I give a short review of the theory of twisted symmetries of differential equations, emphasizing geometrical aspects. Some open problems are also mentioned.
Citation: |
[1] |
D. V. Alekseevsky, A. M. Vinogradov and V. V. Lychagin, Basic ideas and concepts of differential geometry, Geometry, I, 1–264, Encyclopaedia Math. Sci., 28, Springer, Berlin, 1991.
![]() ![]() |
[2] |
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1989.
doi: 10.1007/978-1-4757-2063-1.![]() ![]() ![]() |
[3] |
V. I. Arnold, Ordinary Differential Equations, Springer, 1992.
![]() |
[4] |
V. I. Arnold, Geometrical Methods in the Theory of Differential Equations, Springer, 1983.
![]() ![]() |
[5] |
A. Barco and G. E. Prince, Solvable symmetry structures in differential form, Acta Appl. Math., 66 (2001), 89-121.
doi: 10.1023/A:1010609817442.![]() ![]() ![]() |
[6] |
P. Basarab-Horwath, Integrability by quadratures for systems of involutive vector fields, Ukr. Math. J., 43 (1991), 1236-1242.
doi: 10.1007/BF01061807.![]() ![]() ![]() |
[7] |
J. F. Carinena, M. Del Olmo and P. Winternitz, On the relation between weak and strong invariance of differential equations, Lett. Math. Phys., 29 (1993), 151-163.
doi: 10.1007/BF00749730.![]() ![]() ![]() |
[8] |
D. Catalano Ferraioli, Nonlocal aspects of $\lambda$-symmetries and ODEs reduction, J. Phys. A: Math. Theor., 40 (2007), 5479-5489.
doi: 10.1088/1751-8113/40/21/002.![]() ![]() ![]() |
[9] |
D. Catalano Ferraioli and P. Morando, Local and nonlocal solvable structures in the reduction of ODEs, J. Phys. A: Math. Theor., 42 (2009), 035210 (15pp).
doi: 10.1088/1751-8113/42/3/035210.![]() ![]() ![]() |
[10] |
D. Catalano Ferraioli and P. Morando, Exploiting solvable structures in the integration of variational ordinary differential equations, preprint.
![]() |
[11] |
G. Cicogna, Reduction of systems of first-order differential equations via $\Lambda$-symmetries, Phys. Lett. A, 372 (2008), 3672-3677.
doi: 10.1016/j.physleta.2008.02.041.![]() ![]() ![]() |
[12] |
G. Cicogna, Symmetries of Hamiltonian equations and $\Lambda$-constants of motion, J. Nonlin. Math. Phys., 16 (2009), 43-60.
doi: 10.1142/S1402925109000315.![]() ![]() ![]() |
[13] |
G. Cicogna and G. Gaeta, Symmetry and Perturbation Theory in Nonlinear Dynamics, Springer, 1999.
![]() ![]() |
[14] |
G. Cicogna and G. Gaeta, Partial Lie-point symmetries of differential equations, J. Phys. A, 34 (2001), 491-512.
doi: 10.1088/0305-4470/34/3/312.![]() ![]() ![]() |
[15] |
G. Cicogna and G. Gaeta, Noether theorem for $\mu$-symmetries, J. Phys. A, 40 (2007), 11899-11921.
doi: 10.1088/1751-8113/40/39/013.![]() ![]() ![]() |
[16] |
G. Cicogna, G. Gaeta and P. Morando, On the relation between standard and $\mu$-symmetries for PDEs, J. Phys. A, 37 (2004), 9467-9486.
doi: 10.1088/0305-4470/37/40/010.![]() ![]() ![]() |
[17] |
G. Cicogna, G. Gaeta and S. Walcher, A generalization of $\lambda$-symmetry reduction for systems of ODEs: $\sigma$-symmetries, J. Phys. A, 45 (2012), 355205 (29pp).
doi: 10.1088/1751-8113/45/35/355205.![]() ![]() ![]() |
[18] |
G. Cicogna, G. Gaeta and S. Walcher, Orbital reducibility and a generalization of $\lambda$-symmetries, J. Lie Theory, 23 (2013), 357-381.
![]() ![]() |
[19] |
G. Cicogna, G. Gaeta and S. Walcher, Dynamical systems and $\sigma$-symmetries, J. Phys. A, 46 (2013), 235204 (23pp).
doi: 10.1088/1751-8113/46/23/235204.![]() ![]() ![]() |
[20] |
G. Cicogna, G. Gaeta and S. Walcher, Side conditions for ordinary differential equations, J. Lie Theory, 25 (2015), 125-146.
![]() ![]() |
[21] |
G. Gaeta, Twisted symmetries of differential equations, J. Nonlin. Math. Phys., 16 (2009), S107–S136.
doi: 10.1142/S1402925109000352.![]() ![]() ![]() |
[22] |
G. Gaeta, Simple and collective twisted symmetries, J. Nonlin. Math. Phys., 21 (2014), 593-627.
doi: 10.1080/14029251.2014.975530.![]() ![]() ![]() |
[23] |
G. Gaeta, Smooth changes of frame and prolongations of vector fields, Int. J. Geom. Meths. Mod. Phys., 4 (2007), 807-827.
doi: 10.1142/S0219887807002302.![]() ![]() ![]() |
[24] |
G. Gaeta, A gauge-theoretic description of $\mu$-prolongations, and $\mu$-symmetries of differential equations, J. Geom. Phys., 59 (2009), 519-539.
doi: 10.1016/j.geomphys.2009.01.004.![]() ![]() ![]() |
[25] |
G. Gaeta, Gauge fixing and twisted prolongations, J. Phys. A, 44 (2011), 325203 (9 pp).
doi: 10.1088/1751-8113/44/32/325203.![]() ![]() ![]() |
[26] |
G. Gaeta, Symmetry and Lie-Frobenius reduction of differential equations, J.Phys. A, 48 (2015), 015202, 22pp.
doi: 10.1088/1751-8113/48/1/015202.![]() ![]() ![]() |
[27] |
G. Gaeta, Symmetry of stochastic non-variational differential equations, Phys. Rep., 686 (2017), 1–62 [Erratum, Phys. Rep., 713 (2017), 18].
doi: 10.1016/j.physrep.2017.05.005.![]() ![]() ![]() |
[28] |
G. Gaeta and G. Cicogna, Twisted symmetries and integrable systems, Int. J. Geom. Meths. Mod. Phys., 6 (2009), 1305-1321.
doi: 10.1142/S0219887809004235.![]() ![]() ![]() |
[29] |
G. Gaeta, F. D. Grosshans, J. Scheurle and S. Walcher, Reduction and reconstruction for symmetric ordinary differential equations, J. Diff. Eqs., 244 (2008), 1810-1839.
doi: 10.1016/j.jde.2008.01.009.![]() ![]() ![]() |
[30] |
G. Gaeta and P. Morando, On the geometry of lambda-symmetries and PDE reduction, J. Phys. A, 37 (2004), 6955-6975.
doi: 10.1088/0305-4470/37/27/007.![]() ![]() ![]() |
[31] |
G. Gaeta and S. Walcher, Dimension increase and splitting for Poincaré-Dulac normal forms, J. Nonlin. Math. Phys., 12 (2005), 327-342.
doi: 10.2991/jnmp.2005.12.s1.26.![]() ![]() ![]() |
[32] |
G. Gaeta and S. Walcher, Embedding and splitting ordinary differential equations in normal form, J. Diff. Eqs., 224 (2006), 98-119.
doi: 10.1016/j.jde.2005.06.025.![]() ![]() ![]() |
[33] |
C. Godbillon, Géométrie Différentielle et Mécanique Analitique, Hermann, 1969.
![]() ![]() |
[34] |
K. P. Hadeler and S. Walcher, Reducible ordinary differential equations, J. Nonlinear Sci., 16 (2006), 583-613.
doi: 10.1007/s00332-004-0627-8.![]() ![]() ![]() |
[35] |
T. Hartl and C. Athorne, Solvable structures and hidden symmetries, J. Phys. A: Math. Gen., 27 (1994), 3463-3474.
doi: 10.1088/0305-4470/27/10/022.![]() ![]() ![]() |
[36] |
Y. Kosmann-Schwarzbach, The Noether Theorems. Invariance and Conservation laws in the Twentieth Century, Springer, 2011.
doi: 10.1007/978-0-387-87868-3.![]() ![]() ![]() |
[37] |
I. S. Krasil'schik and A. M. Vinogradov, Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, A.M.S., 1999.
![]() ![]() |
[38] |
D. Levi, M. C. Nucci and M. A. Rodriguez, $\lambda$-symmetries for the reduction of continuous and discrete equations, Acta Appl. Math, 122 (2012), 311-321.
doi: 10.1007/s10440-012-9745-8.![]() ![]() ![]() |
[39] |
D. Levi and M. A. Rodriguez, $\lambda$-symmetries for discrete equations, J. Phys. A, 43 (2010), 292001, 9pp.
doi: 10.1088/1751-8113/43/29/292001.![]() ![]() ![]() |
[40] |
D. Levi and P. Winternitz, Non-classical symmetry reduction: Example of the Boussinesq equation, J. Phys. A, 22 (1989), 2915-2924.
doi: 10.1088/0305-4470/22/15/010.![]() ![]() ![]() |
[41] |
T. Levi Civita, Sulla determinazione di soluzioni particolari di un sistema canonico quando se ne conosce qualche integrale o relazione invariante, R. Accad. Lincei Ser. V, 10 (1901).
![]() |
[42] |
P. Morando, Deformation of Lie derivative and $\mu$-symmetries, J. Phys. A, 40 (2007), 11547-11559.
doi: 10.1088/1751-8113/40/38/007.![]() ![]() ![]() |
[43] |
P. Morando, Reduction by $\lambda$-symmetries and $\sigma$-symmetries: A Frobenius approach, J. Nonlin. Math. Phys., 22 (2015), 47-59.
doi: 10.1080/14029251.2015.996439.![]() ![]() ![]() |
[44] |
C. Muriel and J. L. Romero, New methods of reduction for ordinary differential equations, IMA J. Appl. Math., 66 (2001), 111-125.
doi: 10.1093/imamat/66.2.111.![]() ![]() ![]() |
[45] |
C. Muriel and J. L. Romero, $C^\infty$ symmetries and nonsolvable symmetry algebras, IMA J. Appl. Math., 66 (2001), 477-498.
doi: 10.1093/imamat/66.5.477.![]() ![]() ![]() |
[46] |
C. Muriel and J. L. Romero, Prolongations of vector fields and the invariants-by-derivation property, Theor. Math. Phys., 113 (2002), 1565-1575.
![]() |
[47] |
C. Muriel and J. L. Romero, $C^\infty$-symmetries and integrability of ordinary differential equations, Proceedings of the I Colloquium on Lie Theory and Applications (Vigo), 35 (2002), 143-150.
![]() ![]() |
[48] |
C. Muriel and J. L. Romero, $C^\infty$ symmetries and reduction of equations without Lie point symmetries, J. Lie Theory, 13 (2003), 167-188.
![]() ![]() |
[49] |
C. Muriel and J. L. Romero, $C^\infty$-symmetries and nonlocal symmetries of exponential type, IMA J. Appl. Math., 72 (2007), 191-205.
doi: 10.1093/imamat/hxm001.![]() ![]() ![]() |
[50] |
C. Muriel and J. L. Romero, Integrating factors and lambda-symmetries, J. Nonlin. Math. Phys., 15 (2008), 300-309.
doi: 10.2991/jnmp.2008.15.s3.29.![]() ![]() ![]() |
[51] |
C. Muriel and J. L. Romero, First integrals, integrating factors and $\lambda$-symmetries of second-order differential equations, J. Phys. A, 42 (2009), 365207, 17pp.
doi: 10.1088/1751-8113/42/36/365207.![]() ![]() ![]() |
[52] |
C. Muriel and J. L. Romero, A $\lambda$-symmetry-based method for the linearization and determination of first integrals of a family of second-order ordinary differential equations, J. Phys. A, 44 (2011), 245201, 19pp.
doi: 10.1088/1751-8113/44/24/245201.![]() ![]() ![]() |
[53] |
C. Muriel and J. L. Romero, Second-order differential equations with first integrals of the form $C (t)+ 1/(A (t, x) x + B (t, x))$, J. Nonlin. Math. Phys., 18 (2011), 237-250.
doi: 10.1142/S1402925111001398.![]() ![]() ![]() |
[54] |
C. Muriel and J. L. Romero, Nonlocal symmetries, telescopic vector fields and $\lambda$-symmetries of ordinary differential equations, Symmetry, Integrability and Geometry: Methods and Applications, 8 (2012), Paper 106, 21 pp.
doi: 10.3842/SIGMA.2012.106.![]() ![]() ![]() |
[55] |
C. Muriel and J. L. Romero, The $\lambda$-symmetry reduction method and Jacobi last multipliers, Comm. Nonlin. Science Num. Sim., 19 (2014), 807-820.
doi: 10.1016/j.cnsns.2013.07.027.![]() ![]() ![]() |
[56] |
C. Muriel, J. L. Romero and P. J. Olver, Variational $C^\infty$ symmetries and Euler-Lagrange equations, J. Diff. Eqs., 222 (2006), 164-184.
doi: 10.1016/j.jde.2005.01.012.![]() ![]() ![]() |
[57] |
P. J. Olver, Application of Lie Groups to Differential Equations, Springer, 1986.
doi: 10.1007/978-1-4684-0274-2.![]() ![]() ![]() |
[58] |
P. J. Olver, Equivalence, Invariants and Symmetry, Cambridge University Press, 1995.
doi: 10.1017/CBO9780511609565.![]() ![]() ![]() |
[59] |
P. J. Olver and Ph. Rosenau, The construction of special solutions to partial differential equations, Phys. Lett. A, 114 (1986), 107-112.
doi: 10.1016/0375-9601(86)90534-7.![]() ![]() ![]() |
[60] |
P. J. Olver and Ph. Rosenau, Group-invariant solutions of differential equations, SIAM J. Appl. Math., 47 (1987), 263-278.
doi: 10.1137/0147018.![]() ![]() ![]() |
[61] |
G. Pucacco and K. Rosquist, Configurational invariants of Hamiltonian systems, J. Math. Phys., 46 (2005), 052902, 21pp.
doi: 10.1063/1.1888565.![]() ![]() ![]() |
[62] |
E. Pucci and G. Saccomandi, On the weak symmetry groups of partial differential equations, J. Math. Anal. Appl., 163 (1992), 588-598.
doi: 10.1016/0022-247X(92)90269-J.![]() ![]() ![]() |
[63] |
E. Pucci and G. Saccomandi, On the reduction methods for ordinary differential equations, J. Phys. A, 35 (2002), 6145-6155.
doi: 10.1088/0305-4470/35/29/314.![]() ![]() ![]() |
[64] |
K. Rosquist and G. Pucacco, Invariants at fixed and arbitrary energy. A unified geometric approach, J. Phys. A, 28 (1995), 3235-3252.
doi: 10.1088/0305-4470/28/11/021.![]() ![]() ![]() |
[65] |
A. Ruiz, C. Muriel and P. J. Olver, On the commutator of $C^\infty$ symmetries and the reduction of Euler-Lagrange equations, J. Phys. A, 51 (2018), 145202, 21pp.
doi: 10.1088/1751-8121/aab036.![]() ![]() ![]() |
[66] |
W. Sarlet and F. Cantrijn, Generalizations of Noethers theorem in classical mechanics, SIAM Rev., 23 (1981), 467-494.
doi: 10.1137/1023098.![]() ![]() ![]() |
[67] |
W. Sarlet, P. G. L. Leach and F. Cantrijn, First integrals versus configurational invariants and a weak form of complete integrability, Physica D, 17 (1985), 87-98.
doi: 10.1016/0167-2789(85)90136-8.![]() ![]() ![]() |
[68] |
R. W. Sharpe, Differential Geometry, Springer, 1997.
![]() ![]() |
[69] |
J. Sherring and G. Prince, Geometric aspects of reduction of order, Trans. Am. Math. Soc., 334 (1992), 433-453.
doi: 10.1090/S0002-9947-1992-1149125-6.![]() ![]() ![]() |
[70] |
H. Stephani, Differential Equations. Their Solution Using Symmetries, Cambridge University Press, 1989.
![]() ![]() |
[71] |
S. Sternberg, Lectures on Differential Geometry, Chelsea, 1983.
![]() ![]() |
[72] |
S. Walcher, Multi-parameter symmetries of first order ordinary differential equations, J. Lie Theory, 9 (1999), 249-269.
![]() ![]() |
[73] |
S. Walcher, Orbital symmetries of first order ODEs, in Symmetry and Perturbation Theory (SPT98), Editors: Degasperis A and Gaeta G, World Scientific, Singapore, (1999), 96–113.
![]() ![]() |