SYMMETRY AND HERGLOTZ TYPE CONSERVED QUANTITIES FOR NONHOLONOMIC SYSTEMS WITH TIME DELAY
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摘要: 时滞是自然界和工程实践中常见的一种时间滞后现象, 对力学系统的动力学行为及基本性质都具有深刻的影响. Herglotz广义变分原理推广了经典变分原理, 可用于非保守系统的研究. 故利用Herglotz广义变分原理研究含时滞的非完整系统的对称性与守恒量在理论和应用上均具有重要意义. 文章将Herglotz型Noether定理拓展到含时滞的非完整系统. 首先, 建立含时滞系统的Herglotz型微分变分原理, 利用拉格朗日乘子法, 推导出含时滞的一般非完整系统的Herglotz型Routh方程. 其次, 基于含时滞的Hamilton-Herglotz作用量在无限小变换下的不变性, 给出作用量变分的两个基本公式, 进而定义含时滞非完整系统的Herglotz型Noether对称性, 给出Herglotz型Noether等式. 再次, 建立含时滞一般非完整系统的Herglotz型Noether定理. 讨论特殊情形下的Noether定理: 如果约束都是完整的, 定理退化为含时滞完整系统的Herglotz型Noether定理; 如果系统是保守的, 退化为含时滞非完整保守系统的Herglotz型Noether定理; 如果不考虑时滞, 则退化为非完整系统的Herglotz型Noether定理. 最后, 给出含时滞非完整系统的Herglotz型Noether逆定理. 文末通过对含时滞的算例求解, 说明了理论分析结果, 并通过数值模拟验证方法的可行性及正确性.
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关键词:
- 非完整系统 /
- Herglotz变分原理 /
- Noether定理 /
- 时滞
Abstract: Time delay is a common time delay phenomenon in nature and engineering practice, which has a profound impact on the dynamic behavior and basic properties of mechanical systems. The Herglotz type generalized variational principle extends the classical variational principle and can be used to study nonconservative systems. Therefore, using the Herglotz type generalized variational principle to study the symmetry and conserved quantity of nonholonomic systems with time delay is of great significance both in theory and application. In this paper, the Herglotz type Noether theorem is extended to nonholonomic systems with time delay. Firstly, the Herglotz type differential variational principle of the system with time delay is established. By means of the Lagrange multiplier method, the Routh-type differential equations of motion of the general nonholonomic system with time delay are derived. Secondly, based on the invariance of the Hamilton-Herglotz action with time delay under infinitesimal transformations, two basic formulas for the variation of the action are given, and then the Herglotz type Noether symmetry is defined and the Herglotz type Noether identity is given. Thirdly, the Noether theorem of Herglotz type for general nonholonomic systems with time delay is established. In addition, the Noether theorem in special cases is discussed. If all the constraints are holonomic, the theorem is reduced to the Herglotz type Noether theorem for holonomic systems with time delay. If the system is conservative, it is reduced to the Herglotz type Noether theorem for nonholonomic conservative systems with time delay. If the time delay is not considered, it is reduced to the Herglotz type Noether theorem for nonholonomic systems. Finally, the Noether inverse theorem of Herglotz type for nonholonomic systems with time delay is given. At the end of this paper, the theoretical analysis results are illustrated by solving an example with time delay, and the feasibility and correctness of the proposed method are verified by numerical simulation.-
Keywords:
- nonholonomic system /
- Herglotz variational principle /
- Noether theorem /
- time delay
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引 言
时滞现象广泛存在于自然界及工程实际中. 即使受轻微时滞的影响, 系统的动力学行为都会变得非常复杂[1-2]. 而随着对复杂系统的动力学行为和控制的要求越来越精确, 系统的时滞现象已不可忽略, 即使毫秒级的延迟也可导致整个结论完全错误, 这使得含时滞系统的研究具有极大挑战性. 关于时滞变分问题的极值刻画可追溯至1964年El’sgol’c[3]给出的结果. 1968年, Hughes[4]研究了含时滞的变分问题, 在$ \left( {n + 1} \right) $维空间中导出了类似于经典变分问题的Euler-Lagrange方程. 2012年, Frederico等[5]研究了时滞变分情形的DuBois-Reymond必要最优性条件和Noether定理, 证明了拉格朗日和哈密顿版本的Noether定理. 近年来, 文献[6-9]研究了含时滞的拉格朗日系统, 哈密顿系统, 伯克霍夫系统的Noether定理.
Noether定理是由德国女数学家Noether[10]于1918年所提出, 揭示了对称性与守恒量之间的关系. Noether定理给出了变分原理的物理意义[11]. 对数学和力学尤为重要. 譬如, 在数学中, 可以用来证明麦克斯韦方程组的数学一致性. 在力学中, 用来推导系统的守恒定律. Noether定理对研究复杂系统的动力学行为及其稳定性等具有重要意义[12]. 通过作用量在无限小变换下的不变性可以证明Noether定理[12-14]. Noether定理通常与经典积分变分原理相对应, 但对于非保守系统, 经典积分变分原理已不再适用. 而作为经典变分原理的推广, Herglotz广义变分原理[15-16]提供了非保守系统的一个变分方法.
Herglotz广义变分原理与经典积分变分原理不同, 它的作用量是由微分方程$\dot z = L\left( {t,{q_s},{{\dot q}_s},z} \right)$所定义[16]. Herglotz广义变分原理已被应用于量子系统, 最优控制问题等的动力学建模[17-20]. Santos等[21-22]研究了含时滞的Herglotz型Noether定理. 此后, Zhang等[23-26]研究了含时滞的拉格朗日系统, 哈密顿系统, 伯克霍夫系统以及非完整vakonomic动力学的Herglotz型Noether对称性及其导致的Herglotz型守恒量. 对于非完整系统而言, Zhang等[27-28]基于微分变分原理给出了非完整系统的Herglotz型Noether守恒量, 但没有考虑时滞. 非完整约束是一类不可积分约束[29], 它在轮式系统, 机电系统, 控制系统, 机器人动力学等[30-34]中广泛存在. 本文研究含时滞的非完整力学系统的Herglotz型Noether对称性. 建立含时滞的Herglotz型微分变分原理, Routh方程, 作用量变分, 对称性判据 (Noether等式), Noether定理及其逆定理, 并通过对一个含时滞非完整非保守系统的理论分析和数值计算验证结果的正确性.
本工作采用变分方法, 针对含时滞的非完整力学系统, 建立了系统的Herglotz型Routh方程, 定义Herglotz型Noether对称变换与准对称变换, 建立含时滞的非完整系统的Herglotz型Noether定理及其逆定理, 以期为研究含时滞的约束力学系统的对称性与守恒量提供参考.
1. 含时滞的Herglotz型Routh方程
研究含时滞的非保守系统, 设广义坐标为${q_s}$, Herglotz型拉格朗日函数为$L\left( {t,{q_s},{{\dot q}_s},{q_{s\tau }},{{\dot q}_{s\tau }},z} \right) {\overset{\Delta}{\mathop{=}}} $ $L\left[ {t,{q_s}\left( t \right),{{\dot q}_s}\left( t \right),{q_s}\left( {t - \tau } \right),{{\dot q}_s}\left( {t - \tau } \right),z\left( t \right)} \right]$, $\tau $是时滞量, $z$为Hamilton-Herglotz作用量.
含时滞的Herglotz变分问题为: 确定函数${q_s}$, 使得$z\left( {{t_2}} \right) \to {\text{extr}}$, 其中泛函$z$由方程
$$ \dot z = L\left[ {t,{q_s}\left( t \right),{{\dot q}_s}\left( t \right),{q_s}\left( {t - \tau } \right),{{\dot q}_s}\left( {t - \tau } \right),z\left( t \right)} \right] $$ (1) 定义, 且满足初始条件
$$ {\left. {z\left( t \right)} \right|_{t = {t_1}}} = {z_1} $$ (2) 及边界条件
$$ {q_s}\left( t \right) = {\varPhi _s}\left( t \right),{\text{ }}t \in \left[ {{t_1} - \tau ,{t_1}} \right] $$ (3) $$ {q_s}\left( {{t_2}} \right) = {q_{s2}}\qquad\qquad\qquad $$ (4) 式中, ${\varPhi _s}\left( t \right)$是区间 $\left[ {{t_1} - \tau ,{t_1}} \right]$上分段光滑函数, $\tau < {t_2} - {t_1},$ ${q_{s2}}$和${z_1}$是常数.
对方程 (1) 的两边同时取变分, 可得
$$ \delta \dot z = \frac{{\text{d}}}{{{\text{d}}t}}\delta z = \varPsi + \frac{{\partial L}}{{\partial z}}\delta z $$ (5) 式中, $\varPsi = \dfrac{{\partial L}}{{\partial {q_s}}}\delta {q_s} + \dfrac{{\partial L}}{{\partial {{\dot q}_s}}}\delta {\dot q_s} + \dfrac{{\partial L}}{{\partial {q_{s\tau }}}}\delta {q_{s\tau }}+\dfrac{{\partial L}}{{\partial {{\dot q}_{s\tau }}}}\delta {\dot q_{s\tau }} $. 本文采用爱因斯坦求和约定, 即两个相同指标表示对该指标的取值范围求和.
方程 (5) 是关于$\delta z$的一阶微分方程, 根据解的存在唯一性定理[35], 可解得
$$ \delta z\left( t \right)\rho \left( t \right) - \delta z\left( {{t_1}} \right) = \int_{{t_1}}^t {\rho \left( t \right)\varPsi {\text{d}}t} $$ (6) 式中, $\rho \left( t \right) = {\text{exp}}\left( { - \displaystyle\int_{{t_1}}^t {\dfrac{{\partial L}}{{\partial z}}{\text{d}}t} } \right)$. 对于$\forall t \in \left[ {{t_1},{t_2}} \right]$, 方程 (6) 均成立, 不妨取$t = {t_2}$, 由于$\delta z\left( {{t_2}} \right) = 0$, $\delta z\left( {{t_1}} \right) = 0$, 则式 (6) 成为
$$ \int_{{t_1}}^{{t_2}} {\rho \left( t \right)\varPsi {\text{d}}t} = 0 $$ (7) 对方程 (7) 中含时滞的项进行变量变换, $t = \theta + \tau $, 则方程 (7) 成为
$$ \int_{{t_1}}^{{t_2}} {\left[ {{\phi _s}\left( t \right)\delta {q_s} + {\varphi _s}\left( t \right)\delta {{\dot q}_s}} \right]} {\text{d}}t = 0 $$ (8) 式中
$$ {\phi _s}\left( t \right) = \left\{ \begin{gathered} \frac{{\partial L}}{{\partial {q_s}}}\rho \left( t \right) + \frac{{\partial L}}{{\partial {q_{s\tau }}}}\rho \left( {t + \tau } \right),\quad t \in \left[ {{t_1},{t_2} - \tau } \right] \\ \frac{{\partial L}}{{\partial {q_s}}}\rho \left( t \right),\quad t \in \left( {{t_2} - \tau ,{t_2}} \right] \\ \end{gathered} \right.\quad $$ (9) $$ {\varphi _s}\left( t \right) = \left\{ \begin{gathered} \frac{{\partial L}}{{\partial {{\dot q}_s}}}\rho \left( t \right) + \frac{{\partial L}}{{\partial {{\dot q}_{s\tau }}}}\rho \left( {t + \tau } \right),\quad t \in \left[ {{t_1},{t_2} - \tau } \right] \\ \frac{{\partial L}}{{\partial {{\dot q}_s}}}\rho \left( t \right),\quad t \in \left( {{t_2} - \tau ,{t_2}} \right] \\ \end{gathered} \right. $$ (10) 利用分部积分并考虑到边界条件式 (3) 和式 (4), 可将方程 (8) 写为
$$ 0 = \int_{{t_1}}^{{t_2}} {\left[ {{\phi _s}\left( t \right)\delta {q_s} + {\varphi _s}\left( t \right)\delta {{\dot q}_s}} \right]} {\text{d}}t = \int_{{t_1}}^{{t_2}} {\left[ {{\phi _s}\left( t \right) - {{\dot \varphi }_s}\left( t \right)} \right]\delta {q_s}{\text{d}}t} $$ (11) 考虑到积分区间的任意性, 由式 (11) 可得
$$ \left[ {{\phi _s}\left( t \right) - {{\dot \varphi }_s}\left( t \right)} \right]\delta {q_s} = 0 $$ (12) 式 (12) 为含时滞的非保守系统的Herglotz型微分变分原理.
设系统的运动受$g$个理想非完整约束
$$ {f_\beta }\left( {t,{q_s},{{\dot q}_s}} \right) = 0{\text{ }}\left( {\beta = 1,2, \cdots ,g} \right) $$ (13) 约束对虚位移的限制满足Appell-Cheatev条件[36]
$$ \frac{{\partial {f_\beta }}}{{\partial {{\dot q}_s}}}\delta {q_s} = 0 $$ (14) 由原理式(12) 和条件式(14), 利用拉格朗日乘子法, 得到含时滞的Herglotz型Routh方程
$$ {\phi _s}\left( t \right) - {\dot \varphi _s}\left( t \right) + {\lambda _\beta }\frac{{\partial {f_\beta }}}{{\partial {{\dot q}_s}}} = 0 $$ (15) 式中, ${\lambda _\beta }$为约束乘子, 可由式 (13) 和式 (15) 联立求解得出${\lambda _\beta } = {\lambda _\beta }\left( {t,{q_s},{{\dot q}_s}} \right)$. 考虑到式 (9) 和式 (10), 方程 (15) 可表示为
$$\left. \begin{split} &\rho \left( t \right)\left( {\frac{{\partial L}}{{\partial {q_s}}} - \frac{{\text{d}}}{{{\text{d}}t}}\frac{{\partial L}}{{\partial {{\dot q}_s}}} + \frac{{\partial L}}{{\partial z}}\frac{{\partial L}}{{\partial {{\dot q}_s}}} + {\varLambda _s}} \right) +\\ &\qquad \rho \left( {t + \tau } \right)\left( {\frac{{\partial L}}{{\partial {q_{s\tau }}}} - \frac{{\text{d}}}{{{\text{d}}t}}\frac{{\partial L}}{{\partial {{\dot q}_{s\tau }}}} + } \right.\left. {\frac{{\partial L}}{{\partial z}}\frac{{\partial L}}{{\partial {{\dot q}_{s\tau }}}}} \right)\left( {t + \tau } \right){\text{ = 0}}, \\ &\qquad t \in \left[ {{t_1},{t_2} - \tau } \right] \\ & \rho \left( t \right)\left( {\frac{{\partial L}}{{\partial {q_s}}} - \frac{{\text{d}}}{{{\text{d}}t}}\frac{{\partial L}}{{\partial {{\dot q}_s}}} + \frac{{\partial L}}{{\partial z}}\frac{{\partial L}}{{\partial {{\dot q}_s}}} + {\varLambda _s}} \right) = 0, \\ &\qquad t \in \left( {{t_2} - \tau ,{t_2}} \right] \end{split}\right\} $$ (16) 其中, ${\varLambda _s} = {\lambda _\beta }\dfrac{{\partial {f_\beta }}}{{\partial {{\dot q}_s}}}$表示非完整约束反力.
2. 作用量的变分
引入时间和空间的无限小变换
$$ \bar t = t + \Delta t, {\text{ }}{\bar q_s}\left( {\bar t} \right) = {q_s}\left( t \right) + \Delta {q_s} $$ (17) 及其展开式
$$\left. \begin{split} &\bar t = t + \varepsilon {\xi _0}\left( {t,{q_k},{{\dot q}_k}} \right) \\& {\bar q_s}\left( {\bar t} \right) = {q_s}\left( t \right) + \varepsilon {\xi _s}\left( {t,{q_k},{{\dot q}_k}} \right) \\& \qquad \left( {s,k = 1,2, \cdots ,n} \right) \end{split}\right\} $$ (18) 其中, $ \varepsilon $是无限小参数, $ {\xi _0}和{\xi _s} $为生成元.
对方程 (1) 的两边同时求非等时变分, 可得
$$ \Delta \dot z = \frac{{\partial L}}{{\partial t}}\Delta t + \frac{{\partial L}}{{\partial {q_s}}}\Delta {q_s} + \frac{{\partial L}}{{\partial {{\dot q}_s}}}\Delta {\dot q_s} + $$ $$ \;\;\;\frac{{\partial L}}{{\partial {q_{s\tau }}}}\Delta {q_{s\tau }} + \frac{{\partial L}}{{\partial {{\dot q}_{s\tau }}}}\Delta {\dot q_{s\tau }} + \frac{{\partial L}}{{\partial z}}\Delta z $$ (19) 根据非完整力学系统关于微分和变分运算交换关系的Hölder定义, 设微分运算 ${\text{d}}$ 和等时变分运算 $\delta $可以任意交换, 则对任意函数$h$有[36]
$$\left.\begin{split} &\varDelta \left( h \right) = \delta \left( h \right) + \Delta t\frac{{\text{d}}}{{{\text{d}}t}}h\\ &\frac{{\text{d}}}{{{\text{d}}t}} {\Delta h} = \varDelta \frac{{\text{d}}}{{{\text{d}}t}}h + \frac{{\text{d}}}{{{\text{d}}t}}h\frac{{\text{d}}}{{{\text{d}}t}} {\Delta t} \end{split}\right\} $$ (20) 利用式 (20), 方程 (19) 成为
$$ \frac{{\text{d}}}{{{\text{d}}t}}\Delta z = \varOmega + \frac{{\partial L}}{{\partial z}}\Delta z $$ (21) 式中
$$ \begin{split} &\varOmega = \frac{{\partial L}}{{\partial t}}\Delta t + \frac{{\partial L}}{{\partial {q_s}}}\Delta {q_s} + \frac{{\partial L}}{{\partial {{\dot q}_s}}}\Delta {\dot q_s} + \frac{{\partial L}}{{\partial {q_{s\tau }}}}\Delta {q_{s\tau }} + \\& \qquad \frac{{\partial L}}{{\partial {{\dot q}_{s\tau }}}}\Delta {\dot q_{s\tau }} + L\frac{{\text{d}}}{{{\text{d}}t}}\Delta t \end{split} $$ (22) 方程 (21) 有解
$$ \rho \left( t \right)\Delta z\left( t \right) - \Delta z\left( {{t_1}} \right) = \int_{{t_1}}^t {\rho \left( t \right)\varOmega } {\text{d}}t $$ (23) 这里$\Delta z\left( {{t_1}} \right) = 0$. 令$t = {t_2}$, 式 (23) 成为
$$ \rho \left( {{t_2}} \right)\Delta z\left( {{t_2}} \right) = \int_{{t_1}}^{{t_2}} {\rho \left( t \right)\varOmega } {\text{d}}t $$ (24) 对式 (24) 中含时滞的项进行变量变换, $t = \theta + \tau $, 则方程 (24) 可表示为
$$ \begin{split} &\rho \left( {{t_2}} \right)\Delta z\left( {{t_2}} \right) =\\& \qquad \left. {\int_{{t_1} - \tau }^{{t_1}} {\rho \left( {t + \tau } \right)\left( {\frac{{\partial L}}{{\partial {q_{s\tau }}}}{{\dot q}_{s\tau }} + } \right.} \frac{{\partial L}}{{\partial {{\dot q}_{s\tau }}}}{{\ddot q}_{s\tau }}} \right)\left( {t + \tau } \right)\Delta t{\text{d}}t +\\& \qquad \int_{{t_1}}^{{t_2}} {\left[ {\rho \left( t \right)\left( {\frac{{\partial L}}{{\partial t}}\Delta t + L\frac{{\text{d}}}{{{\text{d}}t}}\Delta t} \right) + {\phi _s}\left( t \right)\Delta {q_s} + {\varphi _s}\left( t \right)\Delta {{\dot q}_s}} \right]} {\text{d}}t \end{split} $$ (25) 方程 (24) 亦可写为
$$ \begin{split} &\rho \left( {{t_2}} \right)\Delta z\left( {{t_2}} \right) = \int_{{t_1}}^{{t_2}} {\Bigg\{ {\frac{{\text{d}}}{{{\text{d}}t}}\left[ {\rho \left( t \right)L\Delta t + {\varphi _s}\left( t \right)\delta {q_s}} \right]}} +\\ & \qquad {\left[ {{\phi _s}\left( t \right) - {{\dot \varphi }_s}\left( t \right)} \right]\delta {q_s}} \Bigg\}{\text{d}}t \end{split} $$ (26) 由于$\Delta t = \varepsilon {\xi _0}$, $\Delta {q_s} = \varepsilon {\xi _s}$, 由式 (20) 可得
$$ \Delta {\dot q_s} = \varepsilon \left( {{{\dot \xi }_s} - {{\dot q}_s}{{\dot \xi }_0}} \right),\quad \delta {q_s} = \varepsilon \left( {{\xi _s} - {{\dot q}_s}{\xi _0}} \right) $$ (27) 将式 (27) 代入式 (25) 和式 (26), 得到
$$ \begin{split} &\rho \left( {{t_2}} \right)\Delta z\left( {{t_2}} \right) =\\& \qquad \int_{{t_1} - \tau }^{{t_1}} {\varepsilon \left[ {\rho \left( {t + \tau } \right)\left( {\frac{{\partial L}}{{\partial {q_{s\tau }}}}{{\dot q}_{s\tau }} + \frac{{\partial L}}{{\partial {{\dot q}_{s\tau }}}}{{\ddot q}_{s\tau }}} \right)\left( {t + \tau } \right)} \right] {\xi _0}{\text{d}}t} + \\& \qquad \int_{{t_1}}^{{t_2}} {\varepsilon \Bigg[ {\rho \left( t \right)\left( {\frac{{\partial L}}{{\partial t}}{\xi _0} + L{{\dot \xi }_0}} \right) + {\phi _s}\left( t \right){\xi _s} + } } \\& \qquad {{\varphi _s}\left( t \right)\left( {{{\dot \xi }_s} - {{\dot q}_s}{{\dot \xi }_0}} \right)} \Bigg]{\text{d}}t\\[-2pt] \end{split} $$ (28) $$ \begin{split} &\rho \left( {{t_2}} \right)\Delta z\left( {{t_2}} \right) = \int_{{t_1}}^{{t_2}} {\varepsilon \Bigg\{ {\frac{{\text{d}}}{{{\text{d}}t}}\left[ {\rho \left( t \right)L{\xi _0} + {\varphi _s}\left( t \right){{\bar \xi }_s}} \right] + } }\\& \qquad {\left[ {{\phi _s}\left( t \right) - {{\dot \varphi }_s}\left( t \right)} \right]{{\bar \xi }_s}} \Bigg\}{\text{d}}t \end{split} $$ (29) 式中, $ {\bar \xi _s} = {\xi _s} - {\dot q_s}{\xi _0} $. 式(28)和式(29)是含时滞的Hamilton-Herglotz作用量的变分公式.
3. Herglotz型Noether对称性
依据含时滞的Hamilton-Herglotz作用量在无限小变换下的不变性, 可定义含时滞非完整系统的Herglotz型Noether对称性.
如果含时滞的Hamilton-Herglotz作用量在无限小变换式(18) 下, 满足
$$ \rho \left( {{t_2}} \right)\Delta z\left( {{t_2}} \right) = 0 $$ (30) 以及Appell-Cheatev条件式 (14), 则称该无限小变换为含时滞非完整系统式(13)和式(15) 的Herglotz型Noether对称变换.
将式 (28) 代入式 (30), 可以得到
$$ \begin{split} &\rho \left( {t + \tau } \right)\left( {\frac{{\partial L}}{{\partial {q_{s\tau }}}}{{\dot q}_{s\tau }} + \frac{{\partial L}}{{\partial {{\dot q}_{s\tau }}}}{{\ddot q}_{s\tau }}} \right)\left( {t + \tau } \right){\xi _0} = 0,\\ &\qquad t \in \left[ {{t_1} - \tau ,{t_1}} \right)\end{split} $$ (31) $$ \begin{split} &\rho \left( t \right)\left( {\frac{{\partial L}}{{\partial t}}{\xi _0} + L{{\dot \xi }_0}} \right) + {\phi _s}\left( t \right){\xi _s} +{\varphi _s}\left( t \right)\left( {{{\dot \xi }_s} - {{\dot q}_s}{{\dot \xi }_0}} \right) = 0,\\ &\qquad t \in \left[ {{t_1},{t_2}} \right]\end{split} $$ (32) 称式(31)和式(32) 为Herglotz型Noether等式.
如果含时滞的Hamilton-Herglotz作用量在无限小变换式 (18) 下, 满足
$$ \rho \left( {{t_2}} \right)\Delta z\left( {{t_2}} \right) = - \int_{{t_1}}^{{t_2}} {\frac{{\text{d}}}{{{\text{d}}t}}\left[ {\rho \left( t \right)\Delta G} \right]} {\text{d}}t $$ (33) 以及Appell-Cheatev条件式(14), 则称该无限小变换为含时滞非完整系统式(13)和式(15) 的Herglotz型Noether准对称变换, 其中$ \Delta G = \varepsilon G $, $ G = G\left( {t,{q_s},{{\dot q}_s}} \right) $为规范函数.
将式 (28) 代入式 (33), 可以得到
$$ \begin{split} &\rho \left( {t + \tau } \right)\left( {\frac{{\partial L}}{{\partial {q_{s\tau }}}}{{\dot q}_{s\tau }} + \frac{{\partial L}}{{\partial {{\dot q}_{s\tau }}}}{{\ddot q}_{s\tau }}} \right)\left( {t + \tau } \right){\xi _0} = 0,\\ &\qquad t \in \left[ {{t_1} - \tau ,{t_1}} \right)\end{split} $$ (34) $$ \begin{split} &\rho \left( t \right)\left( {\frac{{\partial L}}{{\partial t}}{\xi _0} + L{{\dot \xi }_0}} \right) + {\phi _s}\left( t \right){\xi _s} +\\&\qquad {\varphi _s}\left( t \right)\left( {{{\dot \xi }_s} - {{\dot q}_s}{{\dot \xi }_0}} \right) = - \frac{{\text{d}}}{{{\text{d}}t}}\left[ {\rho \left( t \right)G} \right] ,\;\;t \in \left[ {{t_1},{t_2}} \right] \end{split}$$ (35) 式 (34) 和式 (35) 亦称为Herglotz型Noether等式.
4. Herglotz型Noether定理
对于含时滞的非完整系统, 由Herglotz型Noether对称性可导致Herglotz型Noether守恒量, 有如下定理.
定理1 如果无限小变换 (18) 满足Herglotz型Noether等式 (31) 和式 (32), 以及Appell-Cheatev条件
$$ \frac{{\partial {f_\beta }}}{{\partial {{\dot q}_s}}}{\bar \xi _s} = 0 $$ (36) 则含时滞的非完整系统式(13)和式(15) 的Herglotz型Noether对称变换导致Herglotz型Noether守恒量, 形如
$$ I = \rho \left( t \right)L{\xi _0} + {\varphi _s}\left( t \right){\bar \xi _s} = {\text{const}}. $$ (37) 证明: 因变换式(18) 是Herglotz型Noether对称变换, 则$ \rho \left( {{t_2}} \right)\Delta z\left( {{t_2}} \right) = 0 $. 利用式 (29), 得
$$ \int_{{t_1}}^{{t_2}} \varepsilon \left\{ {\frac{{\text{d}}}{{{\text{d}}t}}\left[ {\rho \left( t \right)L{\xi _0} + {\varphi _s}\left( t \right){{\bar \xi }_s}} \right] + } \left[ {{\phi _s}\left( t \right) - {{\dot \varphi }_s}\left( t \right)} \right]{{\bar \xi }_s} \right\}{\text{d}}t = 0 $$ (38) 由Appell-Cheatev条件式(36), 可得
$$ \int_{{t_1}}^{{t_2}} {\varepsilon {\lambda _\beta }\frac{{\partial {f_\beta }}}{{\partial {{\dot q}_s}}}{{\bar \xi }_s}{\text{d}}t} = 0 $$ (39) 将式 (38) 和式 (39) 相加, 得到
$$ \begin{split} &\int_{{t_1}}^{{t_2}} {\varepsilon \left\{ {\frac{{\text{d}}}{{{\text{d}}t}}\left[ {\rho \left( t \right)L{\xi _0} + {\varphi _s}\left( t \right){{\bar \xi }_s}} \right] + } \right.}\\&\qquad \left. {\left[ {{\phi _s}\left( t \right) - {{\dot \varphi }_s}\left( t \right) + {\lambda _\beta }\frac{{\partial {f_\beta }}}{{\partial {{\dot q}_s}}}} \right]{{\bar \xi }_s}} \right\}{\text{d}}t = 0 \end{split}$$ (40) 将方程 (15) 代入方程 (40), 并考虑到参数$\varepsilon $以及积分区间$\left[ {{t_1},{t_2}} \right]$的任意性, 可得
$$ \frac{{\text{d}}}{{{\text{d}}t}}\left[ {\rho \left( t \right)L{\xi _0} + {\varphi _s}\left( t \right){{\bar \xi }_s}} \right] = 0 $$ (41) 对式 (41) 积分, 即得守恒量式(37).
定理2 如果无限小变换 (18) 满足Herglotz型Noether等式(34) 和式(35), 以及Appell-Cheatev条件式(36), 则含时滞的非完整系统式(13)和式(15) 的Herglotz型Noether准对称变换导致Herglotz型Noether守恒量, 形如
$$ I = \rho \left( t \right)L{\xi _0} + {\varphi _s}\left( t \right){\bar \xi _s} + \rho \left( t \right)G = {\text{const}}. $$ (42) 证明: 因变换 (18) 是Herglotz型Noether准对称变换, 则$ \rho \left( {{t_2}} \right)\Delta z\left( {{t_2}} \right) = - \int_{{t_1}}^{{t_2}} {\dfrac{{\text{d}}}{{{\text{d}}t}}\left[ {\rho \left( t \right)\Delta G} \right]} {\text{d}}t $. 利用式 (29), 类似于定理1的证明, 可得
$$ \begin{split} &\int_{{t_1}}^{{t_2}} {\varepsilon \left\{ {\frac{{\text{d}}}{{{\text{d}}t}}\left[ {\rho \left( t \right)L{\xi _0} + {\varphi _s}\left( t \right){{\bar \xi }_s} + \rho \left( t \right)G} \right] + } \right.} \\& \qquad\left. {\left[ {{\phi _s}\left( t \right) - {{\dot \varphi }_s}\left( t \right) + {\lambda _\beta }\frac{{\partial {f_\beta }}}{{\partial {{\dot q}_s}}}} \right]{{\bar \xi }_s}} \right\}{\text{d}}t = 0 \end{split}$$ (43) 将方程 (15) 代入方程 (43), 并考虑到参数$\varepsilon $以及积分区间$\left[ {{t_1},{t_2}} \right]$的任意性, 可得
$$ \frac{{\text{d}}}{{{\text{d}}t}}\left[ {\rho \left( t \right)L{\xi _0} + {\varphi _s}\left( t \right){{\bar \xi }_s} + \rho \left( t \right)G} \right] = 0 $$ (44) 积分方程 (44), 便得守恒量式(42).
定理1和定理2称为含时滞的非完整系统的Herglotz型Noether定理. 通过求解Noether等式和Appell-Cheatev条件可以得到系统的Noether对称变换或准对称变换, 从而获得系统的Herglotz型守恒量, 为寻找非保守非完整系统的守恒量提供了一个新方法.
如果系统是完整的, 则方程 (15) 退化为含时滞完整非保守系统的Lagrange方程, 即
$$ {\phi _s}\left( t \right) - {\dot \varphi _s}\left( t \right) = 0 $$ (45) 定理2退化为含时滞完整非保守系统的Herglotz型Noether定理, 即有定理3.
定理3 对于含时滞完整非保守系统式(45), 如果无限小变换式(18) 满足Herglotz型Noether等式(34)和式(35), 则系统存在形如式 (42) 的Herglotz型Noether守恒量.
如果系统是保守的, 则方程 (15) 退化为经典的含时滞非完整保守系统的Routh方程
$$ \tilde{\phi}_s t-\dot{\tilde{\varphi}}_s t + \lambda_\beta \frac{\partial f_\beta}{\partial \dot{q}_s} = 0 $$ (46) 式中
$$ {\tilde \phi _s}\left( t \right) = \left\{ \begin{gathered} \frac{{\partial \tilde L}}{{\partial {q_s}}} + \frac{{\partial \tilde L}}{{\partial {q_{s\tau }}}},{\text{ }}t \in \left[ {{t_1},{t_2} - \tau } \right] \\ \frac{{\partial \tilde L}}{{\partial {q_s}}},{\text{ }}t \in \left( {{t_2} - \tau ,{t_2}} \right] \\ \end{gathered} \right. $$ (47) $$ {\tilde \varphi _s}\left( t \right) = \left\{ \begin{gathered} \frac{{\partial \tilde L}}{{\partial {{\dot q}_s}}} + \frac{{\partial \tilde L}}{{\partial {{\dot q}_{s\tau }}}},{\text{ }}t \in \left[ {{t_1},{t_2} - \tau } \right] \\ \frac{{\partial \tilde L}}{{\partial {{\dot q}_s}}},{\text{ }}t \in \left( {{t_2} - \tau ,{t_2}} \right] \\ \end{gathered} \right. $$ (48) 这里$\tilde L = \tilde L\left( {t,{q_s},{{\dot q}_s},{q_{s\tau }},{{\dot q}_{s\tau }}} \right)$是经典含时滞的拉格朗日函数. 定理2退化为含时滞非完整保守系统的Herglotz型Noether定理, 即有定理4.
定理4 对于含时滞非完整保守系统式(13)、式(46), 如果无限小变换式(18) 满足如下Noether等式, 即当$t \in \left[ {{t_1} - \tau ,{t_1}} \right)$时, 有
$$ \left( {\frac{{\partial \tilde L}}{{\partial {q_{s\tau }}}}{{\dot q}_{s\tau }} + \frac{{\partial \tilde L}}{{\partial {{\dot q}_{s\tau }}}}{{\ddot q}_{s\tau }}} \right)\left( {t + \tau } \right){\xi _0} = 0 $$ (49) 当$t \in \left[ {{t_1},{t_2}} \right]$时, 有
$$ \frac{{\partial \tilde L}}{{\partial t}}{\xi _0} + \tilde L{\dot \xi _0} + {\varLambda _s}{\bar \xi _s} + {\tilde \phi _s}\left( t \right){\xi _s} + {\tilde \varphi _s}\left( t \right)\left( {{{\dot \xi }_s} - {{\dot q}_s}{{\dot \xi }_0}} \right) = - \frac{{\text{d}}}{{{\text{d}}t}}G $$ (50) 以及Appell-Cheatev条件式(36), 则系统存在含时滞的经典Noether守恒量, 形如
$$ I = \tilde L{\xi _0} + \tilde \varphi \left( t \right){\bar \xi _s} + G = {\text{const}}. $$ (51) 如果系统不含时滞, 则方程 (15) 退化为非完整非保守系统的Herglotz型Lagrange方程, 即
$$ \rho \left( t \right)\left( {\frac{{\partial L}}{{\partial {q_s}}} - \frac{{\text{d}}}{{{\text{d}}t}}\frac{{\partial L}}{{\partial {{\dot q}_s}}} + } \right.\left. {\frac{{\partial L}}{{\partial z}}\frac{{\partial L}}{{\partial {{\dot q}_s}}} + {\lambda _\beta }\frac{{\partial {f_\beta }}}{{\partial {{\dot q}_s}}}} \right) = 0 $$ (52) 定理2退化为非完整非保守系统的Herglotz型Noether定理, 即有定理5.
定理5 对于非完整非保守系统式(13)和式(52), 如果无限小变换式(18) 满足如下Herglotz型Noether等式, 即
$$ \rho \left( t \right)\left[ {\frac{{\partial L}}{{\partial t}}{\xi _0} + L{{\dot \xi }_0} + \frac{{\partial L}}{{\partial {q_s}}}{\xi _s} + } \right. \left. {\frac{{\partial L}}{{\partial {{\dot q}_s}}}\left( {{{\dot \xi }_s} - {{\dot q}_s}{{\dot \xi }_0}} \right)} \right] = - \frac{{\text{d}}}{{{\text{d}}t}}\left[ {\rho \left( t \right)G} \right] $$ (53) 以及Appell-Cheatev条件式(36), 则系统存在Herglotz型Noether守恒量
$$ I = \rho \left( t \right)\left( {L{\xi _0} + \frac{{\partial L}}{{\partial {{\dot q}_s}}}{{\bar \xi }_s} + G} \right) = {\text{const}}. $$ (54) 定理3和定理5分别为含时滞完整非保守系统和不考虑时滞时非完整非保守系统的Herglotz型Noether定理, 定理4为含时滞非完整保守系统经典Noether定理.
5. Herglotz型Noether逆定理
假设含时滞非完整系统式(13)和式(15) 有守恒量
$$ I = I\left( {t,{q_s},{{\dot q}_s},{q_{s\tau }},{{\dot q}_{s\tau }}} \right) = {\text{const}} .$$ (55) 求解生成元${\xi _0}$和${\xi _s}$, 使它们成为Herglotz型Noether准对称变换.
首先, 对式 (55) 求导, 可得
$$ \frac{{{\text{d}}I}}{{{\text{d}}t}} = \frac{{\partial I}}{{\partial t}} + \frac{{\partial I}}{{\partial {q_s}}}{\dot q_s} + \frac{{\partial I}}{{\partial {{\dot q}_s}}}{\ddot q_s} + \frac{{\partial I}}{{\partial {q_{s\tau }}}}{\dot q_{s\tau }} + \frac{{\partial I}}{{\partial {{\dot q}_{s\tau }}}}{\ddot q_{s\tau }} $$ (56) 再将运动方程 (15) 乘以${\bar \xi _s}$, 并对$s$求和, 可得
$$ \left[ {{\phi _s}\left( t \right) - {{\dot \varphi }_s}\left( t \right) + {\lambda _\beta }\frac{{\partial {f_\beta }}}{{\partial {{\dot q}_s}}}} \right]{\bar \xi _s} = 0 $$ (57) 其次, 分别在$ \left[ {{t_1},{t_2} - \tau } \right] $及$\left( {{t_2} - \tau ,{t_2}} \right]$上比较含${\ddot q_k}$项的系数, 可以得到
$$ \begin{split} &\frac{{\partial I}}{{\partial {{\dot q}_k}}} + \frac{{\partial I}}{{\partial {{\dot q}_{k\tau }}}} = \rho \left( t \right)\left( { - \frac{{{\partial ^2}L}}{{\partial {{\dot q}_s}\partial {{\dot q}_k}}}} \right){\bar \xi _s} +\\& \qquad \rho \left( {t + \tau } \right)\left( { - \frac{{{\partial ^2}L}}{{\partial {{\dot q}_{s\tau }}\partial {{\dot q}_{k\tau }}}}} \right)\left( {t + \tau } \right){\bar \xi _s} \end{split} $$ (58) $$ \frac{{\partial I}}{{\partial {{\dot q}_k}}} = - \rho \left( t \right)\frac{{{\partial ^2}L}}{{\partial {{\dot q}_s}\partial {{\dot q}_k}}}{\bar \xi _s} $$ (59) 最后, 令
$$ \rho \left( t \right)L{\xi _0} + {\varphi _s}\left( t \right){\bar \xi _s} + \rho \left( t \right)G = I $$ (60) 那么, 通过式 (58) ~ 式(60) 可找到相应的无限小生成元及规范函数. 有以下定理.
定理6 对于含时滞的非完整系统式(13)和式(15), 若系统存在守恒量式(55), 则由式 (58) ~ 式(60) 可寻得无限小变换的时间和空间的生成元${\xi _0}$, ${\xi _s}$及规范函数$G$.
定理6是含时滞的非完整系统的Herglotz型Noether定理的逆定理. 通过对已知守恒量求解, 可以求出相应的生成元及规范函数.
6. 算例
假设含时滞非保守系统的Herglotz型拉格朗日函数为
$$ L = \frac{1}{2}\left[ {\dot q_1^2\left( t \right) + \dot q_1^2\left( {t - \tau } \right)} \right] + \frac{1}{2}\left[ {\dot q_2^2\left( t \right) + \dot q_2^2\left( {t - \tau } \right)} \right] - z $$ (61) 非保守力为
$$ {Q_1} = - {\dot q_1}, \quad {Q_2} = - {\dot q_2} $$ (62) 受有非完整约束
$$ f = {\dot q_1} + bt{\dot q_2} = 0 $$ (63) 试研究系统的对称性与Herglotz型Noether守恒量.
Herglotz型Routh方程 (15) 给出
$$ \left.\begin{split} &- {\ddot q_1} - {\dot q_1} + \lambda + \left[ { - {{\ddot q}_{1\tau }}\left( {t + \tau } \right) - {{\dot q}_{1\tau }}\left( {t + \tau } \right)} \right]{{\mathrm{e}}^\tau } = 0\\ & - {\ddot q_2} - {\dot q_2} + bt\lambda + \left[ { - {{\ddot q}_{2\tau }}\left( {t + \tau } \right) - } \right. \\ &\qquad \left[ {{{\dot q}_{2\tau }}\left( {t + \tau } \right)} \right]{{\mathrm{e}}^\tau } = 0,\quad t \in \left[ {{t_1},{t_2} - \tau } \right]\\ & - {\ddot q_1} - {\dot q_1} + \lambda = 0 \\ & - {\ddot q_2} - {\dot q_2} + bt\lambda = 0 , \quad{\text{ }}t \in \left( {{t_2} - \tau ,{t_2}} \right] \end{split}\right\}$$ (64) 由式(63) 和式 (64), 可以求得
$$ \lambda = \left\{ \begin{split} & { - \dfrac{{b{{\dot q}_2}\left( {1 + {{\mathrm{e}}^\tau }} \right)}}{{1 + {b^2}{t^2}}}},\quad {t \in \left[ {{t_1},{t_2} - \tau } \right]} \\ & { - \dfrac{{b{{\dot q}_2}}}{{1 + {b^2}{t^2}}}},\quad {t \in \left( {{t_2} - \tau ,{t_2}} \right]} \end{split} \right. $$ (65) 将式 (65) 代入方程 (64), 简化可得
$$ \left.\begin{split} &{\ddot q_1} = - {\dot q_1} - \frac{{b{{\dot q}_2}}}{{1 + {b^2}{t^2}}}\\ & {\ddot q_2} = - {\dot q_2} - \frac{{{b^2}t{{\dot q}_2}}}{{1 + {b^2}{t^2}}}, {\text{ }}t \in \left[ {{t_1},{t_2}} \right] \end{split} \right\}$$ (66) 取$b = 1$, 则广义坐标$ {q_1} $和${q_2}$运动轨迹如图1所示.
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图 1 ${q_1}$和${q_2}$在$t \in \left[ {0,8} \right]$上的值Figure 1. Simulation of ${q_1}$ and ${q_2}$ on $t \in \left[ {0,8} \right]$Appell-Chetaev条件式(36) 给出
$$ {\xi _1} + bt{\xi _2} - \left( {{{\dot q}_1} + bt{{\dot q}_2}} \right){\xi _0} = 0 $$ (67) 根据定理2, 有
$$ \left( {{{\dot q}_1}{{\ddot q}_1} + {{\dot q}_2}{{\ddot q}_2}} \right){\xi _0} = 0, \quad t \in \left[ {{t_1} - \tau ,{t_1}} \right) $$ (68) $$ \begin{split} &{\dot q_1}\left( {{{\dot \xi }_1} - {{\dot q}_1}{{\dot \xi }_0}} \right) + {\dot q_2}\left( {{{\dot \xi }_2} - {{\dot q}_2}{{\dot \xi }_0}} \right) + L{\dot \xi _0} + G + \\ &\qquad \dot G + {{\mathrm{e}}^\tau }\left\{ {{{\dot q}_{1\tau }}\left( {t + \tau } \right)\left[ {{{\dot \xi }_1} - {{\dot q}_{1\tau }}\left( {t + \tau } \right){{\dot \xi }_0}} \right] + } \right. \\ &\qquad \left. {{{\dot q}_{2\tau }}\left( {t + \tau } \right)\left[ {{{\dot \xi }_2} - {{\dot q}_{2\tau }}\left( {t + \tau } \right){{\dot \xi }_0}} \right]} \right\} = 0,\\ &\qquad t \in \left[ {{t_1},{t_2} - \tau } \right] \end{split} $$ (69) $$ \begin{aligned} &{\dot q_1}\left( {{{\dot \xi }_1} - {{\dot q}_1}{{\dot \xi }_0}} \right) + {\dot q_2}\left( {{{\dot \xi }_2} - {{\dot q}_2}{{\dot \xi }_0}} \right) + L{\dot \xi _0} + G + \dot G = 0,\\ &\qquad t \in \left( {{t_2} - \tau ,{t_2}} \right]\end{aligned} $$ (70) 当$t \in \left[ {{t_1} - \tau ,{t_1}} \right)$, 由方程 (68) 易知${\xi _0}$恒为0. 当$t \in \left[ {{t_1},{t_2} - \tau } \right]$, 由方程 (67) 和式 (69), 可解得
$$\left.\begin{split} & \xi _0^1 = 0,\;\; \xi _1^1 = \frac{{ - bt{{\mathrm{e}}^{{t_1}}}}}{{\sqrt {1 + {b^2}{t^2}} }} + {\dot q_1},\;\; \xi _2^1 = \frac{{{{\mathrm{e}}^{{t_1}}}}}{{\sqrt {1 + {b^2}{t^2}} }} + {\dot q_2}\\ & {G^1} = - \left( {\dot q_1^2 + \dot q_2^2} \right)\left( {1 + {{\mathrm{e}}^\tau }} \right) \end{split} \right\}$$ (71) $$\left. \begin{split} &\xi _0^2 = 0, \;\; \xi _1^2 = \frac{{ - bt{{\mathrm{e}}^{ - t + {t_1}}}}}{{\left( {1 + {b^2}{t^2}} \right){{\dot q}_2}}},\;\; \xi _2^2 = \frac{{{{\mathrm{e}}^{ - t + {t_1}}}}}{{\left( {1 + {b^2}{t^2}} \right){{\dot q}_2}}}\\ &{G^2} = {{\mathrm{e}}^{ - t + {t_1}}}\left( {1 + {{\mathrm{e}}^\tau }} \right)\left[ {{\text{ln}}{{\dot q}_2} + t + \frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right)} \right] \end{split}\right\} $$ (72) 由定理2, 与生成元式(71) 和式(72) 相应的守恒量为
$$ {I_1} = \left( {1 + {{\mathrm{e}}^\tau }} \right){{\mathrm{e}}^t}\frac{{{{\dot q}_2} - bt{{\dot q}_1}}}{{\sqrt {1 + {b^2}{t^2}} }}\qquad\qquad\qquad\quad $$ (73) $$ {I_2} = 1 + \left( {1 + {{\mathrm{e}}^\tau }} \right)\left[ {{\text{ln}}{{\dot q}_2} + t + \frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right)} \right] $$ (74) 取$\tau = 1$, $b = 1$, 则表达式 (73) 和式(74) 随时间$t$ 的变化规律如图2所示, 数值结果表明, ${I_1}$和${I_2}$ 确为所论系统的守恒量.
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图 2 $\tau = 1$时${I_1}$和${I_2}$在$t \in \left[ {0,2000} \right]$上的值Figure 2. Simulation of ${I_1}$ and ${I_2}$ on $t \in \left[ {0,2000} \right]$ with $\tau = 1$下面分析时滞对守恒量的影响. 取$b = 1$, 图3给出了在时刻$t = 20$, $ 100\;{\text{s}}$时, 时滞量变化对守恒量${I_1}$的影响. 数值结果表明时滞的改变会对力学系统产生较大影响, 是不可忽略的.
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图 3 守恒量${I_1}$在$\tau \in \left[ {0,1} \right]$上的值Figure 3. Simulation of ${I_1}$ on $\tau \in \left[ {0,1} \right]$当$t \in \left( {{t_2} - \tau ,{t_2}} \right]$, 由方程 (67) 和式 (70) 解得
$$\left.\begin{split} &\xi _0^1 = 0,\;\;\xi _1^1 = \frac{{ - bt{{\mathrm{e}}^{{t_1}}}}}{{\sqrt {1 + {b^2}{t^2}} }} + {\dot q_1},\;\;\xi _2^1 = \frac{{{{\mathrm{e}}^{{t_1}}}}}{{\sqrt {1 + {b^2}{t^2}} }} + {\dot q_2}\\ & {G^1} = - \left( {\dot q_1^2 + \dot q_2^2} \right) \end{split}\right\}$$ (75) $$ \left.\begin{split} &\xi _0^2 = 0,\;\;\xi _1^2 = \frac{{ - bt{{\mathrm{e}}^{ - t + {t_1}}}}}{{\left( {1 + {b^2}{t^2}} \right){{\dot q}_2}}},\;\; \xi _2^2 = \frac{{{{\mathrm{e}}^{ - t + {t_1}}}}}{{\left( {1 + {b^2}{t^2}} \right){{\dot q}_2}}}\\ & {G^2} = {{\mathrm{e}}^{ - t + {t_1}}}\left[ {{\text{ln}}{{\dot q}_2} + t + \frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right)} \right] \end{split}\right\} $$ (76) 由定理2, 与生成元式(75) 和式(76)相应的守恒量为
$$ {I_1} = {{\mathrm{e}}^t}\frac{{{{\dot q}_2} - bt{{\dot q}_1}}}{{\sqrt {1 + {b^2}{t^2}} }} \qquad\qquad\quad\;\;$$ (77) $$ {I_2} = 1 + {\text{ln}}{\dot q_2} + t + \frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right) $$ (78) 由式 (77) 和式 (78) 可看出, 当$t \in \left( {{t_2} - \tau ,{t_2}} \right]$时, 此时时滞对守恒量没有影响.
其次, 研究逆问题. 当$t \in \left[ {{t_1},{t_2} - \tau } \right]$, 假设系统有守恒量
$$ I = 1 + \left( {1 + {{\mathrm{e}}^\tau }} \right)\left[ {{\text{ln}}{{\dot q}_2} + t + \frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right)} \right] $$ (79) 由定理6可知
$$ \frac{{1 + {{\mathrm{e}}^\tau }}}{{{{\dot q}_2}}} = - {{\mathrm{e}}^{t - {t_1}}}\left( {1 + {{\mathrm{e}}^\tau }} \right)\left( {{\xi _2} - {{\dot q}_2}{\xi _0}} \right) $$ (80) $$ \begin{split} & 1 + \left( {1 + {{\mathrm{e}}^\tau }} \right)\left[ {{\text{ln}}{{\dot q}_2} + t + \frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right)} \right] =\\&\qquad {{\mathrm{e}}^{t - {t_1}}}\left( {1 + {{\mathrm{e}}^\tau }} \right)\left[ {{{\dot q}_1}\left( {{\xi _1} - {{\dot q}_1}{\xi _0}} \right) + } \right.\\&\qquad \left. {{{\dot q}_2}\left( {{\xi _2} - {{\dot q}_2}{\xi _0}} \right)} \right] + {{\mathrm{e}}^{t - {t_1}}}\left( {L{\xi _0} + G} \right) \end{split} $$ (81) 方程 (80) 和方程 (81), 2个方程4个未知量, 因此方程的解不唯一.
令${\xi _0} = 0$, 则解得
$$ \left.\begin{split} &{\xi _1} = \frac{{bt{{\mathrm{e}}^{ - t + {t_1}}}}}{{{{\dot q}_2}}},\quad {\xi _2} = - \frac{{{{\mathrm{e}}^{ - t + {t_1}}}}}{{{{\dot q}_2}}}\\ & G = {{\mathrm{e}}^{ - t + {t_1}}}\Bigg\{ {1 + \left( {1 + {{\mathrm{e}}^\tau }} \right)} \Bigg[ {{\text{ln}}{{\dot q}_2} + t + } \\ &\qquad{ {\frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right) + 1 + {b^2}{t^2}} \Bigg]} \Bigg\} \end{split}\right\} $$ (82) 令${\xi _0} = 1$, 则解得
$$\left.\begin{split} & {\xi _1} = \frac{{bt{{\mathrm{e}}^{ - t + {t_1}}}}}{{{{\dot q}_2}}} - bt{\dot q_2},\quad {\xi _2} = - \frac{{{{\mathrm{e}}^{ - t + {t_1}}}}}{{{{\dot q}_2}}} + {\dot q_2}\\ & G = {{\mathrm{e}}^{ - t + {t_1}}}\Bigg\{ {1 + \left( {1 + {{\mathrm{e}}^\tau }} \right)\Bigg[ {{\text{ln}}{{\dot q}_2} + } } t + \\ &\qquad { {\frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right) + 1 + {b^2}{t^2}} \Bigg]} \Bigg\} - L \end{split}\right\}$$ (83) 生成元式(82) 和式(83) 都满足Appell-Cheatev条件 (67), 因此它们都相应于系统的准对称变换.
当$t \in \left( {{t_2} - \tau ,{t_2}} \right]$, 假设系统有守恒量
$$ I = 1 + {\text{ln}}{\dot q_2} + t + \frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right) $$ (84) 由定理6, 可知
$$ \frac{1}{{{{\dot q}_2}}} = - {{\mathrm{e}}^{t - {t_1}}}\left( {{\xi _2} - {{\dot q}_2}{\xi _0}} \right) \qquad\qquad\qquad\qquad\qquad\;\;$$ (85) $$ \begin{split} & 1 + \left[ {{\text{ln}}{{\dot q}_2} + t + \frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right)} \right] =\\ &\qquad {{\mathrm{e}}^{t - {t_1}}}\left[ {{{\dot q}_1}\left( {{\xi _1} - {{\dot q}_1}{\xi _0}} \right) + {{\dot q}_2}\left( {{\xi _2} - {{\dot q}_2}{\xi _0}} \right) + L{\xi _0} + G} \right] \end{split}$$ (86) 显然方程式(85)和式(86) 的解不唯一. 如令${\xi _0} = 0$, 则解得
$$\left. \begin{split} &{\xi _1} = \frac{{bt{{\mathrm{e}}^{ - t + {t_1}}}}}{{{{\dot q}_2}}},\quad {\xi _2} = - \frac{{{{\mathrm{e}}^{ - t + {t_1}}}}}{{{{\dot q}_2}}}\\ & G = {{\mathrm{e}}^{ - t + {t_1}}}\left\{ {1 + \left[ {{\text{ln}}{{\dot q}_2} + t + \frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right) + 1 + {b^2}{t^2}} \right]} \right\} \end{split}\right\} $$ (87) 如${\xi _0} = 1$, 则解得
$$ \left.\begin{split} &{\xi _1} = \frac{{bt{{\mathrm{e}}^{ - t + {t_1}}}}}{{{{\dot q}_2}}} - bt{\dot q_2},\quad {\xi _2} = - \frac{{{{\mathrm{e}}^{ - t + {t_1}}}}}{{{{\dot q}_2}}} + {\dot q_2} \\ & G = {{\mathrm{e}}^{ - t + {t_1}}}\Bigg\{ {1 + \Bigg[ {{\text{ln}}{{\dot q}_2} + t + } } \\ &\qquad { {\frac{1}{2}{\text{ln}}\left( {1 + {b^2}{t^2}} \right) + 1 + {b^2}{t^2}} \Bigg]} \Bigg\} - L \end{split}\right\} $$ (88) 生成元式(87) 和式(88) 亦满足Appell-Chetaev条件式(67).
7. 结论
Herglotz广义变分原理为非保守系统变分方法的研究提供了新思路. 文章从含时滞的Herglotz变分问题出发研究含时滞非完整系统的Noether对称性与守恒量. 主要工作一是建立了含时滞的Herglotz型微分变分原理, 结合Appell-Cheatev条件, 采用拉格朗日乘子法, 建立含时滞的非完整力学系统的Herglotz型Routh方程; 二是依据非完整系统关于微分和变分运算交换关系的Hölder定义, 推导出了含时滞的Hamilton-Herglotz作用量的变分公式, 定义Herglotz型Noether对称变换和准对称变换, 给出Herglotz型Noether等式; 三是建立并证明了含时滞的非完整系统的Herglotz型Noether定理及其逆定理, 并通过算例分析和数值计算, 验证了结果的正确性, 演示了时滞对守恒量的影响. 文章方法和结果可进一步推广到含时滞任意阶非完整约束系统等.
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图 1 ${q_1}$和${q_2}$在$t \in \left[ {0,8} \right]$上的值
Figure 1. Simulation of ${q_1}$ and ${q_2}$ on $t \in \left[ {0,8} \right]$
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图 2 $\tau = 1$时${I_1}$和${I_2}$在$t \in \left[ {0,2000} \right]$上的值
Figure 2. Simulation of ${I_1}$ and ${I_2}$ on $t \in \left[ {0,2000} \right]$ with $\tau = 1$
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