HOJMAN CONSERVED QUANTITY FOR TIME SCALES LAGRANGE SYSTEMS
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摘要: 利用对称性和守恒律, 可以简化动力学问题甚至求解力学系统的精确解, 更好地理解其动力学行为. 时间尺度分析将连续和离散动力学模型统一并拓展到时间尺度框架, 既避免了重复研究又可揭示两者之区别和联系. 因此, 通过对称性来探寻在时间尺度的框架下新的守恒定律很有必要. 本文首先建立了时间尺度上Lagrange方程, 利用时间尺度微积分性质导出了时间尺度上Lagrange系统的两个重要关系式; 其次, 依据微分方程在单参数Lie变换群下的不变性, 建立了时间尺度上Lie对称性的定义和确定方程; 最后, 建立了时间尺度上Lie对称性定理并利用上述关系式给出了证明, 得到了时间尺度上Lagrange系统的新守恒量. 当时间尺度取为实数集时, 该守恒量退化为著名的Hojman守恒量. 文末考察了一个两自由度时间尺度Lagrange系统, 在3种不同时间尺度情形下得到了该系统的Hojman守恒量, 数值计算结果验证了定理的正确性.
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关键词:
- Lagrange系统 /
- Lie对称性 /
- Hojman守恒量 /
- 时间尺度
Abstract: By using symmetry and conservation laws, we can simplify dynamical problem and even obtain the exact solution of mechanical system, and better understand the dynamical behavior of system. Time scales analysis unifies and extends the continuous and discrete dynamics models to the time scales framework, which not only avoids repeated studies but also reveals the differences and connections between them. Therefore, it is necessary to explore new conservation laws in the framework of time scale through symmetry. Firstly, the Lagrange equations on time scales are established, and two important relations of time scales Lagrange system are derived by using the properties of time scales calculus. Secondly, according to the invariance of differential equation under the one-parameter Lie group of transformations, the definition of Lie symmetry on time scales and its determining equation are established. Thirdly, the Lie symmetry theorem on time scales is established and proved by using the above relations, and the new conservation laws of time scales Lagrange system are obtained. When the time scale is taken to the set of real numbers, the conservation laws degenerate to the famous Hojman conserved quantity. Finally, a two-degree-of-freedom time scales Lagrange system is investigated, and its Hojman conserved quantities are obtained in three different time scales, and the correctness of the theorem we obtained is verified by numerical calculation.-
Key words:
- Lagrange system /
- Lie symmetry /
- Hojman conserved quantity /
- time scales
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图 1 时间尺度
$ {{\mathbb{T}}} = {2^{{{\mathbb{N}}_0}}} $ 上$ {q_1},{q_2},{I_1},{I_2} $ 的值Figure 1. Simulations of
$ {q_1},{q_2},{I_1},{I_2} $ on the time scale$ {{\mathbb{T}}} = {2^{{{\mathbb{N}}_0}}} $ 
图 2 时间尺度
${{\mathbb{T}}} = {{\mathbb{Z}}_ + }$ 上$ {q_1},{q_2},{I_1},{I_2} $ 的值Figure 2. Simulations of
$ {q_1},{q_2},{I_1},{I_2} $ on the time scale${{\mathbb{T}}} = {{\mathbb{Z}}_ + }$ 表 1 时间尺度
$ {{\mathbb{T}}} = {2^{{{\mathbb{N}}_0}}} $ 上$ {q_1},{q_2},{I_1},{I_2} $ 的值Table 1. The values of
$ {q_1},{q_2},{I_1},{I_2} $ on the time scale$ {{\mathbb{T}}} = {2^{{{\mathbb{N}}_0}}} $ $ {\mathbb{T}}/{\rm{s}} $ $ {q_1}/{\rm{m}} $ $ {q_2}/{\rm{m}} $ $ {I_1} $ $ {I_2} $ 1 1 0 −9.33 6 2 2 1 −9.33 6 4 0 −5 −9.33 6 8 −20 −81 −9.33 6 16 −124 −745 −9.33 6 32 −588 −6169 −9.33 6 64 −2540 −49785 −9.33 6 
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表 2 时间尺度
${{\mathbb{T}}} = {{\mathbb{Z}}_ + }$ 上$ {q_1},{q_2},{I_1},{I_2} $ 的值.Table 2. The values of
$ {q_1},{q_2},{I_1},{I_2} $ on the time scale${{\mathbb{T}}} = {{\mathbb{Z}}_ + }$ $ {\mathbb{T}}/{\rm{s}} $ $ {q_1} /{\rm{m}}$ $ {q_2}/{\rm{m}} $ $ {I_1} $ $ {I_2} $ 1 1 0 −4 4 2 2 1 −4 4 3 2 0 −4 4 4 1 −4 −4 4 5 −1 −12 −4 4 6 −4 −25 −4 4 7 −8 −44 −4 4 8 −13 −70 −4 4 9 −19 −104 −4 4 10 −26 −147 −4 4
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