sigma ^\nabla }\frac{{\partial {Y^{\left( 0 \right)}}\left( B \right)}}{{\partial t}} \\ =\;& {\sigma ^\nabla }\Big\{ {Y^{\left( 0 \right)}}\left[ {{Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)} \right]a_\beta ^\Delta - {Y^{\left( 0 \right)}}\left[ {{Y^{\left( 0 \right)}}\left( B \right)} \right] + {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)\xi _\beta ^\Delta - {Y^{\left( 0 \right)}}\left( B \right)\xi _0^\Delta + G_{\text{M}}^\Delta \Big\} \\ & + {\xi _\beta }\Bigg\{ \frac{\nabla }{{\nabla t}}{Y^{\left( 0 \right)}}\left( {{R_\beta }} \right) + {\sigma ^\nabla }\Bigg[ \frac{{\partial {Y^{\left( 0 \right)}}\left( B \right)}}{{\partial {a_\beta }}} - \frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\gamma }} \right)}}{{\partial {a_\beta }}}a_\gamma ^\Delta \Bigg]\Bigg\} . \end{split}$$ (30) $$ \frac{\nabla }{{\nabla t}}{I_{\text{M}}} = 0 . $$ (31) 因此, (28)式是系统的守恒量. 证毕.
定理1可称为时间尺度上非迁移Birkhoff系统(8)的Mei对称性定理, (28)式称为Mei守恒量.
4.2 广义Birkhoff系统
定理2 假设变换(15)满足判据方程(22), 则时间尺度上非迁移广义Birkhoff系统(10)存在新型守恒量
$$\begin{split} {I_{\text{M}}} =\;& {Y^{(0)}}\left( {{R_\beta }} \right)\xi _\beta ^\sigma - {Y^{(0)}}( B )\xi _0^\sigma + G_{\text{M}}^\sigma \\ & + \int_{{t_1}}^t {\xi _0}\bigg[-{\sigma ^\nabla }{Y^{(0)}}\left( {{\varPhi _\beta }} \right)a_\beta ^\Delta + {\sigma ^\nabla }\frac{{\partial {Y^{(0)}}\left( {{R_\beta }} \right)}}{{\partial t}}a_\beta ^\Delta + \frac{\nabla }{{\nabla t}}{Y^{(0)}}(B) - {\sigma ^\nabla }\frac{{\partial {Y^{(0)}}(B)}}{{\partial t}} \bigg]\nabla t , \\[-15pt]\end{split}$$ (32) 其中
${G_{\text{M}}}$ 是规范函数, 满足$$\begin{split} &{Y^{(0)}}\big[{{Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)}\big] a_\beta ^\Delta - {Y^{(0)}}\big[{{Y^{(0)}}(B)}\big] + {Y^{( 0 )}}\left( {{R_\beta }} \right)\xi _\beta ^\Delta - {Y^{(0)}}(B)\xi _0^\Delta + {Y^{(0)}}\left( {{\varPhi _\beta }} \right)\left( {{\xi _\beta } - a_\beta ^\Delta {\xi _0}} \right) + G_{\text{M}}^\Delta = 0 . \end{split}$$ (33) 证明
$$\begin{split} \frac{\nabla }{{\nabla t}}{I_{\text{M}}} =\;& {\xi _\beta }\frac{\nabla }{{\nabla t}}{Y^{(0)}}\left( {{R_\beta }} \right) + {\sigma ^\nabla }\xi _\beta ^\Delta {Y^{(0)}}\left( {{R_\beta }} \right) - {\xi _0}\frac{\nabla }{{\nabla t}}{Y^{(0)}}(B) - {\sigma ^\nabla }\xi _0^\Delta {Y^{(0)}}(B) + \frac{\nabla }{{\nabla t}}G_{\text{M}}^\sigma - {\xi _0}{\sigma ^\nabla }{Y^{(0)}}\left( {{\varPhi _\beta }} \right)a_\beta ^\Delta \\ &+ {\xi _0}{\sigma ^\nabla }a_\beta ^\Delta \frac{{\partial {Y^{(0)}}\left( {{R_\beta }} \right)}}{{\partial t}} + {\xi _0}\frac{\nabla }{{\nabla t}}{Y^{(0)}}(B) - {\xi _0}{\sigma ^\nabla }\frac{{\partial {Y^{(0)}}(B)}}{{\partial t}}\\ =\;& {\sigma ^\nabla }\Big\{ {Y^{(0)}}\big[ {{Y^{(0)}}\left( {{R_\beta }} \right)} \big]a_\beta ^\Delta - {Y^{(0)}}\big[{{Y^{(0)}}(B)} \big] + {Y^{(0)}}\left( {{R_\beta }} \right)\xi _\beta ^\Delta - {Y^{(0)}}(B)\xi _0^\Delta + {Y^{(0)}}\left( {{\varPhi _\beta }} \right)( {{\xi _\beta } - a_\beta ^\Delta {\xi _0}} ) + G_{\text{M}}^\Delta \Big\} \\&+ {\xi _\beta }\Bigg\{ \frac{\nabla }{{\nabla t}}{Y^{(0)}}\left( {{R_\beta }} \right) + {\sigma ^\nabla }\Bigg[ \frac{{\partial {Y^{(0)}}(B)}}{{\partial {a_\beta }}} - \frac{{\partial {Y^{(0)}}\left( {{R_\gamma }} \right)}}{{\partial {a_\beta }}}a_\gamma ^\Delta \Bigg] - {\sigma ^\nabla }{Y^{(0)}}\left( {{\Phi _\beta }} \right) \Bigg\}.\\[-12pt] \end{split}$$ (34) 将方程(22)和方程(33)代入(34)式, 得到
$\dfrac{\nabla }{{\nabla t}}{I_{\text{M}}} = 0$ , 于是(32)式是系统的守恒量.定理2可称为时间尺度上非迁移广义Birkhoff系统(10)的Mei对称性定理, (32)式称为Mei守恒量. 证毕.
4.3 约束Birkhoff系统
定理3 假设变换(15)满足判据方程(25), 则时间尺度上与约束Birkhoff系统(13)和(11)相应的自由Birkhoff系统(14)存在新型守恒量
$$\begin{split} {I_{\text{M}}} =\;& {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)\xi _\beta ^\sigma - {Y^{\left( 0 \right)}}\left( B \right)\xi _0^\sigma + G_{\text{M}}^\sigma\\&+ \int_{{t_1}}^t {\xi _0}\Bigg[ - {\sigma ^\nabla }{Y^{(0)}}\left( {{P_\beta }} \right)a_\beta ^\Delta + {\sigma ^\nabla }a_\beta ^\Delta \frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)}}{{\partial t}}\\& + \frac{\nabla }{{\nabla t}}{Y^{\left( 0 \right)}}\left( B \right) - {\sigma ^\nabla }\frac{{\partial {Y^{\left( 0 \right)}}\left( B \right)}}{{\partial t}} \Bigg]\nabla t , \\[-18pt]\end{split}$$ (35) 其中
${G_{\text{M}}}$ 是规范函数, 满足$$\begin{split}& {Y^{\left( 0 \right)}}\left[ {{Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)} \right]a_\beta ^\Delta - {Y^{\left( 0 \right)}}\left[ {{Y^{\left( 0 \right)}}\left( B \right)} \right] \\& + {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)\xi _\beta ^\Delta - {Y^{\left( 0 \right)}}\left( B \right)\xi _0^\Delta \\& + {Y^{\left( 0 \right)}}\left( {{P_\beta }} \right)\left( {{\xi _\beta } - a_\beta ^\Delta {\xi _0}} \right) + G_{\text{M}}^\Delta = 0 . \end{split}$$ (36) 定理4 假设变换(15)满足判据方程(25)和限制条件(27)式, 则时间尺度上约束Birkhoff系统(13)和(11)存在新型守恒量(35), 其中规范函数
${G_{\text{M}}}$ 满足方程(36).定理3为时间尺度上与约束Birkhoff系统(13)和(11)相应的自由Birkhoff系统(14)的Mei对称性定理. 定理4为时间尺度上非迁移约束Birkhoff系统的Mei对称性定理, (35)式是Mei守恒量.
5. 算 例
例1 研究时间尺度上Birkhoff系统, 设Birkhoff函数和Birkhoff函数组为
$$\begin{split} &B = \frac{1}{2}\left[ {{{\left( {{a_3}} \right)}^2} + 2{a_2}{a_3} - {{\left( {{a_4}} \right)}^2}} \right]\text{, } \\& {R_1} = {a_2} + {a_3}\text{, }{R_2} = 0\text{, }{R_3} = {a_4}\text{, }{R_4} = 0. \end{split}$$ (37) 试研究该系统的Mei对称性与守恒量.
由方程(8)得到
$$\begin{split} &\frac{\nabla }{{\nabla t}}{a_2} + \frac{\nabla }{{\nabla t}}{a_3} = 0\text{, } - {\sigma ^\nabla }\left( {a_1^\Delta - {a_3}} \right) = 0\text{, } \\&\frac{\nabla }{{\nabla t}}{a_4} - {\sigma ^\nabla }\left( {a_1^\Delta - {a_3} - {a_2}} \right) = 0\text{, } \\& - {\sigma ^\nabla }\left( {a_3^\Delta + {a_4}} \right) = 0. \end{split}$$ (38) 如取
${\mathbb{T}} = {\mathbb{R}}$ , 则方程(38)成为$$ \begin{split} & {\dot a_2} + {\dot a_3} = 0\text{, } {\dot a_1} - {a_3} =0\text{, } \\ & {\dot a_4} - {\dot a_1} + {a_3} + {a_2} = 0\text{, } {\dot a_3} + {a_4} = 0. \end{split}$$ (39) 这是著名的Hojman-Urrutia问题[3,4]. 该问题本质上不是自伴随的, 因此没有Lagrange结构或Hamilton结构.
下面来计算Mei对称性. 经计算, 有
$$\begin{split}& {Y^{\left( 0 \right)}}\left( B \right) = {\xi _2}{a_3} + {\xi _3}\left( {{a_2} + {a_3}} \right) - {\xi _4}{a_4} \text{, } \\ &{Y^{\left( 0 \right)}}\left( {{R_1}} \right) = {\xi _2} + {\xi _3} \text{, } ~~{Y^{\left( 0 \right)}}\left( {{R_2}} \right) = 0 \text{, }\\ &{Y^{\left( 0 \right)}}\left( {{R_3}} \right) = {\xi _4} \text{, }~~ {Y^{\left( 0 \right)}}\left( {{R_4}} \right) = 0 . \end{split}$$ (40) 取生成函数为
$$\begin{split} {\xi _0} =\;& 1\text{, }~~{\xi _1} = a_2^\rho + a_3^\rho + \rho \left( t \right)\text{, }\\ {\xi _2} =\;& 2\left( {1 + \frac{{{a_3}}}{{{a_2}}}} \right)\text{, }~~{\xi _3} = - \frac{{2{a_3}}}{{{a_2}}}\text{, }~~{\xi _4} = 0, \end{split}$$ (41) 则
$$\begin{split} &\;{Y^{\left( 0 \right)}}\left( B \right) = 0 \text{, }~~ {Y^{\left( 0 \right)}}\left( {{R_1}} \right) = 2 \text{, }~~ \\& {Y^{\left( 0 \right)}}\left( {{R_2}} \right) = {Y^{\left( 0 \right)}}\left( {{R_3}} \right) = {Y^{\left( 0 \right)}}\left( {{R_4}} \right) = 0 . \end{split}$$ (42) 生成函数(41)满足判据方程(19), 因此它相应于系统的Mei对称性. 将(41)式代入方程(29), 可解得
$$ {G_{\text{M}}} = - 2\rho \left( t \right). $$ (43) 由定理1, 系统有Mei守恒量, 形如
$$ {I_{\text{M}}} = 2\left( {{a_2} + {a_3}} \right) = {\text{const}}. $$ (44) (44)式表明, 对于任意的时间尺度, (44)式都是Birkhoff系统(37)的守恒量. 如取生成函数为
$$\begin{split} {\xi _0} =\;& 0\text{, }~~{\xi _1} = a_2^\rho \sin \rho \left( t \right) + a_4^\rho \cos \rho \left( t \right) + \rho \left( t \right)\text{, }\\{\xi _2} =\;& 1 + \frac{{{a_3}}}{{{a_2}}}\text{, }~~{\xi _3} = - \frac{{{a_3}}}{{{a_2}}}\text{, }~~{\xi _4} = 0, \end{split}$$ (45) $$ {G_{\text{M}}} = a_3^\rho - {a_3} - \rho \left( t \right). $$ (46) 由定理1, 得到Mei守恒量
$$ {I_{\text{M}}} = {a_2}\sin t + {a_4}\cos t + {a_3} - a_3^\sigma = {\text{const}}. $$ (47) 对于守恒量(47), 如果系统是通常的Birkhoff系统, 即取
${\mathbb{T}} = {\mathbb{R}}$ , 则$\sigma \left( t \right) = t$ , 从而(47)式给出$$ {I_{\text{M}}} = {a_2}\sin t + {a_4}\cos t = {\text{const}}. $$ (48) 这是通常意义下Hojman-Urrutia问题的守恒量[3]. 如果是离散情形, 即取
${\mathbb{T}} = h{\mathbb{Z}}$ , 这里$h > 0$ , 则$\sigma \left( t \right) = $ $ t + h$ , 从而(47)式成为$$ \begin{split}{I_{\text{M}}} =\;& {a_2}\left( t \right)\sin t + {a_4}\left( t \right)\cos t + {a_3}\left( t \right) \\&- {a_3}\left( {t + h} \right) = {\text{const}}. \end{split}$$ (49) 这是步长为
$h$ 的离散版本的Mei守恒量.例2 研究时间尺度上广义Birkhoff系统
$$\begin{split}& B = \frac{1}{2}{\left( {{a_3}} \right)^2} + {a_2}\text{, }~~ {R_1} = {a_3}\text{, }~~ {R_2} = {a_4}\text{, }\\ & {R_3} = {R_4} = 0\text{, } ~~ {\varPhi _1} = {\varPhi _2} = {\varPhi _3} = 0\text{, }~~{\varPhi _4} = - {a_4} \end{split}$$ (50) 的Mei对称性与守恒量.
广义Birkhoff方程(10)给出
$$\begin{split} &\frac{\nabla }{{\nabla t}}{a_3} = 0\text{, } ~~\frac{\nabla }{{\nabla t}}{a_4} + {\sigma ^\nabla } = 0\text{, } \\& - {\sigma ^\nabla }\left( {a_1^\Delta - {a_3}} \right) = 0\text{, } ~~ - {\sigma ^\nabla }a_2^\Delta = - {\sigma ^\nabla }{a_4}. \end{split}$$ (51) 计算Mei对称性, 由于
$$\begin{split} &{Y^{\left( 0 \right)}}\left( B \right) = {\xi _3}{a_3} + {\xi _2} \text{, } ~~{Y^{\left( 0 \right)}}\left( {{R_1}} \right) = {\xi _3} \text{, }\\ &{Y^{\left( 0 \right)}}\left( {{R_2}} \right) = {\xi _4} \text{, } ~~{Y^{\left( 0 \right)}}\left( {{R_3}} \right) = {Y^{\left( 0 \right)}}\left( {{R_4}} \right) = 0 \text{, } \\ &{Y^{\left( 0 \right)}}\left( {{\varPhi _1}} \right) = {Y^{\left( 0 \right)}}\left( {{\varPhi _2}} \right) = {Y^{\left( 0 \right)}}\left( {{\varPhi _3}} \right) = 0 \text{, }\\&{Y^{\left( 0 \right)}}\left( {{\varPhi _4}} \right) = - {\xi _4} , \end{split}$$ (52) $$\begin{split} & {\xi _0} = 0\text{, }~~\,{\xi _1} = a_4^\rho + 2t\text{, } ~~\,{\xi _2} = - {a_3}\text{, } ~~ {\xi _3} = 1\text{, }~~{\xi _4} = 0, \end{split} $$ (53) $$\begin{split} & {\xi _0} = 0\text{, }~~\,{\xi _1} = a_3^\rho + 2t\text{, }~~\,{\xi _2} = - {a_3}\text{, }~~ {\xi _3} = 1\text{, }~~{\xi _4} = 0. \end{split}$$ (54) (53)式和(54)式相应于系统的Mei对称性. 将(53)式代入方程(33), 解得
$$ {G_{\text{M}}} = - t. $$ (55) 由定理2, 系统有Mei守恒量, 形如
$$ {I_{\text{M}}} = {a_4} + \sigma \left( t \right) = {\text{const}}. $$ (56) 同理, 相应于生成函数(54),
${G_{\text{M}}} = - 2 t$ , 由定理2得$$ {I_{\text{M}}} = {a_3} = {\text{const}}. $$ (57) (56)式和(57)式是由Mei对称性(53)和(54)导致的Mei守恒量.
例3 研究时间尺度上约束Birkhoff系统
$$\begin{split}& B = \frac{1}{2}{\left( {{a_3}} \right)^2} + {\left( {{a_4}} \right)^2} - g{a_1}\sin \varphi \text{, }\\ &{R_1} = {a_3}\text{, }{R_2} = {a_4}\text{, }{R_3} = {R_4} = 0, \end{split}$$ (58) 约束为
$$\begin{split} & {f_1} = {a_1} - {a_2} = 0\text{, }\\ & {f_2} = {a_3} - 2{a_4} = 0. \end{split} $$ (59) 试研究其Mei对称性与守恒量, 其中
$g$ 和$\varphi $ 是常数.方程(13)给出
$$\begin{split} &\frac{\nabla }{{\nabla t}}{a_3} - {\sigma ^\nabla }g\sin \varphi = {\sigma ^\nabla }{\lambda _1} \text{, } ~~ \frac{\nabla }{{\nabla t}}{a_4} = - {\sigma ^\nabla }{\lambda _1} \text{, } \\& - {\sigma ^\nabla }\left( {a_1^\Delta - {a_3}} \right) = {\sigma ^\nabla }{\lambda _2}\text{, } \\&- {\sigma ^\nabla }\left( {a_2^\Delta - 2{a_4}} \right) = - 2{\sigma ^\nabla }{\lambda _2}. \end{split}$$ (60) $$ {\lambda _1} = - \frac{1}{3}g\sin \varphi \text{, }~~{\lambda _2} = \frac{1}{3}\left( {{a_3} - 2{a_4}} \right), $$ (61) 因此有
$$\begin{split} {P_1} = \;& - \frac{1}{3}g\sin \varphi \text{, }~~{P_2} = \frac{1}{3}g\sin \varphi \text{, }\\{P_3} =\;& \frac{1}{3}\left( {{a_3} - 2{a_4}} \right)\text{, }~~{P_4} = - \frac{2}{3}\left( {{a_3} - 2{a_4}} \right). \end{split}$$ (62) 做计算
$$\begin{split}& {Y^{\left( 0 \right)}}\left( B \right) = {\xi _3}{a_3} + {\xi _4}{a_4} - {\xi _1}g\sin \varphi \text{, } ~~ {Y^{\left( 0 \right)}}\left( {{R_1}} \right) = {\xi _3} \text{, }\\& {Y^{\left( 0 \right)}}\left( {{R_2}} \right) = {\xi _4} \text{, } ~~{Y^{\left( 0 \right)}}\left( {{R_3}} \right) = {Y^{\left( 0 \right)}}\left( {{R_4}} \right) = 0 \text{, } \\& {Y^{\left( 0 \right)}}\left( {{P_1}} \right) = {Y^{\left( 0 \right)}}\left( {{P_2}} \right) = 0 \text{, } ~~{Y^{\left( 0 \right)}}\left( {{P_3}} \right) = \frac{1}{3}\left( {{\xi _3} - 2{\xi _4}} \right) \text{, } \\&{Y^{\left( 0 \right)}}\left( {{P_4}} \right) = - \frac{2}{3}\left( {{\xi _3} - 2{\xi _4}} \right) . \\[-15pt]\end{split}$$ (63) 取生成函数为
$$\begin{split} & {\xi _0} = 0\text{, }~~ {\xi _1} = \frac{{2a_3^\rho + a_4^\rho }}{{g\sin \varphi }}\text{, } \\ & {\xi _2} = \frac{{2a_3^\rho + a_4^\rho }}{{g\sin \varphi }}\text{, }~~{\xi _3} = 2\text{, }~~{\xi _4} = 1, \end{split} $$ (64) 则
$$\begin{split} & {Y^{\left( 0 \right)}}\left( B \right) = 2{a_3} + {a_4} - 2a_3^\rho - a_4^\rho \\ =&\; 2\nu \left( t \right)a_3^\nabla + \nu \left( t \right)a_4^\nabla = \frac{5}{3}g\mu \left( t \right)\sin \varphi,\\ & {Y^{\left( 0 \right)}}\left( {{R_1}} \right) = 2 \text{, }\\ & {Y^{\left( 0 \right)}}\left( {{R_2}} \right) = 1 \text{, }~~ {Y^{\left( 0 \right)}}\left( {{R_3}} \right) = {Y^{\left( 0 \right)}}\left( {{R_4}} \right) = 0 , \\ & {Y^{\left( 0 \right)}}\left( {{P_\beta }} \right) = 0~\,\,\left( {\beta = 1, \cdots ,4} \right) , \end{split}$$ (65) 其中
$ \mu \left( t \right) = \sigma \left( t \right) - t $ 为向前互差函数,$\nu \left( t \right) = t - $ $ \rho \left( t \right)$ 为向后互差函数. 由判据4, 生成函数(64)相应于系统的Mei对称性. 将(65)式代入方程(36), 解得$$ {G_{\text{M}}} = - 5t. $$ (66) 由定理4, 系统有Mei守恒量, 形如
$$ {I_{\text{M}}} = \frac{3}{{g\sin \varphi }}\left( {2{a_3} + {a_4}} \right) - 5\sigma \left( t \right) = {\text{const}}. $$ (67) 6. 讨 论
如果取时间尺度
${\mathbb{T}} = {\mathbb{R}}$ , 则前跳算子$\sigma \left( t \right) = t$ , 互差函数$\mu \left( t \right) = 0$ , 因此上述结果退化为通常意义下Birkhoff系统、广义Birkhoff系统和约束Birkhoff系统连续版本的变分原理、Birkhoff方程和Mei对称性定理.例如, 对于自由Birkhoff系统, 当取
${\mathbb{T}} = {\mathbb{R}}$ 时, 原理(2)成为$$ \begin{split} {\text{δ}}A = {\text{δ }}\int_{{t_1}}^{{t_2}} \left[ {R_\beta }\left( {t,{a_\gamma }\left( t \right)} \right){{\dot a}_\beta } - B\left( {t,{a_\gamma }\left( t \right)} \right) \right] \,{\text{d}}t = 0. \end{split} $$ (68) 方程(8)成为
$$ \frac{{\text{d}}}{{{\text{d}}t}}{R_\beta } + \frac{{\partial B}}{{\partial {a_\beta }}} - \frac{{\partial {R_\gamma }}}{{\partial {a_\beta }}}{\dot a_\gamma } = 0 . $$ (69) 由判据方程(19)容易得到
$$\begin{split} &{\sigma ^\nabla }a_\beta ^\Delta \frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)}}{{\partial t}} + \frac{\nabla }{{\nabla t}}{Y^{\left( 0 \right)}}\left( B \right) - {\sigma ^\nabla }\frac{{\partial {Y^{\left( 0 \right)}}\left( B \right)}}{{\partial t}} \\=\;& {\dot a_\beta }\frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)}}{{\partial t}} + \frac{{\text{d}}}{{{\text{d}}t}}{Y^{\left( 0 \right)}}\left( B \right) - \frac{{\partial {Y^{\left( 0 \right)}}\left( B \right)}}{{\partial t}} \\ =\;& \left[ {\frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\gamma }} \right)}}{{\partial {a_\beta }}} - \frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)}}{{\partial {a_\gamma }}}} \right]{\dot a_\beta }{\dot a_\gamma } = 0 , \\[-15pt]\end{split}$$ (70) 于是, 定理1退化为下述推论1.
推论1 假设变换(15)满足判据方程(19), 则自由Birkhoff系统(69)的Mei对称性导致如下形式的守恒量:
$$ {I_{\text{M}}} = {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right){\xi _\beta } - {Y^{\left( 0 \right)}}\left( B \right){\xi _0} + {G_{\text{M}}} , $$ (71) 其中
${G_{\text{M}}}$ 是规范函数, 满足$$\begin{split} &{Y^{\left( 0 \right)}}\big[ {{Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)} \big]{\dot a_\beta } - {Y^{\left( 0 \right)}}\big[ {{Y^{\left( 0 \right)}}\left( B \right)} \big] \\ & + {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right){\dot \xi _\beta } - {Y^{\left( 0 \right)}}\left( B \right){\dot \xi _0} + {G_{\text{M}}} = 0 . \end{split}$$ (72) 推论1是通常意义下自由Birkhoff系统连续版本的Mei对称性与守恒量定理[7]. 而方程(68)、方程(69)和方程(71)就是通常意义下自由Birkhoff系统连续版本的Pfaff-Birkhoff原理、Birkhoff方程和Mei守恒量.
如果取时间尺度
${\mathbb{T}} = h{\mathbb{Z}}$ , 常数$h > 0$ , 则前跳算子$\sigma \left( t \right) = t + h$ , 互差函数$\mu \left( t \right) = h$ . 此时, 原理(2)成为$$\begin{split} {\text{δ}}A =\;& {\text{δ }}\sum\limits_{\tau = {t_1}}^{{t_2} - h} \big\{ {R_\beta }\left( {\tau ,{a_\gamma }\left( \tau \right)} \right)\left[ {{a_\beta }\left( {\tau + h} \right) - {a_\beta }\left( \tau \right)} \right] \\&- B\left( {\tau ,{a_\gamma }\left( \tau \right)} \right)h \big\} = 0, \\[-10pt]\end{split}$$ (73) 方程(8)成为
$$\begin{split}& {R_\beta }\left( {t,{a_\gamma }\left( t \right)} \right) - {R_\beta }\left( {t - h,{a_\gamma }\left( {t - h} \right)} \right)\\& + \frac{{\partial B}}{{\partial {a_\beta }}}\left( {\tau ,{a_\gamma }\left( \tau \right)} \right)h - \frac{{\partial {R_\alpha }}}{{\partial {a_\beta }}}\left( {\tau ,{a_\gamma }\left( \tau \right)} \right)\\&\times\left[ {{a_\alpha }\left( {\tau + h} \right) - {a_\alpha }\left( \tau \right)} \right] = 0, \end{split}$$ (74) 则定理1退化为下述推论2.
推论2 假设变换(15)满足判据方程(19), 则自由Birkhoff系统(74)的Mei对称性导致如下形式的守恒量:
$$\begin{split} {I_{\text{M}}} = \;&{Y^{\left( 0 \right)}}\left( {{R_\beta }} \right){\xi _\beta }\left( {t + h} \right) - {Y^{\left( 0 \right)}}\left( B \right){\xi _0}\left( {t + h} \right) + {G_{\text{M}}}\left( {t + h} \right) \\ & + \sum\limits_{\tau = {t_1}}^{t - h} {\xi _0}\left( \tau \right)\bigg\{ \big[ {a_\beta }(\tau + h) - {a_\beta}(\tau) ] \frac{{\partial {Y^{\left( 0 \right)}}\left( {{R_\beta }} \right)}}{{\partial \tau }}\left( {\tau ,{a_\gamma }\left( \tau \right)} \right) + {Y^{\left( 0 \right)}}\left( B \right)\left( {\tau ,{a_\gamma }\left( \tau \right)} \right) \\& - {Y^{\left( 0 \right)}} \left( B \right)\left( {\tau - h,{a_\gamma }\left( {\tau - h} \right)} \right) - h\frac{{\partial {Y^{\left( 0 \right)}}\left( B \right)}}{{\partial \tau }}\left( {\tau ,{a_\gamma }\left( \tau \right)} \right) \bigg\}, \end{split}$$ (75) 其中
${G_{\text{M}}}$ 是规范函数, 满足$$\begin{split}&{Y^{(0)}}\left[ {{Y^{(0)}}\left( {{R_\beta }} \right)} \right]\big[ a_\beta(t + h ) - {a_\beta }(t) \big] - {Y^{(0)}}\big[ {{Y^{(0)}}(B)} \big]h + {Y^{(0)}} \big(R_\beta\big) \big[{\xi _\beta }(t + h) \\& - {\xi_\beta}(t) \big] - {Y^{(0)}} (B) \big[ \xi _0(t + h) - \xi_0(t) \big] + \big[{G_{\text{M}}}(t + h) - {G_{\text{M}}}(t) \big]=0. \end{split}$$ (76) 推论2是通常意义下自由Birkhoff系统离散版本的Mei对称性与守恒量定理. 而方程(73)—(75)就是通常意义下自由Birkhoff系统离散版本步长为
$h$ 的Pfaff-Birkhoff原理、Birkhoff方程和Mei守恒量.7. 结 论
对称性和守恒量一直是分析力学研究的一个重要方面. 经典的对称性主要有Noether对称性和Lie对称性. Mei对称性是本质上不同于前两种对称性的一种不变性, 它可以导致Mei守恒量. Mei守恒量不同于Noether守恒量, 是一种新的守恒量. 本文提出并研究了时间尺度上非迁移Birkhoff系统的Mei对称性定理.
一是提出了时间尺度上非迁移Pfaff-Birkhoff原理和广义Pfaff-Birkhoff原理, 导出了时间尺度上Birkhoff系统, 包括自由Birkhoff系统、广义Birkhoff系统和约束Birkhoff系统的动力学方程. 主要结果是原理(2)和(5), Birkhoff方程(8), (10)和(13).
二是定义了时间尺度上非迁移Birkhoff系统的Mei对称性, 并导出了它的判据方程. 主要结果是4个定义和4个判据.
三是提出并证明了时间尺度上非迁移Birkhoff系统、非迁移广义Birkhoff系统和非迁移约束Birkhoff系统的Mei对称性定理. 主要结果是4个定理, Mei守恒量(28), (32)和(35).
当取时间尺度
${\mathbb{T}} = {\mathbb{R}}$ 和${\mathbb{T}} = h{\mathbb{Z}}$ 时, 文中定理给出通常意义下自由Birkhoff系统、广义Birkhoff系统和约束Birkhoff系统的连续版本和离散版本的Mei对称性与守恒量定理. 由于除了实数集和整数集以外, 时间尺度还可以有很多选择, 因此时间尺度上Birkhoff系统的Mei对称性定理具有一般性.[1] Birkhoff G D 1927 Dynamical Systems (Providence: AMS College Publ. ) pp59–96
[2] Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer-Verlag) pp1–280
[3] Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoffian System (Beijing: Beijing Institute of Technology Press) pp1–228
[4] Galiullin A S, Gafarov G G, Malaishka R P, Khwan A M 1997 Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems (Moscow: UFN) pp118–226
[5] Mei F X 2013 Dynamics of Generalized Birkhoffian Systems (Beijing: Science Press) pp1–206
[6] 梅凤翔, 吴惠彬, 李彦敏, 陈向炜 2016 力学学报 48 263Google Scholar
Mei F X, Wu H B, Li Y M, Chen X W 2016 J. Theor. Appl. Mech. 48 263Google Scholar
[7] 梅凤翔 2004 约束力学系统的对称性与守恒量 (北京: 北京理工大学出版社) 第1—482页
Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) pp1–482 (in Chinese)
[8] Wang P, Xue Y, Liu Y L 2012 Chin. Phys. B 21 070203Google Scholar
[9] Zhang Y, Zhai X H 2015 Nonlinear Dyn. 81 469Google Scholar
[10] Zhang H B, Chen H B 2017 J. Math. Anal. Appl. 456 1442Google Scholar
[11] Zhang Y 2018 Int. J. Non-Linear Mech. 101 36Google Scholar
[12] 徐鑫鑫, 张毅 2020 物理学报 69 220401Google Scholar
Xu X X, Zhang Y 2020 Acta Phys. Sin. 69 220401Google Scholar
[13] Zhang L J, Zhang Y 2020 Commun. Nonlinear Sci. Numer. Simul. 91 105435Google Scholar
[14] Guo Y X, Liu C, Liu S X 2010 Commun. Math. 18 21
[15] Liu S X, Liu C, Guo Y X 2011 Chin. Phys. B 20 034501Google Scholar
[16] Chen X W, Li Y M 2013 Acta Mech. 224 1593Google Scholar
[17] Luo S K, He J M, Xu Y L 2016 Int. J. Non-Linear Mech. 78 105Google Scholar
[18] 刘世兴, 刘畅, 郭永新 2011 物理学报 60 064501Google Scholar
Liu S X, Liu C, Guo Y X 2011 Acta Phys. Sin. 60 064501Google Scholar
[19] Liu S X, Liu C, Hua W, Guo Y X 2016 Chin. Phys. B 25 114501Google Scholar
[20] Kong X L, Wu H B, Mei F X 2013 Appl. Math. Comput. 225 326
[21] 孔新雷, 吴惠彬 2017 物理学报 66 084501Google Scholar
Kong X L, Wu H B 2017 Acta Phys. Sin. 66 084501Google Scholar
[22] He L, Wu H B, Mei F X 2017 Nonlinear Dyn. 87 2325Google Scholar
[23] Hilger S 1990 Results Math. 18 18Google Scholar
[24] Bohner M, Peterson A 2001 Dynamic Equations on Time Scales (Boston: Birkhäuser) pp1–353
[25] Bohner M, Georgiev S G 2016 Multivariable Dynamic Calculus on Time Scales (Switzerland: Springer International Publishing AG) pp1–600
[26] Georgiev S G 2018 Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales (Switzerland: Springer International Publishing AG) pp1–357
[27] Atici F M, Biles D C, Lebedinsky A 2006 Math. Comput. Modell. 43 718Google Scholar
[28] Bohner M 2004 Dyn. Syst. Appl. 13 339
[29] Bartosiewicz Z, Torres D F M 2008 J. Math. Anal. Appl. 342 1220Google Scholar
[30] Benkhettou N, Brito da Cruz A M C, Torres D F M 2015 Signal Process. 107 230Google Scholar
[31] Dryl M, Torres D F M 2017 Springer Proceedings in Mathematics & Statistics 195 223
[32] 韩振来, 孙书荣 2014 时间尺度上动态方程振动理论 (济南: 山东大学出版社) 第1—232页
Han Z L, Sun S R 2014 Oscillation Theory of Dynamic Equations on Time Scales (Jinan: Shandong University Press) pp1–232 (in Chinese)
[33] Bourdin L 2014 J. Math. Anal. Appl. 411 543Google Scholar
[34] Anerot B, Cresson J, Belgacem K H, Pierret F 2020 J. Math. Phys. 61 113502Google Scholar
[35] Song C J, Zhang Y 2015 J. Math. Phys. 56 102701Google Scholar
[36] Song C J, Zhang Y 2017 J. Nonlinear Sci. Appl. 10 2268Google Scholar
[37] Song C J, Cheng Y 2020 Appl. Math. Comput. 374 125086
[38] Zhang Y 2019 Chaos, Solitons Fractals 128 306Google Scholar
[39] Zhang Y, Zhai X H 2019 Commun. Nonlinear Sci. Numer. Simul. 75 251Google Scholar
期刊类型引用(5)
1. 宋传静,侯爽. 时间尺度上约束力学系统的Noether型绝热不变量. 力学学报. 2024(08): 2397-2407 . 百度学术 2. ZHANG Yi. A Study on Time Scale Non-Shifted Hamiltonian Dynamics and Noether's Theorems. Wuhan University Journal of Natural Sciences. 2023(02): 106-116 . 必应学术 3. 金世欣,李彦敏. 基于非标准Lagrange函数下非完整系统的广义Chaplygin方程. 力学季刊. 2023(02): 353-361 . 百度学术 4. 王璐,张毅. 基于分数阶模型的非完整系统的Mei对称性及其守恒量. 力学季刊. 2023(03): 633-642 . 百度学术 5. SONG Chuanjing,WANG Jiahang. Conserved Quantity for Fractional Constrained Hamiltonian System. Wuhan University Journal of Natural Sciences. 2022(03): 201-210 . 必应学术 其他类型引用(2)
-
[1] Birkhoff G D 1927 Dynamical Systems (Providence: AMS College Publ. ) pp59–96
[2] Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer-Verlag) pp1–280
[3] Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoffian System (Beijing: Beijing Institute of Technology Press) pp1–228
[4] Galiullin A S, Gafarov G G, Malaishka R P, Khwan A M 1997 Analytical Dynamics of Helmholtz, Birkhoff and Nambu Systems (Moscow: UFN) pp118–226
[5] Mei F X 2013 Dynamics of Generalized Birkhoffian Systems (Beijing: Science Press) pp1–206
[6] 梅凤翔, 吴惠彬, 李彦敏, 陈向炜 2016 力学学报 48 263Google Scholar
Mei F X, Wu H B, Li Y M, Chen X W 2016 J. Theor. Appl. Mech. 48 263Google Scholar
[7] 梅凤翔 2004 约束力学系统的对称性与守恒量 (北京: 北京理工大学出版社) 第1—482页
Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) pp1–482 (in Chinese)
[8] Wang P, Xue Y, Liu Y L 2012 Chin. Phys. B 21 070203Google Scholar
[9] Zhang Y, Zhai X H 2015 Nonlinear Dyn. 81 469Google Scholar
[10] Zhang H B, Chen H B 2017 J. Math. Anal. Appl. 456 1442Google Scholar
[11] Zhang Y 2018 Int. J. Non-Linear Mech. 101 36Google Scholar
[12] 徐鑫鑫, 张毅 2020 物理学报 69 220401Google Scholar
Xu X X, Zhang Y 2020 Acta Phys. Sin. 69 220401Google Scholar
[13] Zhang L J, Zhang Y 2020 Commun. Nonlinear Sci. Numer. Simul. 91 105435Google Scholar
[14] Guo Y X, Liu C, Liu S X 2010 Commun. Math. 18 21
[15] Liu S X, Liu C, Guo Y X 2011 Chin. Phys. B 20 034501Google Scholar
[16] Chen X W, Li Y M 2013 Acta Mech. 224 1593Google Scholar
[17] Luo S K, He J M, Xu Y L 2016 Int. J. Non-Linear Mech. 78 105Google Scholar
[18] 刘世兴, 刘畅, 郭永新 2011 物理学报 60 064501Google Scholar
Liu S X, Liu C, Guo Y X 2011 Acta Phys. Sin. 60 064501Google Scholar
[19] Liu S X, Liu C, Hua W, Guo Y X 2016 Chin. Phys. B 25 114501Google Scholar
[20] Kong X L, Wu H B, Mei F X 2013 Appl. Math. Comput. 225 326
[21] 孔新雷, 吴惠彬 2017 物理学报 66 084501Google Scholar
Kong X L, Wu H B 2017 Acta Phys. Sin. 66 084501Google Scholar
[22] He L, Wu H B, Mei F X 2017 Nonlinear Dyn. 87 2325Google Scholar
[23] Hilger S 1990 Results Math. 18 18Google Scholar
[24] Bohner M, Peterson A 2001 Dynamic Equations on Time Scales (Boston: Birkhäuser) pp1–353
[25] Bohner M, Georgiev S G 2016 Multivariable Dynamic Calculus on Time Scales (Switzerland: Springer International Publishing AG) pp1–600
[26] Georgiev S G 2018 Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales (Switzerland: Springer International Publishing AG) pp1–357
[27] Atici F M, Biles D C, Lebedinsky A 2006 Math. Comput. Modell. 43 718Google Scholar
[28] Bohner M 2004 Dyn. Syst. Appl. 13 339
[29] Bartosiewicz Z, Torres D F M 2008 J. Math. Anal. Appl. 342 1220Google Scholar
[30] Benkhettou N, Brito da Cruz A M C, Torres D F M 2015 Signal Process. 107 230Google Scholar
[31] Dryl M, Torres D F M 2017 Springer Proceedings in Mathematics & Statistics 195 223
[32] 韩振来, 孙书荣 2014 时间尺度上动态方程振动理论 (济南: 山东大学出版社) 第1—232页
Han Z L, Sun S R 2014 Oscillation Theory of Dynamic Equations on Time Scales (Jinan: Shandong University Press) pp1–232 (in Chinese)
[33] Bourdin L 2014 J. Math. Anal. Appl. 411 543Google Scholar
[34] Anerot B, Cresson J, Belgacem K H, Pierret F 2020 J. Math. Phys. 61 113502Google Scholar
[35] Song C J, Zhang Y 2015 J. Math. Phys. 56 102701Google Scholar
[36] Song C J, Zhang Y 2017 J. Nonlinear Sci. Appl. 10 2268Google Scholar
[37] Song C J, Cheng Y 2020 Appl. Math. Comput. 374 125086
[38] Zhang Y 2019 Chaos, Solitons Fractals 128 306Google Scholar
[39] Zhang Y, Zhai X H 2019 Commun. Nonlinear Sci. Numer. Simul. 75 251Google Scholar
期刊类型引用(5)
1. 宋传静,侯爽. 时间尺度上约束力学系统的Noether型绝热不变量. 力学学报. 2024(08): 2397-2407 . 百度学术 2. ZHANG Yi. A Study on Time Scale Non-Shifted Hamiltonian Dynamics and Noether's Theorems. Wuhan University Journal of Natural Sciences. 2023(02): 106-116 . 必应学术 3. 金世欣,李彦敏. 基于非标准Lagrange函数下非完整系统的广义Chaplygin方程. 力学季刊. 2023(02): 353-361 . 百度学术 4. 王璐,张毅. 基于分数阶模型的非完整系统的Mei对称性及其守恒量. 力学季刊. 2023(03): 633-642 . 百度学术 5. SONG Chuanjing,WANG Jiahang. Conserved Quantity for Fractional Constrained Hamiltonian System. Wuhan University Journal of Natural Sciences. 2022(03): 201-210 . 必应学术 其他类型引用(2)
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