ARTICLE
Euler-Lagrange equations and Noether’s theorem of multi-scale mechano-electrophysiological coupling model of neuron membrane dynamics
1
School of Civil Engineering and Architecture, University of Jinan, Jinan, Shangdong, P.R. China
Submission date: 2022-08-24
Final revision date: 2022-11-30
Acceptance date: 2023-09-04
Online publication date: 2023-10-29
Publication date: 2023-10-30
Corresponding author
Journal of Theoretical and Applied Mechanics 2023;61(4):847-856
KEYWORDS
TOPICS
ABSTRACT
Noether’s theorem is applied into a multi-scale mechno-electrophysiological coupling model
of neuron membrane dynamics. The Euler-Lagrange equations in generalized coordinates of
this model are deduced by the nonconservative Hamilton principle. The Noether symmetry
criterion and conserved quantities based on the Lie point transformation group are given. The
influence of external non-potential forces and material parameters on the forms of Noether
conserved quantities is detailed discussed, which indicates that the conserved quantities are
very depending on the loading rate and mechanical parameters of the membrane.
REFERENCES (25)
1.
Bluman G.W., Anco S.C., 2002, Symmetry and Integration Methods for Differential Equation, Springer-Verlag, Berlin.
2.
Chen X.W., Li Y.M., Zhao Y.H., 2005, Lie symmetries, perturbation to symmetries and adiabatic invariants of Lagrange system, Physics Letters A, 337, 274-278.
3.
Dorodnitsyn V., 2001, Noether-type theorems for difference equations, Appllied Numerical Mathematics, 39, 307-321.
4.
Drapaca C.S., 2015, An electromechanical model of neuronal dynamics using Hamilton’s principle, Frontiers in Cellular Neuroscience, 9, 1-8.
5.
Drapaca C.S., 2017, Fractional calculus in neuronal electromechanics, Journal of Mechanics of Materials and Structures, 12, 1, 35-55.
6.
Fang J.H., Ding N., Wang P., 2007, A new type of conserved quantity of Mei symmetry for Lagrange system, Chinese Physics, 16, 887.
7.
Heimburg T., Jackson A.D., 2005, On soliton propagation in biomembranes and nerves, Proceedings of the National Academy of Sciences, 102, 28, 9790-9795.
8.
Hydon P., Mansfield E., 2011, Extensions of Noether’s second theorem: from continuous to discrete systems, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 467, 3206-3221.
9.
Kosmann-Schwarzbach Y., Schwarzbach B.E., 2011, The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century, Springer.
10.
Luo S.K., Yang M.J., Zhang X.T., Dai Y., 2018, Basic theory of fractional Mei symmetrical perturbation and its applications, Acta Mechanica, 229, 1833-1848.
11.
Mansfield E., Rojo-Echeburúa A., Hydon P., Peng L., 2019, Moving frames and Noether’s finite difference conservation laws I, Transactions of Mathematics and its Applications, 3.
12.
Mei F.X., 1993, Noether’s theory of Birkhoffian systems, Science in China, Serie A, 36, 12, 1456-1467.
13.
Mei F.X., 2000, Form invariance of Lagrange system, Journal of Beijing Institute of Technology, 9, 120-124.
14.
Mei F.X., 2004, Symmetries and Conserved Quantities of Constrained Mechanical Systems (in Chinese), Beijing Institute of Technology Press, Beijing.
15.
Noether E., 1918, Invariante Variationsprobleme, Nachr. Akad. Wiss. Göttingen, Mathematical Physic, 2, 235-257.
16.
Olver P.J., 1986, Applications of Lie Groups to Differential Equations, Springer, New York.
17.
Peng L., 2017, Symmetries, conservation laws, and Noether’s theorem for differential-difference equations, Studies in Applied Mathematics, 139, 457-502.
18.
Peng L., Hydon P., 2022, Transformations, symmetries and Noether theorems for differential-difference equations, Proceedings of the Royal Society, A: Mathematical, Physical and Engineering Sciences, 478, 20210944.
19.
Wang P., 2011, Perturbation to Noether symmetry and Noether adiabatic invariants of discrete mechanico-electrical systems, Chinese Physics Letters, 28, 040203-4.
20.
Wang P., 2012, Perturbation to symmetry and adiabatic invariants of discrete nonholonomic nonconservative mechanical system, Nonlinear Dynamics, 68, 53-62.
21.
Wang P., 2018, Conformal invariance and conserved quantities of mechanical system with unilateral constraints, Communications in Nonlinear Science and Numerical Simulation, 59, 463-471.
22.
Wang P., Xue Y., 2016, Conformal invariance of Mei symmetry and conserved quantities of Lagrange equation of thin elastic rod, Nonlinear Dynamics, 83, 1815-1822.
23.
Wang P., Zhu H.J., 2011, Perturbation to symmetry and adiabatic invariants of general discrete holonomic dynamical systems, Acta Physica Polonica A, 119, 298-303.
24.
Zhang Y., 2022, Nonshifted dynamics of constrained systems on time scales under Lagrange framework and its Noether’s theorem, Communications in Nonlinear Science and Numerical Simulation, 108, 106214.
25.
Zhang H.B., Chen H.B., 2018, Noether’s theorem of Hamiltonian systems with generalized fractional derivative operators, International Journal of Non-Linear Mechanics, 107, 34-41.
| eISSN: | 2543-6309 |
| ISSN: | 1429-2955 |
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