Abstract
In this paper, Noether’s theorem and its inverse theorem are proved for the fractional variational problems based on logarithmic Lagrangian systems. The Hamilton principle of the systems is derived. And the definitions and the criterions of Noether’s symmetry and Noether’s quasi-symmetry of the systems based on logarithmic Lagrangians are given. The intrinsic relation between Noether’s symmetry and the conserved quantity is established. At last an example is given to illustrate the application of the results.
Supported by the National Natural Science Foundation of China (61473338) and Hubei Province Key Laboratory of Systems Science in Metallurgical Process (Wuhan University of Science and Technology) (Y201514)
References
[1] Noether A E. Invariante variationsprobleme. Nachr Akad Wiss Gott Math Phys, 1918, 2: 235–237.10.1007/978-3-642-39990-9_13Search in Google Scholar
[2] Frederico G S F, Torres D F M. A formulation of Noether’s theorem for fractional problems of the calculus of variations. Journal of Mathematical Analysis and Applications, 2007, 334(2): 834–846.10.1016/j.jmaa.2007.01.013Search in Google Scholar
[3] Atanackovic T M, Konjik S, Pilipovic S, et al. Variational problems with fractional derivatives: Invariance conditions and Noether’s theorem. Nonlinear Analysis, 2009, 71: 1504–1517.10.1016/j.na.2008.12.043Search in Google Scholar
[4] Zhang Y, Zhai X H. Noether symmetries and conserved quantities for fractional Birkhoffian systems. Non-linear Dynamics, 2015, 81: 469–480.10.1007/s11071-015-2005-5Search in Google Scholar
[5] Frederico G S F, Lazo M J. Fractional Noether’s theorem with classical and Caputo derivatives: Constants of motion for non-conservative systems. Nonlinear Dynamics, 2016, 85: 839–851.10.1007/s11071-016-2727-zSearch in Google Scholar
[6] Frederico G S F, Torres D F M. Noether’s symmetry theorem for variational and optimal control problems with time delay. Numerical Algebra, Control & Optimization, 2012, 2(3): 619–630.10.3934/naco.2012.2.619Search in Google Scholar
[7] Frederico G S F, Odzijewicz T, Torres D F M. Noether’s theorem for non-smooth extremals of variational problems with time delay. Applicable Analysis, 2014, 93(1): 153–170.10.1080/00036811.2012.762090Search in Google Scholar
[8] Lin S X, Zhang Y. Noether symmetries for nonconservative Lagrange systems with time delay based on fractional model. Nonlinear Dynamics, 2015, 79(2): 1169–1183.10.1007/s11071-014-1734-1Search in Google Scholar
[9] Arnold V I. Mathematical methods of classical mechanics. New York: Springer, 1978.10.1007/978-1-4757-1693-1Search in Google Scholar
[10] El-Nabulsi A R. Nonlinear dynamics with nonstandard Lagrangians. Qualitative Theory of Dynamical Systems, 2012, 12(2): 273–291.10.1007/s12346-012-0074-0Search in Google Scholar
[11] El-Nabulsi A R. Non-standard fractional Lagrangians. Nonlinear Dynamics, 2013, 74(1): 381–394.10.1007/s11071-013-0977-6Search in Google Scholar
[12] El-Nabulsi A R. Non-standard Lagrangians in rotational dynamics and the modified Navier-Stokes equation. Nonlinear Dynamics, 2015, 79(3): 2055–2068.10.1007/s11071-014-1794-2Search in Google Scholar
[13] Zhang Y, Zhou X S. Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians. Nonlinear Dynamics, 2016, 84: 1867–1876.10.1007/s11071-016-2611-xSearch in Google Scholar
[14] Zhou Y, Zhang Y. Fractional Pfaff-Birkhoff principle and fractional Birkoff’s equations of Riemann-Liouville derivatives. Bull Sci Technol, 2013, 29: 4–10 (in Chinese).Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Irrational-Behavior-Proof Conditions Based on Limit Characteristic Functions
- Integrated Online Consumer Preference Mining for Product Improvement with Online Reviews
- The Selection and Pricing of Mixed Multi-Channel Marketing Model for Mid-High Wines Under Experience Driven
- SE2IR Invest Market Rumor Spreading Model Considering Hesitating Mechanism
- Aggregation Similarity Measure Based on Hesitant Fuzzy Closeness Degree and Its Application to Clustering Analysis
- Noether’s Symmetries and Its Inverse for Fractional Logarithmic Lagrangian Systems
Articles in the same Issue
- Irrational-Behavior-Proof Conditions Based on Limit Characteristic Functions
- Integrated Online Consumer Preference Mining for Product Improvement with Online Reviews
- The Selection and Pricing of Mixed Multi-Channel Marketing Model for Mid-High Wines Under Experience Driven
- SE2IR Invest Market Rumor Spreading Model Considering Hesitating Mechanism
- Aggregation Similarity Measure Based on Hesitant Fuzzy Closeness Degree and Its Application to Clustering Analysis
- Noether’s Symmetries and Its Inverse for Fractional Logarithmic Lagrangian Systems