Noether’s Theorem for Constrained Hamiltonian System Under Generalized Operators
-
摘要:
研究了广义算子下奇异系统的Noether对称性与守恒量。首先，建立了广义算子下奇异系统的Lagrange方程，并导出该系统的初级约束，然后引入Lagrange乘子建立了广义算子下约束Hamilton方程以及相容性条件。其次，基于Hamilton作用量在无限小变换下的不变性，建立了广义算子下约束Hamilton系统的Noether定理，并给出了该系统的对称性及相应的守恒量。在特定条件下，广义算子下约束Hamilton系统的Noether守恒量可以退化为整数阶约束Hamilton系统的Noether守恒量。最后举例说明了结果的应用。
Abstract:Noether’s symmetry and conserved quantity of singular systems under generalized operators were studied. Firstly, the Lagrangian equation of singular systems under generalized operators was established, and the primary constraints on the system were derived. Then the Lagrangian multiplier was introduced to establish the constrained Hamilton equation and the compatibility condition under generalized operators. Secondly, based on the invariance of the Hamilton action under the infinitesimal transformation, Noether’s theorem for constrained Hamiltonian systems under generalized operators was established, and the symmetry and corresponding conserved quantity of the system were given. Under certain conditions, Noether’s conservation of constrained Hamiltonian systems under generalized operators can be reduced to Noether’s conservation of integer-order constrained Hamiltonian systems. Finally, an example illustrates the application of the results.
-
[1] 李子平. 经典和量子约束系统及其对称性质[M]. 北京: 北京工业大学出版社, 1993.LI Ziping. Classical and Quantal Dynamics of Constrained Systems and Their Symmetrical Properties[M]. Beijing: Beijing Polytechnic University Press, 1993. (in Chinese) [2] 李子平. 约束哈密顿系统及其对称性质[M]. 北京: 北京工业大学出版社, 1999.LI Ziping. Constrained Hamiltonian Systems and Their Symmetrical Properties[M]. Beijing: Beijing Polytechnic University Press, 1999. (in Chinese) [3] NAMBU Y. Generalized Hamiltonian dynamics[J]. Physical Reviewed, 1973, 7(8): 2405-2412. [4] BERGMANN P G, GOLDBERG J. Dirac bracket transformations in phase space[J]. Physical Review, 1955, 98(2): 531-538. doi: 10.1103/PhysRev.98.531 [5] NOETHER E. Invariant variations problems[C]//Nachrichten von der Gesellschaft der Wissenschaftenzu Göttingen. 1918: 235-257. [6] DJUKIC D S, VUJANOVIC B D. Noether’s theory in classical nonconservative mechanics[J]. Acta Mechanica, 1975, 23(1): 17-27. [7] 梅凤翔. 约束力学系统的对称性与守恒量[M]. 北京: 北京理工大学出版社, 2004.MEI Fengxiang. Symmetry and Conserved Quantities of Constrained Mechanical Systems[M]. Beijing: Beijing University of Technology Press, 2004. (in Chinese) [8] 郑明亮, 刘洁, 邓斌. 覆冰输电导线舞动的Noether对称性和守恒量[J]. 应用数学和力学, 2021, 42(3): 275-281ZHENG Mingliang, LIU Jie, DENG Bin. The Noether symmetry and conserved quantity of galloping iced power transmission lines[J]. Applied Mathematics and Mechanics, 2021, 42(3): 275-281.(in Chinese) [9] 张毅. 弱非线性动力学方程的Noether准对称性与近似Noether守恒量[J]. 力学学报, 2020, 52(6): 1765-1773 doi: 10.6052/0459-1879-20-242ZHANG Yi. Noether quasi-symmetry and approximate Noether conservation laws for weakly nonlinear dynamical equations[J]. Chinese Journal of Applied Mechanics, 2020, 52(6): 1765-1773.(in Chinese) doi: 10.6052/0459-1879-20-242 [10] 罗绍凯. 相对论Birkhoff系统的形式不变性与Noether守恒量[J]. 应用数学和力学, 2003, 24(4): 414-422 doi: 10.3321/j.issn:1000-0887.2003.04.012LUO Shaokai. Form invariance and Noether symmetrical conserved quantity of relativistic Birkhoffian systems[J]. Applied Mathematics and Mechanics, 2003, 24(4): 414-422.(in Chinese) doi: 10.3321/j.issn:1000-0887.2003.04.012 [11] ZHAI X H, ZHANG Y. Noether symmetries and conserved quantities for Birkhoffian systems with time delay[J]. Nonlinear Dynamics, 2014, 77: 73-86. doi: 10.1007/s11071-014-1274-8 [12] 李子平. 非完整非保守奇异系统正则形式的Noether定理及其逆定理[J]. 科学通报, 1992, 23: 2204-2205LI Ziping. Noether theorem and its inverse theorem of regular form for nonholonomic nonconservative singular systems[J]. Chinese Science Bulletin, 1992, 23: 2204-2205.(in Chinese) [13] OLDHAM K B, SPANIER J. The Fractional Calculus[M]. San Diego: Academic Press, 1974. [14] SUN Y, YANG X, ZHENG C, et al. Modelling long-term deformation of granular soils incorporating the concept of fractional calculus[J]. Acta Mechanica Sinica, 2016, 32: 112-124. doi: 10.1007/s10409-015-0490-x [15] 黄飞, 马永斌. 移动热源作用下基于分数阶应变的三维弹性体热-机响应[J]. 应用数学和力学, 2021, 42(4): 373-384HUANG Fei, MA Yongbin. Thermomechanical responses of 3D media under moving heat sources based on fractional-order strains[J]. Applied Mathematics and Mechanics, 2021, 42(4): 373-384.(in Chinese) [16] GU Y J, WANG H, YU Y G. Stability and synchronization for Riemann-Liouville fractional-order time-delayed inertial neural networks[J]. Neurocomputing, 2019, 340: 270-280. doi: 10.1016/j.neucom.2019.03.005 [17] VELLAPPANDI M, KUMAR P, GOVINDARAJ V, et al. An optimal control problem for mosaic disease via Caputo fractional derivative[J]. Alexandria Engineering Journal, 2022, 61(10): 8027-8037. doi: 10.1016/j.aej.2022.01.055 [18] SONG C J, ZHANG Y. Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications[J]. Fractional Calculus and Applied Analysis, 2018, 21(2): 509-526. doi: 10.1515/fca-2018-0028 [19] ALMEIDA R. Fractional variational problems with the Riesz-Caputo derivative[J]. Applied Mathematics Letters, 2012, 25(2): 142-148. doi: 10.1016/j.aml.2011.08.003 [20] AGRAWAL O P. Generalized variational problems and Euler-Lagrange equations[J]. Computers & Mathematics With Applications, 2010, 59(5): 1852-1864. [21] RIEWE F. Nonconservative Lagrangian and Hamiltonian mechanics[J]. Physical Review E, 1996, 53(2): 1890-1899. doi: 10.1103/PhysRevE.53.1890 [22] RIEWE F. Mechanics with fractional derivatives[J]. Physical Review E, 1997, 55(3): 3581-3592. doi: 10.1103/PhysRevE.55.3581 [23] FREDERICO G S F, TORRES D F M. A formulation of Noether’s theorem for fractional problems of the calculus of variations[J]. Journal of Mathematical Analysis and Applications, 2007, 334(2): 834-846. doi: 10.1016/j.jmaa.2007.01.013 [24] FREDERICO G S F, TORRES D F M. Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem[J]. International Mathematical Forum, 2008, 3(10): 479-493. [25] ZHOU Y, ZHANG Y. Noether’s theorems of a fractional Birkhoffian system within Riemann-Liouville derivatives[J]. Chinese Physics B, 2014, 23(12): 285-292. [26] SONG C J. Noether symmetry for fractional Hamiltonian system[J]. Physics Letters A, 2019, 383(29): 125914. doi: 10.1016/j.physleta.2019.125914 [27] ZHAI X H, ZHANG Y. Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay[J]. Communications in Nonlinear Science and Numerical Simulation, 2016, 36: 81-97. doi: 10.1016/j.cnsns.2015.11.020 [28] 田雪, 张毅. Caputo Δ型分数阶时间尺度Noether定理[J]. 力学学报, 2021, 53(7): 2010-2022 doi: 10.6052/0459-1879-21-108TIAN Xue, ZHANG Yi. Caputo Δ-type fractional time-scales Noether theorem[J]. Chinese Journal of Applied Mechanics, 2021, 53(7): 2010-2022.(in Chinese) doi: 10.6052/0459-1879-21-108 [29] ZHOU S, FU H, FU J L. Symmetry theories of Hamiltonian systems with fractional derivatives[J]. Science China:Physics, Mechanics & Astronomy, 2011, 54(10): 1847-1853. [30] LUO S K, LI L. Fractional generalized Hamiltonian equations and its integral invariants[J]. Nonlinear Dynamics, 2013, 73(1): 339-346. [31] 张宏彬. 基于广义分数阶算子Birkhoff系统Noether定理[J]. 动力学与控制学报, 2019, 17(5): 458-462 doi: 10.6052/1672-6553-2019-064ZHANG Hongbin. Noether’s theorem of Birkhoffian systems with generalized fractional operators[J]. Journal of Dynamics and Control, 2019, 17(5): 458-462.(in Chinese) doi: 10.6052/1672-6553-2019-064 [32] SONG C J, SHEN S L. Noether symmetry method for Birkhoffian systems in terms of generalized fractional operators[J]. Theoretical & Applied Mechanics Letters, 2021, 11(6): 330-335. [33] SONG C J, CHENG Y. Noethersymmetry method for Hamiltonian mechanics involving generalized operators[J]. Advances in Mathematical Physics, 2021, 2021: 1959643. -
计量
- 文章访问数: 499
- HTML全文浏览量: 203
- PDF下载量: 45
- 被引次数: 0
下载:
渝公网安备50010802005915号