메뉴 건너뛰기




Volumn 2, Issue 2, 2008, Pages 146-157

The Ermakov equation: A commentary

Author keywords

Ermakov; Invariant; Lewis; Pinney; sl(2, R)

Indexed keywords


EID: 65049089300     PISSN: 14528630     EISSN: None     Source Type: Journal    
DOI: 10.2298/AADM0802146L     Document Type: Article
Times cited : (101)

References (106)
  • 1
    • 0039940823 scopus 로고    scopus 로고
    • Complete symmetry groups of ordinary differential equations and their integrals: Some basic considerations
    • K. Andriopoulos, P. G. L. Leach, G. P. Flessas: Complete symmetry groups of ordinary differential equations and their integrals: some basic considerations. Journal of Mathematical Analysis and Applications, 262 (2001), 256-273.
    • (2001) Journal of Mathematical Analysis and Applications , vol.262 , pp. 256-273
    • Andriopoulos, K.1    Leach, P.G.L.2    Flessas, G.P.3
  • 2
    • 0042198533 scopus 로고    scopus 로고
    • The economy of complete symmetry groups for linear higher-dimensional systems
    • Second Supplement
    • K. Andriopoulos, P. G. L. Leach: The economy of complete symmetry groups for linear higher-dimensional systems. Journal of Nonlinear Mathematical Physics, 9 Second Supplement (2002), 10-23.
    • (2002) Journal of Nonlinear Mathematical Physics , vol.9 , pp. 10-23
    • Andriopoulos, K.1    Leach, P.G.L.2
  • 4
    • 0011573017 scopus 로고
    • Polyhedra Ermakov systems
    • C. Athorne: Polyhedra Ermakov systems. Physics Letters A, 151 (1990), 407-411.
    • (1990) Physics Letters A , vol.151 , pp. 407-411
    • Athorne, C.1
  • 10
    • 33744769865 scopus 로고    scopus 로고
    • Factorisations and transformations of differential equations. Methods and Applications
    • L. M. Berkovich: Factorisations and transformations of differential equations. Methods and Applications. Regular and Chaotic Dynamics 464 (2002).
    • (2002) Regular and Chaotic Dynamics , vol.464
    • Berkovich, L.M.1
  • 12
    • 78650949364 scopus 로고    scopus 로고
    • Method of factorising of ordinary differential equations and some of its applications
    • Lev M. Berkovich: Method of factorising of ordinary differential equations and some of its applications. Applicable Analysis and Discrete Mathematics, 1 (2007), 122-149.
    • (2007) Applicable Analysis and Discrete Mathematics , vol.1 , pp. 122-149
    • Berkovich, L.M.1
  • 14
    • 83055176229 scopus 로고    scopus 로고
    • Recent applications of the theory of Lie systems in Ermakov systems
    • 031
    • Jośe F. Carineña, Javier De Lucas, Manuel F. Reñada: Recent applications of the theory of Lie systems in Ermakov systems. SIGMA, 4 031 (2008), 18 pages.
    • (2008) SIGMA , vol.4 , pp. 18
    • Carineña, J.F.1    De Lucas, J.2    Reñada, M.F.3
  • 15
    • 45149099903 scopus 로고    scopus 로고
    • Nonlinear waves in Bose-Einstein condensates: Physical relevance and mathematical techniques
    • R. Carretero-González, D. J. Frantzeskakis, P. G. Kevrekidis: Nonlinear waves in Bose-Einstein condensates: physical relevance and mathematical techniques. Nonlinearity, 21 (2008), R139-R202.
    • (2008) Nonlinearity , vol.21
    • Carretero-González, R.1    Frantzeskakis, D.J.2    Kevrekidis, P.G.3
  • 18
    • 0041154594 scopus 로고    scopus 로고
    • Two discretisations of the Ermakov-Pinney equation
    • A. K. Common, M. Musette: Two discretisations of the Ermakov-Pinney equation. Physics Letters A, 235 (1997), 574-580.
    • (1997) Physics Letters A , vol.235 , pp. 574-580
    • Common, A.K.1    Musette, M.2
  • 19
    • 33846053142 scopus 로고    scopus 로고
    • Inverse scattering transform for the Camassa-Holm equation
    • Adrian Constantin, Vladimir S. Gerdjikov, Rossen I. Ivanov: Inverse scattering transform for the Camassa-Holm equation. Inverse Problems, 22 (2006), 2197-2207.
    • (2006) Inverse Problems , vol.22 , pp. 2197-2207
    • Constantin, A..1    Gerdjikov, V.S..2    Ivanov, R.I..3
  • 20
    • 34548698665 scopus 로고    scopus 로고
    • Generalised Fourier transform for the Camassa-Holm hierarchy
    • Adrian Constantin, Vladimir S. Gerdjikov, Rossen I. Ivanov: Generalised Fourier transform for the Camassa-Holm hierarchy. Inverse Problems, 23 (2007), 1565-1597.
    • (2007) Inverse Problems , vol.23 , pp. 1565-1597
    • Constantin, A.1    Gerdjikov, V.S.2    Ivanov, R.I.3
  • 25
    • 0001593957 scopus 로고    scopus 로고
    • Nonlinear excitations in two-dimensional molecular structures with the impurities
    • Yu B. Gaididei, K. O. Rasmussen, P. L. Christiansen: Nonlinear excitations in two-dimensional molecular structures with the impurities. Physical Review E, 52 (1997), 2951-2962.
    • (1997) Physical Review E , vol.52 , pp. 2951-2962
    • Gaididei, Y.B.1    Rasmussen, K.O.2    Christiansen, P.L.3
  • 27
    • 0001003567 scopus 로고
    • Second constant of motion for generalised Ermakov systems
    • J. Goedert: Second constant of motion for generalised Ermakov systems. Physics Letters A, 136 (1989), 391-394.
    • (1989) Physics Letters A , vol.136 , pp. 391-394
    • Goedert, J.1
  • 30
    • 0038898231 scopus 로고
    • Generalized Ermakov systems in terms of sl(2, R) invariants
    • K. S. Govinder, P. G. L. Leach: Generalized Ermakov systems in terms of sl(2, R) invariants. Quæstiones Mathematicæ, 16 (1993), 405-412.
    • (1993) Quæstiones Mathematicæ , vol.16 , pp. 405-412
    • Govinder, K.S.1    Leach, P.G.L.2
  • 31
    • 0011570579 scopus 로고
    • Ermakov systems: A group theoretic approach
    • K. S. Govinder, P. G. L. Leach: Ermakov systems: a group theoretic approach. Physics Letters A, 186 (1994), 391-395.
    • (1994) Physics Letters A , vol.186 , pp. 391-395
    • Govinder, K.S.1    Leach, P.G.L.2
  • 33
    • 77954604100 scopus 로고
    • Algebraic properties of angular momentum type first integrals
    • K. S. Govinder, P. G. L. Leach: Algebraic properties of angular momentum type first integrals. Lie Groups & their Applications, 1 (1994), 95-102.
    • (1994) Lie Groups & their Applications , vol.1 , pp. 95-102
    • Govinder, K.S.1    Leach, P.G.L.2
  • 34
    • 78650945608 scopus 로고
    • Infinite symmetries and the integrability of generalized Ermakov systems
    • W. F. Ames, ed (Georgia Institute of Technology, Atlanta)
    • K. S. Govinder, P. G. L. Leach: Infinite symmetries and the integrability of generalized Ermakov systems. Proceedings 14th IMACS World Congress on Computational and Applied Mathematics. W. F. Ames, ed (Georgia Institute of Technology, Atlanta), (1994), 196-199.
    • (1994) Proceedings 14th IMACS World Congress on Computational and Applied Mathematics , pp. 196-199
    • Govinder, K.S.1    Leach, P.G.L.2
  • 35
    • 0346269026 scopus 로고    scopus 로고
    • On the equivalence of linear third-order differential equations under nonlocal transformations
    • K. S. Govinder, P. G. L. Leach: On the equivalence of linear third-order differential equations under nonlocal transformations. Journal of Mathematical Analysis and Applications, 287 (2003), 402-407.
    • (2003) Journal of Mathematical Analysis and Applications , vol.287 , pp. 402-407
    • Govinder, K.S.1    Leach, P.G.L.2
  • 36
    • 36749109227 scopus 로고
    • Generalized invariants for the time-dependent harmonic oscillator
    • N. J. Günther, P. G. L. Leach: Generalized invariants for the time-dependent harmonic oscillator. Journal of Mathematical Physics, 18 (1977), 572-576.
    • (1977) Journal of Mathematical Physics , vol.18 , pp. 572-576
    • Günther, N.J.1    Leach, P.G.L.2
  • 38
    • 20644461514 scopus 로고    scopus 로고
    • Stochastic quantisation of time-dependent systems by the Haba and Kleinert method
    • DOI: 10.1007/s10773-005-3987-4
    • F. Haas: Stochastic quantisation of time-dependent systems by the Haba and Kleinert method. International Journal of Theoretical Physics, 44 (2005), DOI: 10.1007/s10773-005-3987-4.
    • (2005) International Journal of Theoretical Physics , vol.44
    • Haas, F.1
  • 39
    • 0037101379 scopus 로고    scopus 로고
    • Ermakov-Pinney equation in scaler field cosmologies
    • Rachel M. Hawkins, James E. Lidsey: Ermakov-Pinney equation in scaler field cosmologies. Physical Review D, 66 (2002), 023523.
    • (2002) Physical Review D , vol.66 , pp. 023523
    • Hawkins, R.M.1    Lidsey, J.E.2
  • 40
    • 34347388636 scopus 로고    scopus 로고
    • From Feshbach-resonance managed Bose-Einstein condensates to anisotropic universes: Applications of the Ermakov-Pinney equation with time-dependent nonline-arity
    • G. Herring, P. G. Kevrekidis, F. Williams, T. Christodoulakis, D. J. Frantz-eskakis: From Feshbach-resonance managed Bose-Einstein condensates to anisotropic universes: Applications of the Ermakov-Pinney equation with time-dependent nonline-arity. Physics Letters A, 367 (2007), 140-148.
    • (2007) Physics Letters A , vol.367 , pp. 140-148
    • Herring, G.1    Kevrekidis, P.G.2    Williams, F.3    Christodoulakis, T.4    Frantzeskakis, D.J.5
  • 41
    • 37449033007 scopus 로고    scopus 로고
    • From Feshbach-resonance managed Bose-Einstein condensates to anisotropic universes: Applications of the Ermakov-Pinney equation with time-dependent nonlin-earity
    • (arXiv:cond-mat/0701756v1, 30 Jan. 2007
    • G. Herring, P. G. Kevrekidis, F. Williams, T. Christodoulakis, D. J. Frantz-eskakis: From Feshbach-resonance managed Bose-Einstein condensates to anisotropic universes: Applications of the Ermakov-Pinney equation with time-dependent nonlin-earity. Physics Letters A, 372 (2007), 277-283 (arXiv:cond-mat/0701756v1, 30 Jan. 2007).
    • (2007) Physics Letters A , vol.372 , pp. 277-283
    • Herring, G.1    Kevrekidis, P.G.2    Williams, F.3    Christodoulakis, T.4    Frantzeskakis, D.J.5
  • 44
    • 0347244494 scopus 로고    scopus 로고
    • Exact discreet discretisation of the Ermakov-Pinney equation
    • A. N. W. Hone: Exact discreet discretisation of the Ermakov-Pinney equation. Physics Letters A, 263 (1999), 347-354.
    • (1999) Physics Letters A , vol.263 , pp. 347-354
    • Hone, A.N.W.1
  • 45
    • 0037275227 scopus 로고    scopus 로고
    • Prolongation algebras and Hamiltonian operators for peakon equations
    • Andrew N. W. Hone, Jing Ping Wang: Prolongation algebras and Hamiltonian operators for peakon equations. Inverse Problems, 19 (2003), 129-145.
    • (2003) Inverse Problems , vol.19 , pp. 129-145
    • Hone, A.N.W.1    Wang, J.P..2
  • 46
    • 33846082735 scopus 로고    scopus 로고
    • An integrable hierarchy with a perturbed H́enon-Heiles system
    • A. N. W. Hone, V. Novikov, C. Verhoeven: An integrable hierarchy with a perturbed H́enon-Heiles system. Inverse Problems, 22 (2006), 2001-2020.
    • (2006) Inverse Problems , vol.22 , pp. 2001-2020
    • Hone, A.N.W.1    Novikov, V.2    Verhoeven, C.3
  • 47
    • 0038581249 scopus 로고    scopus 로고
    • Nonlinear supersymmetric (Darboux) covariance of the Ermakov-Milne-Pinney equation
    • M. V. Ioffe, H. J. Korsch: Nonlinear supersymmetric (Darboux) covariance of the Ermakov-Milne-Pinney equation. Physics Letters A, 311 (2003), 200-205.
    • (2003) Physics Letters A , vol.311 , pp. 200-205
    • Ioffe, M.V.1    Korsch, H.J.2
  • 49
    • 0032363040 scopus 로고    scopus 로고
    • Construction of exact invariants for time dependent classical dynamical systems
    • R. S. Kaushal: Construction of exact invariants for time dependent classical dynamical systems. International Journal of Theoretical Physics, 37 (1998), 1793-1856.
    • (1998) International Journal of Theoretical Physics , vol.37 , pp. 1793-1856
    • Kaushal, R.S.1
  • 51
    • 33846912566 scopus 로고    scopus 로고
    • Nonlinearity from linearity: The Ermakov-Pinney equation revisited
    • Panayotis Kevrekidis, Yannis Drossinos: Nonlinearity from linearity: The Ermakov-Pinney equation revisited. Mathematics and Computers in Simulation, 74 (2007), 196-202.
    • (2007) Mathematics and Computers in Simulation , vol.74 , pp. 196-202
    • Kevrekidis, P.1    Drossinos, Y.2
  • 52
    • 21844489158 scopus 로고
    • On the complete symmetry group of the classical Kepler system
    • J. Krause: On the complete symmetry group of the classical Kepler system. Journal of Mathematical Physics, 35 (1994), 5734-5748.
    • (1994) Journal of Mathematical Physics , vol.35 , pp. 5734-5748
    • Krause, J.1
  • 54
    • 33644697094 scopus 로고
    • Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic
    • M. Kruskal: Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic. Journal of Mathematical Physics, 3 (1962), 806-828.
    • (1962) Journal of Mathematical Physics , vol.3 , pp. 806-828
    • Kruskal, M.1
  • 55
    • 0040733925 scopus 로고
    • Berry's phase and wave functions for time-dependent Hamiltonian systems
    • P. G. L. Leach: Berry's phase and wave functions for time-dependent Hamiltonian systems. Journal of Physics A: Mathematical and General, 23 (1990),2695-2699.
    • (1990) Journal of Physics A: Mathematical and General , vol.23 , pp. 2695-2699
    • Leach, P.G.L.1
  • 56
    • 33846593036 scopus 로고
    • Generalized Ermakov Systems
    • P. G. L. Leach: Generalized Ermakov Systems. Physics Letters, 158A (1991), 102-106.
    • (1991) Physics Letters , vol.158 A , pp. 102-106
    • Leach, P.G.L.1
  • 58
    • 0242641531 scopus 로고    scopus 로고
    • The Ermanno-Bernoulli constants and representations of the complete symmetry group of the Kepler Problem
    • P. G. L. Leach, K. Andriopoulos, M. C. Nucci: The Ermanno-Bernoulli constants and representations of the complete symmetry group of the Kepler Problem. Journal of Mathematical Physics, 44 (2003), 4090-4106.
    • (2003) Journal of Mathematical Physics , vol.44 , pp. 4090-4106
    • Leach, P.G.L.1    Andriopoulos, K.2    Nucci, M.C.3
  • 59
    • 0041351950 scopus 로고    scopus 로고
    • Equivalence classes of second-order ordinary differential equations with only a three-dimensional Lie algebra of point symmetries and linearisation
    • P. G. L. Leach: Equivalence classes of second-order ordinary differential equations with only a three-dimensional Lie algebra of point symmetries and linearisation. Journal of Mathematical Analysis and Applications, 284 (2003), 31-48.
    • (2003) Journal of Mathematical Analysis and Applications , vol.284 , pp. 31-48
    • Leach, P.G.L.1
  • 61
    • 85032070192 scopus 로고    scopus 로고
    • Erma-kov's superintegrable toy and nonlocal symmetries
    • Paper 018
    • P. G. L. Leach, A. Karasu (Kalkanli), M. C. Nucci, K. Andriopoulos: Erma-kov's superintegrable toy and nonlocal symmetries. SIGMA, 1 (2005), Paper 018, 15 pages.
    • (2005) SIGMA , vol.1 , pp. 15
    • Leach, P.G.L.1    Karasu Kalkanli, A.2    Nucci, M.C.3    Andriopoulos, K.4
  • 62
    • 37549001763 scopus 로고    scopus 로고
    • An invariant for the doubly generalised classical Ermakov-Pinney system and its quantal equivalent
    • 015002, DOI:10.1088/0031-8949/77/01/015002
    • P. G. L. Leach, K. Andriopoulos: An invariant for the doubly generalised classical Ermakov-Pinney system and its quantal equivalent. Physica Scripta, 77 015002 (2008), (7 pages) DOI:10.1088/0031-8949/77/01/015002.
    • (2008) Physica Scripta , vol.77 , pp. 7
    • Leach, P.G.L.1    Andriopoulos, K.2
  • 63
    • 23044454169 scopus 로고    scopus 로고
    • Twist character of the fourth order resonant periodic solution
    • DOI: 10.1007/s10884-005-2937-4
    • Jinzhi Lei, J. Pedro Torres, Meirong Zhang: Twist character of the fourth order resonant periodic solution. Journal of Dynamics and Differential Equations, 17 (2005), DOI: 10.1007/s10884-005-2937-4.
    • (2005) Journal of Dynamics and Differential Equations , vol.17
    • Lei, J.1    Pedro Torres, J.2    Zhang, M.3
  • 65
    • 25944457986 scopus 로고
    • Classical and quantum systems with time-dependent harmonic oscillator-type Hamiltonians
    • Ralph H. Lewis Jr: Classical and quantum systems with time-dependent harmonic oscillator-type Hamiltonians. Physical Review Letters, 18 (1967), 510-512.
    • (1967) Physical Review Letters , vol.18 , pp. 510-512
    • Lewis R.H., Jr.1
  • 66
    • 0001367064 scopus 로고
    • Motion of a time-dependent harmonic oscillator and of a charged particle in a time-dependent, axially symmetric, electromagnetic field
    • Ralph N. Lewis Jr: Motion of a time-dependent harmonic oscillator and of a charged particle in a time-dependent, axially symmetric, electromagnetic field. Physical Review, 172 (1968), 1313-1315.
    • (1968) Physical Review , vol.172 , pp. 1313-1315
    • Lewis, R.H.Jr.1
  • 67
    • 34547918663 scopus 로고
    • Class of exact invariants for classical and quantum time-dependent harmonic oscillators
    • Ralph H. Lewis Jr: Class of exact invariants for classical and quantum time-dependent harmonic oscillators. Journal of Mathematical Physics, 9 (1968), 1976-1986.
    • (1968) Journal of Mathematical Physics , vol.9 , pp. 1976-1986
    • Lewis R.H., Jr.1
  • 68
    • 36749021775 scopus 로고
    • An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field
    • Ralph H. Lewis Jr, W. B. Riesenfeld: An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field. Journal of Mathematical Physics, 10 (1969), 1458-1473.
    • (1969) Journal of Mathematical Physics , vol.10 , pp. 1458-1473
    • Lewis, R.H.Jr.1    Riesenfeld, W.B.2
  • 69
    • 36749116721 scopus 로고
    • Exact invariants for a class of time-dependent nonlinear Hamiltonian systems
    • Ralph H. Lewis, P. G. L. Leach: Exact invariants for a class of time-dependent nonlinear Hamiltonian systems. Journal of Mathematical Physics, 23 (1982), 165-175.
    • (1982) Journal of Mathematical Physics , vol.23 , pp. 165-175
    • Lewis, R.H.1    Leach, P.G.L.2
  • 70
    • 36749115178 scopus 로고
    • A direct approach to finding exact invariants for one-dimensional time-dependent classical Hamiltonians
    • Ralph H. Lewis, P. G. L. Leach: A direct approach to finding exact invariants for one-dimensional time-dependent classical Hamiltonians' Journal of Mathematical Physics, 23 (1982), 2371-2374.
    • (1982) Journal of Mathematical Physics , vol.23 , pp. 2371-2374
    • Lewis, R.H.1    Leach, P.G.L.2
  • 71
    • 77956966755 scopus 로고
    • Exact invariants for time-dependent nonlinear Hamiltonian systems
    • A. R. Bishop, D. K. Campbell, B. Nicolenko, edd (North Holland, New York)
    • Ralph H. Lewis, P. G. L. Leach: Exact invariants for time-dependent nonlinear Hamiltonian systems. Nonlinear Problems Present and Future, A. R. Bishop, D. K. Campbell, B. Nicolenko, edd (North Holland, New York) (1982), 133-145.
    • (1982) Nonlinear Problems Present and Future , pp. 133-145
    • Lewis, R.H.1    Leach, P.G.L.2
  • 72
    • 0041021384 scopus 로고
    • A resonance formulation for invariants of particle motion in a one-dimensional time-dependent potential
    • H. R. Lewis, P. G. L. Leach: A resonance formulation for invariants of particle motion in a one-dimensional time-dependent potential. Annals of Physics, 164 (1985), 47-76.
    • (1985) Annals of Physics , vol.164 , pp. 47-76
    • Lewis, H.R.1    Leach, P.G.L.2
  • 73
    • 0000091417 scopus 로고
    • Representations of one-dimensional Hamiltonians in terms of their invariants
    • Ralph H. Lewis, P. G. L. Leach, Serge Bouquet, Mark R. Feix: Representations of one-dimensional Hamiltonians in terms of their invariants. Journal of Mathematical Physics, 33 (1992), 591-598.
    • (1992) Journal of Mathematical Physics , vol.33 , pp. 591-598
    • Lewis, R.H.1    Leach, P.G.L.2    Bouquet, S.3    Feix, M.R.4
  • 74
    • 78650933838 scopus 로고
    • Representation of time-dependent one-dimensional Hamiltonians in terms of their first integrals
    • Dubna (World Scientific Publications, Singapore)
    • Ralph H. Lewis, P. G. L. Leach, S. Bouquet, M. R. Feix: Representation of time-dependent one-dimensional Hamiltonians in terms of their first integrals. Proceedings of the NEEDS' 92 Workshop, Dubna (World Scientific Publications, Singapore) (1992).
    • (1992) Proceedings of the NEEDS' 92 Workshop
    • Lewis, R.H.1    Leach, P.G.L.2    Bouquet, S.3    Feix, M.R.4
  • 75
    • 1342310904 scopus 로고    scopus 로고
    • Cosmic dynamics of Bose-Einstein condensates
    • James E. Lidsey: Cosmic dynamics of Bose-Einstein condensates. Classical and Quantum Gravity, 21 (2004), 777-785.
    • (2004) Classical and Quantum Gravity , vol.21 , pp. 777-785
    • Lidsey, J.E.1
  • 76
    • 35349022640 scopus 로고    scopus 로고
    • The method of reduction of order and linearisation of the two-dimensional Ermakov system
    • A. Maharaj, P. G. L. Leach: The method of reduction of order and linearisation of the two-dimensional Ermakov system. Mathematical Methods in the Applied Sciences, 30 (2007), 2125-2145.
    • (2007) Mathematical Methods in the Applied Sciences , vol.30 , pp. 2125-2145
    • Maharaj, A.1    Leach, P.G.L.2
  • 78
    • 0034656529 scopus 로고    scopus 로고
    • Lie symmetry analysis and approximate solutions for nonlinear radial oscillations of an incompressible Mooney-Rivlin cylindrical tube
    • D. P. Mason, N. Roussos: Lie symmetry analysis and approximate solutions for nonlinear radial oscillations of an incompressible Mooney-Rivlin cylindrical tube. Journal of Mathematical Analysis and Applications, 245 (2000), 346-392.
    • (2000) Journal of Mathematical Analysis and Applications , vol.245 , pp. 346-392
    • Mason, D.P.1    Roussos, N.2
  • 79
    • 34248184475 scopus 로고    scopus 로고
    • Nonlinear radial oscillations of a transversely isotropic hyperelastic incompressible tube
    • D. P. Mason, G. H. Maluleke: Nonlinear radial oscillations of a transversely isotropic hyperelastic incompressible tube. Journal of Mathematical Analysis and Applications, 333 (2007), 365-380.
    • (2007) Journal of Mathematical Analysis and Applications , vol.333 , pp. 365-380
    • Mason, D.P.1    Maluleke, G.H.2
  • 83
    • 21244489297 scopus 로고    scopus 로고
    • Jacobi's last multiplier and the complete symmetry group of the Ermakov-Pinney equation
    • M. C. Nucci, P. G. L. Leach: Jacobi's last multiplier and the complete symmetry group of the Ermakov-Pinney equation. Journal of Nonlinear Mathematical Physics, 12 (2005), 305-320.
    • (2005) Journal of Nonlinear Mathematical Physics , vol.12 , pp. 305-320
    • Nucci, M.C.1    Leach, P.G.L.2
  • 86
    • 0000120150 scopus 로고
    • More exact invariants for the time-dependent harmonic oscillator
    • J. R. Ray, J. L. Reid: More exact invariants for the time-dependent harmonic oscillator. Physics Letters A, 71 (1979), 317-318.
    • (1979) Physics Letters A , vol.71 , pp. 317-318
    • Ray, J.R.1    Reid, J.L.2
  • 87
    • 0011633552 scopus 로고
    • Nonlinear superposition law for generalized Ermakov systems
    • J. R. Ray: Nonlinear superposition law for generalized Ermakov systems. Physics Letters A, 78 (1980), 4-6.
    • (1980) Physics Letters A , vol.78 , pp. 4-6
    • Ray, J.R.1
  • 88
    • 33646188435 scopus 로고
    • Noether's theorem and Ermakov systems for nonlinear equations of motion
    • J. R. Ray, J. L. Reid: Noether's theorem and Ermakov systems for nonlinear equations of motion. Nuovo Cimento, 59A (1980), 134-140.
    • (1980) Nuovo Cimento , vol.59 A , pp. 134-140
    • Ray, J.R.1    Reid, J.L.2
  • 89
    • 36749120227 scopus 로고
    • Ermakov systems, nonlinear superposition principles and solutions of nonlinear equations of motion
    • J. R. Ray, J. L. Reid: Ermakov systems, nonlinear superposition principles and solutions of nonlinear equations of motion. Journal of Mathematical Physics, 21 (1980), 1583-1587.
    • (1980) Journal of Mathematical Physics , vol.21 , pp. 1583-1587
    • Ray, J.R.1    Reid, J.L.2
  • 90
    • 0011568226 scopus 로고
    • Ermakov systems, velocity dependent potentials and nonlinear superposition
    • J. R. Ray, J. L. Reid: Ermakov systems, velocity dependent potentials and nonlinear superposition. Journal of Mathematical Physics, 22 (1981), 91-95.
    • (1981) Journal of Mathematical Physics , vol.22 , pp. 91-95
    • Ray, J.R.1    Reid, J.L.2
  • 91
    • 0011696069 scopus 로고
    • New nonlinear dynamical systems possessing invariants
    • J. R. Ray, J. L. Reid, M. Lutzky: New nonlinear dynamical systems possessing invariants. Physics Letters A, 84 (1981), 42-44.
    • (1981) Physics Letters A , vol.84 , pp. 42-44
    • Ray, J.R.1    Reid, J.L.2    Lutzky, M.3
  • 92
    • 0013195675 scopus 로고
    • Invariants for forced time-dependent oscillators and generalisations
    • J. R. Ray, J. L. Reid: Invariants for forced time-dependent oscillators and generalisations. Physical Review A, 26 (1982), 1042-1047.
    • (1982) Physical Review A , vol.26 , pp. 1042-1047
    • Ray, J.R.1    Reid, J.L.2
  • 94
    • 0042357459 scopus 로고    scopus 로고
    • Nonlinear radial oscillations of thin-walled double-layer hyperelastic cylindrical tube
    • N. Roussos, D. P. Mason: Nonlinear radial oscillations of thin-walled double-layer hyperelastic cylindrical tube. International Journal of Nonlinear Mechanics, 33 (1998), 507-530.
    • (1998) International Journal of Nonlinear Mechanics , vol.33 , pp. 507-530
    • Roussos, N.1    Mason, D.P.2
  • 95
    • 11144351632 scopus 로고    scopus 로고
    • Radial oscillations of thin cylindrical and spherical shells: Investigation of Lie point symmetries for arbitrary strain-energy functions
    • N. Roussos, D. P. Mason: Radial oscillations of thin cylindrical and spherical shells: investigation of Lie point symmetries for arbitrary strain-energy functions. Communications in Nonlinear Science and Numerical Simulation, 10 (2005), 139-150.
    • (2005) Communications in Nonlinear Science and Numerical Simulation , vol.10 , pp. 139-150
    • Roussos, N.1    Mason, D.P.2
  • 96
    • 36749119410 scopus 로고
    • Classification scheme for two-dimensional Ermakov systems and generalizations
    • W. Sarlet, J. R. Ray: Classification scheme for two-dimensional Ermakov systems and generalizations. Journal of Mathematical Physics, 22 (1981), 2504-2511.
    • (1981) Journal of Mathematical Physics , vol.22 , pp. 2504-2511
    • Sarlet, W.1    Ray, J.R.2
  • 97
    • 0033229605 scopus 로고    scopus 로고
    • n with applications to nonstatic spherically symmetric perfect fluid solutions
    • n with applications to nonstatic spherically symmetric perfect fluid solutions. Classical and Quantum Gravity, 16 (1999), 3553-3566.
    • (1999) Classical and Quantum Gravity , vol.16 , pp. 3553-3566
    • Wafo, S.C.1    Mahomed, F.M.2
  • 99
    • 0037490630 scopus 로고    scopus 로고
    • On (2 + 1)-dimensional Friedman-Robertson-Walker universes: An Ermakov-Pinney equation approach
    • F. L. Williams, P. G. Kevrekidis: On (2 + 1)-dimensional Friedman-Robertson-Walker universes: an Ermakov-Pinney equation approach. Classical and Quantum Gravity, 20 (2003), L177-L184.
    • (2003) Classical and Quantum Gravity , vol.20
    • Williams, F.L.1    Kevrekidis, P.G.2
  • 101
    • 78650964217 scopus 로고
    • Additions to Kamke's treatise: Nonlinear third-order differential equations
    • Jovan D. Kečkić: Additions to Kamke's treatise: nonlinear third-order differential equations. Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz., No. 396 (1972), 81-84.
    • (1972) Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz. , vol.396 , pp. 81-84
    • Kečkić, J.D.1
  • 102
    • 78650926597 scopus 로고
    • Additions to Kamke's treatise, V: A remark on the generalised Emden's equation
    • Jovan D. Kečkić: Additions to Kamke's treatise, V: a remark on the generalised Emden's equation. Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz., No. 503 (1975), 39-40.
    • (1975) Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz. , vol.503 , pp. 39-40
    • Kečkić, J.D.1
  • 103
    • 84897905397 scopus 로고
    • Additions to Kamke's treatise, VII: Variation of parameters for nonlinear second-order differential equations
    • Jovan D. Kečkić: Additions to Kamke's treatise, VII: variation of parameters for nonlinear second-order differential equations. Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz., No. 549 (1976), 31-36.
    • (1976) Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz. , vol.549 , pp. 31-36
    • Kečkić, J.D.1
  • 104
  • 105
    • 78650962861 scopus 로고
    • Conditions for the integrability of second-order nonlinear differential equation
    • Vlajko Lj. Kocic: Conditions for the integrability of second-order nonlinear differential equation, II. Publications de l'Institut Math́ematique (nouvelle śerie), 34 (1983), 95-101.
    • (1983) II. Publications de l'Institut Math́ematique (nouvelle śerie) , vol.34 , pp. 95-101
    • Kocic, V.L.1
  • 106
    • 78650956012 scopus 로고
    • Compléments au traité de Kamke's, XIV: Applications of the variation of parameters methods to nonlinear second-order differential equations
    • Dragoslav S. Mitrinović, Jovan D. Kečkić: Compléments au traité de Kamke's, XIV: applications of the variation of parameters methods to nonlinear second-order differential equations. Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz., No. 734 (1976), 3-7.
    • (1976) Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz. , vol.734 , pp. 3-7
    • Mitrinović, D.S.1    Kečkić, J.D.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.