American Institute of Mathematical Sciences

Advanced
September  2020, 12(3): 541-551. doi: 10.3934/jgm.2020009

A summary on symmetries and conserved quantities of autonomous Hamiltonian systems

Narciso Román-Roy

Departament of Mathematics. Universidad Politécnica de Cataluña, Edificio C-3, Campus Norte UPC. C/ Jordi Girona 1. 08034 Barcelona, Spain

Received  April 2019 Revised  December 2019 Published  March 2020

A complete geometric classification of symmetries of autonomous Hamiltonian systems is established; explaining how to obtain their associated conserved quantities in all cases. In particular, first we review well-known results and properties about the symmetries of the Hamiltonian and of the symplectic form and then some new kinds of non-symplectic symmetries and their conserved quantities are introduced and studied.

Keywords: Symmetries, conserved quantities, Hamiltonian systems, Noether theorem, symplectic manifolds.
Mathematics Subject Classification: Primary: 37K05, 70H05, 70S10; Secondary: 37J15, 53D05, 70S05.
Citation: Narciso Román-Roy. A summary on symmetries and conserved quantities of autonomous Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (3) : 541-551. doi: 10.3934/jgm.2020009
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Second edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. doi: 10.1090/chel/364.  Google Scholar

[2]

A. Arancibia and M. S. Plyushchay, Chiral asymmetry in propagation of soliton defects in crystalline backgrounds, Phys. Rev. D, 92 (2015), 105009, 20 pp. doi: 10.1103/PhysRevD.92.105009.  Google Scholar

[3]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[4]

P. Birtea and R. M. Tudoran, Non-Noether conservation laws, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1220004, 5 pp. doi: 10.1142/S0219887812200046.  Google Scholar

[5]

A. V. Bolsinov, Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution, Math. USSR-Izvestiya, 38 (1992), 69-90.  doi: 10.1070/IM1992v038n01ABEH002187.  Google Scholar

[6]

A. V. Bolsinov and A. V. Borisov, Compatible Poisson brackets on Lie algebras, Math. Notes, 72 (2002), 10-30.  doi: 10.1023/A:1019856702638.  Google Scholar

[7]

J. F. Cariñena and L. A. Ibort, Non-Noether constants of motion, J. Phys. A, 16 (1983), 1-7.  doi: 10.1088/0305-4470/16/1/010.  Google Scholar

[8]

J. F. Cariñena, G. Marmo and M. F. Rañada, Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator, J. Phys. A, 35 (2002), L679–L686. doi: 10.1088/0305-4470/35/47/101.  Google Scholar

[9]

G. Chavchanidze, Non-Noether symmetries and their influence on phase space geometry, J. Geom. Phys., 48 (2003), 190-202.  doi: 10.1016/S0393-0440(03)00040-8.  Google Scholar

[10]

G. Chavchanidze, Non-Noether symmetries in Hamiltonian dynamical systems, Mem. Diff. Eqs. Math. Phys., 36 (2005), 81-134.   Google Scholar

[11]

M. Crampin, Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16 (1977), 741-754.  doi: 10.1007/BF01807231.  Google Scholar

[12]

M. Crampin, A note on non-Noether constants of motion, Phys. Lett. A, 95 (1983), 209-212.  doi: 10.1016/0375-9601(83)90605-9.  Google Scholar

[13]

M. Crampin, W. Sarlet and G. Thompson, Bi-differential calculi and bi-Hamiltonian systems, J. Phys. A, 33 (2000), L177–180. doi: 10.1088/0305-4470/33/20/101.  Google Scholar

[14]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Reduction of presymplectic manifolds with symmetry, Rev. Math. Phys., 11 (1999), 1209-1247.  doi: 10.1142/S0129055X99000386.  Google Scholar

[15]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484.  doi: 10.1088/0305-4470/32/48/309.  Google Scholar

[16]

G. Falqui, F. Magri and M. Pedroni, Bihamiltonian geometry and separation of variables for Toda lattices, J. Nonlinear Math. Phys., 8 (2001), suppl., 118–127. doi: 10.2991/jnmp.2001.8.s.21.  Google Scholar

[17]

J. Gaset, P. D. Prieto-Martínez and N. Román-Roy, Variational principles and symmetries on fibered multisymplectic manifolds, Comm. Math., 24 (2016), 137-152.  doi: 10.1515/cm-2016-0010.  Google Scholar

[18]

B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theor. App. Mechanics, 43 (2016), 255-273.  doi: 10.2298/TAM160121009J.  Google Scholar

[19]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications, 35. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[20]

C. López, E. Martínez and M. F. Rañada, Dynamical symmetries, non-Cartan symmetries and superintegrability of the $n$-dimensional harmonic oscillator, J. Phys. A, 32 (1999), 1241-1249.  doi: 10.1088/0305-4470/32/7/013.  Google Scholar

[21]

F. A. Lunev, Analog of Noether's theorem for non-Noether and nonlocal symmetries, Theor. Math. Phys., 84 (1990), 816-820.  doi: 10.1007/BF01017679.  Google Scholar

[22]

M. Lutzky, Origin of non-Noether invariants, Phys. Lett. A, 75 (1980), 8-10.  doi: 10.1016/0375-9601(79)90258-5.  Google Scholar

[23]

M. Lutzky, New classes of conserved quantities associated with non-Noether symmetries, J. Phys. A, 15 (1982), L87–L91. doi: 10.1088/0305-4470/15/3/001.  Google Scholar

[24]

C. M. Marle and J. Nunes da Costa, Master symmetries and bi-Hamiltonian structures for the relativistic Toda lattice, J. Phys. A, 30 (1997), 7551-7556.  doi: 10.1088/0305-4470/30/21/025.  Google Scholar

[25]

G. Marmo and N. Mukunda, Symmetries and constants of the motion in the Lagrangian formalism on $TQ$: Beyond point transformations, Nuovo Cim B, 92 (1986), 1-12.  doi: 10.1007/BF02729691.  Google Scholar

[26]

G. Marmo, E. J. Saletan, A. Simoni and B. Vitale, Dynamical Systems, a Differential Geometric Approach to Symmetry and Reduction, John Wiley & Sons, Ltd., Chichester, 1985.  Google Scholar

[27]

J. C. Marrero, N. Román-Roy, M. Salgado and S. Vilariño, Reduction of polysymplectic manifolds, J. Phys. A, 48 (2015), 055206, 43 pp. doi: 10.1088/1751-8113/48/5/055206.  Google Scholar

[28]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[29]

J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar

[30]

J. Mateos-Guilarte and M. S. Plyushchay, Perfectly invisible $\mathcal{PT}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry, J. High Energy Phys., 2017 (2017), 061, front matter+35 pp. doi: 10.1007/JHEP12(2017)061.  Google Scholar

[31]

M. F. Rañada, Integrable three-particle systems, hidden symmetries and deformation of the Calogero-Moser system, J. Math. Phys., 36 (1995), 3541-3558.  doi: 10.1063/1.530980.  Google Scholar

[32]

M. F. Rañada, Superintegrable $n = 2$ systems, quadratic constants of motion, and potential of Drach, J. Math. Phys., 38 (1997), 4165-4178.  doi: 10.1063/1.532089.  Google Scholar

[33]

M. F. Rañada, Dynamical symmetries, bi-Hamiltonian structures and superintegrable $n = 2$ systems, J. Math. Phys., 41 (2000), 2121-2134.  doi: 10.1063/1.533230.  Google Scholar

[34]

N. Román-Roy, M. Salgado and S. Vilariño, Higher-order Noether symmetries in $k$-symplectic Hamiltonian field theory, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1360013, 9 pp. doi: 10.1142/S021988781360013X.  Google Scholar

[35]

G. Rosensteel and J. P. Draayer, Symmetry algebra of the anisotropic harmonic oscillator with commensurate frequencies, J. Phys. A, 22 (1989), 1323-1327.  doi: 10.1088/0305-4470/22/9/021.  Google Scholar

[36]

W. Sarlet and F. Cantrijn, Higher-order Noether symmetries and constants of the motion, J. Phys. A, 14 (1981), 479-492.  doi: 10.1088/0305-4470/14/2/023.  Google Scholar

[37]

W. Sarlet and F. Cantrijn, Generalizations of Noether's theorem in classical mechanics, SIAM Rev., 23 (1981), 467-494.  doi: 10.1137/1023098.  Google Scholar

[38]

Yu. B. Suris, On the bi-Hamiltonian structure of Toda and relativistic Toda lattices, Phys. Lett. A, 180 (1993), 419-429.  doi: 10.1016/0375-9601(93)90293-9.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Second edition, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. doi: 10.1090/chel/364.  Google Scholar

[2]

A. Arancibia and M. S. Plyushchay, Chiral asymmetry in propagation of soliton defects in crystalline backgrounds, Phys. Rev. D, 92 (2015), 105009, 20 pp. doi: 10.1103/PhysRevD.92.105009.  Google Scholar

[3]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[4]

P. Birtea and R. M. Tudoran, Non-Noether conservation laws, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1220004, 5 pp. doi: 10.1142/S0219887812200046.  Google Scholar

[5]

A. V. Bolsinov, Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution, Math. USSR-Izvestiya, 38 (1992), 69-90.  doi: 10.1070/IM1992v038n01ABEH002187.  Google Scholar

[6]

A. V. Bolsinov and A. V. Borisov, Compatible Poisson brackets on Lie algebras, Math. Notes, 72 (2002), 10-30.  doi: 10.1023/A:1019856702638.  Google Scholar

[7]

J. F. Cariñena and L. A. Ibort, Non-Noether constants of motion, J. Phys. A, 16 (1983), 1-7.  doi: 10.1088/0305-4470/16/1/010.  Google Scholar

[8]

J. F. Cariñena, G. Marmo and M. F. Rañada, Non-symplectic symmetries and bi-Hamiltonian structures of the rational harmonic oscillator, J. Phys. A, 35 (2002), L679–L686. doi: 10.1088/0305-4470/35/47/101.  Google Scholar

[9]

G. Chavchanidze, Non-Noether symmetries and their influence on phase space geometry, J. Geom. Phys., 48 (2003), 190-202.  doi: 10.1016/S0393-0440(03)00040-8.  Google Scholar

[10]

G. Chavchanidze, Non-Noether symmetries in Hamiltonian dynamical systems, Mem. Diff. Eqs. Math. Phys., 36 (2005), 81-134.   Google Scholar

[11]

M. Crampin, Constants of the motion in Lagrangian mechanics, Int. J. Theor. Phys., 16 (1977), 741-754.  doi: 10.1007/BF01807231.  Google Scholar

[12]

M. Crampin, A note on non-Noether constants of motion, Phys. Lett. A, 95 (1983), 209-212.  doi: 10.1016/0375-9601(83)90605-9.  Google Scholar

[13]

M. Crampin, W. Sarlet and G. Thompson, Bi-differential calculi and bi-Hamiltonian systems, J. Phys. A, 33 (2000), L177–180. doi: 10.1088/0305-4470/33/20/101.  Google Scholar

[14]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Reduction of presymplectic manifolds with symmetry, Rev. Math. Phys., 11 (1999), 1209-1247.  doi: 10.1142/S0129055X99000386.  Google Scholar

[15]

A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector field formulation of Hamiltonian field theories: Equations and symmetries, J. Phys. A: Math. Gen., 32 (1999), 8461-8484.  doi: 10.1088/0305-4470/32/48/309.  Google Scholar

[16]

G. Falqui, F. Magri and M. Pedroni, Bihamiltonian geometry and separation of variables for Toda lattices, J. Nonlinear Math. Phys., 8 (2001), suppl., 118–127. doi: 10.2991/jnmp.2001.8.s.21.  Google Scholar

[17]

J. Gaset, P. D. Prieto-Martínez and N. Román-Roy, Variational principles and symmetries on fibered multisymplectic manifolds, Comm. Math., 24 (2016), 137-152.  doi: 10.1515/cm-2016-0010.  Google Scholar

[18]

B. Jovanović, Noether symmetries and integrability in Hamiltonian time-dependent mechanics, Theor. App. Mechanics, 43 (2016), 255-273.  doi: 10.2298/TAM160121009J.  Google Scholar

[19]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications, 35. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[20]

C. López, E. Martínez and M. F. Rañada, Dynamical symmetries, non-Cartan symmetries and superintegrability of the $n$-dimensional harmonic oscillator, J. Phys. A, 32 (1999), 1241-1249.  doi: 10.1088/0305-4470/32/7/013.  Google Scholar

[21]

F. A. Lunev, Analog of Noether's theorem for non-Noether and nonlocal symmetries, Theor. Math. Phys., 84 (1990), 816-820.  doi: 10.1007/BF01017679.  Google Scholar

[22]

M. Lutzky, Origin of non-Noether invariants, Phys. Lett. A, 75 (1980), 8-10.  doi: 10.1016/0375-9601(79)90258-5.  Google Scholar

[23]

M. Lutzky, New classes of conserved quantities associated with non-Noether symmetries, J. Phys. A, 15 (1982), L87–L91. doi: 10.1088/0305-4470/15/3/001.  Google Scholar

[24]

C. M. Marle and J. Nunes da Costa, Master symmetries and bi-Hamiltonian structures for the relativistic Toda lattice, J. Phys. A, 30 (1997), 7551-7556.  doi: 10.1088/0305-4470/30/21/025.  Google Scholar

[25]

G. Marmo and N. Mukunda, Symmetries and constants of the motion in the Lagrangian formalism on $TQ$: Beyond point transformations, Nuovo Cim B, 92 (1986), 1-12.  doi: 10.1007/BF02729691.  Google Scholar

[26]

G. Marmo, E. J. Saletan, A. Simoni and B. Vitale, Dynamical Systems, a Differential Geometric Approach to Symmetry and Reduction, John Wiley & Sons, Ltd., Chichester, 1985.  Google Scholar

[27]

J. C. Marrero, N. Román-Roy, M. Salgado and S. Vilariño, Reduction of polysymplectic manifolds, J. Phys. A, 48 (2015), 055206, 43 pp. doi: 10.1088/1751-8113/48/5/055206.  Google Scholar

[28]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Second edition, Texts in Applied Mathematics, 17. Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[29]

J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-130.  doi: 10.1016/0034-4877(74)90021-4.  Google Scholar

[30]

J. Mateos-Guilarte and M. S. Plyushchay, Perfectly invisible $\mathcal{PT}$-symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry, J. High Energy Phys., 2017 (2017), 061, front matter+35 pp. doi: 10.1007/JHEP12(2017)061.  Google Scholar

[31]

M. F. Rañada, Integrable three-particle systems, hidden symmetries and deformation of the Calogero-Moser system, J. Math. Phys., 36 (1995), 3541-3558.  doi: 10.1063/1.530980.  Google Scholar

[32]

M. F. Rañada, Superintegrable $n = 2$ systems, quadratic constants of motion, and potential of Drach, J. Math. Phys., 38 (1997), 4165-4178.  doi: 10.1063/1.532089.  Google Scholar

[33]

M. F. Rañada, Dynamical symmetries, bi-Hamiltonian structures and superintegrable $n = 2$ systems, J. Math. Phys., 41 (2000), 2121-2134.  doi: 10.1063/1.533230.  Google Scholar

[34]

N. Román-Roy, M. Salgado and S. Vilariño, Higher-order Noether symmetries in $k$-symplectic Hamiltonian field theory, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1360013, 9 pp. doi: 10.1142/S021988781360013X.  Google Scholar

[35]

G. Rosensteel and J. P. Draayer, Symmetry algebra of the anisotropic harmonic oscillator with commensurate frequencies, J. Phys. A, 22 (1989), 1323-1327.  doi: 10.1088/0305-4470/22/9/021.  Google Scholar

[36]

W. Sarlet and F. Cantrijn, Higher-order Noether symmetries and constants of the motion, J. Phys. A, 14 (1981), 479-492.  doi: 10.1088/0305-4470/14/2/023.  Google Scholar

[37]

W. Sarlet and F. Cantrijn, Generalizations of Noether's theorem in classical mechanics, SIAM Rev., 23 (1981), 467-494.  doi: 10.1137/1023098.  Google Scholar

[38]

Yu. B. Suris, On the bi-Hamiltonian structure of Toda and relativistic Toda lattices, Phys. Lett. A, 180 (1993), 419-429.  doi: 10.1016/0375-9601(93)90293-9.  Google Scholar

[1]

Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029
[2]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
[3]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233
[4]

Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341
[5]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[6]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453
[7]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028
[8]

Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
[9]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[10]

Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161
[11]

Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
[12]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017
[13]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513
[14]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451
[15]

Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021022
[16]

F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605
[17]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[18]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[19]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[20]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (221)
  • HTML views (370)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences