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Higher order normal modes
1. | Dipartimento di Matematica, Università degli Studi di Milano, v. Saldini 50, I-20133 Milano, Italy |
2. | SMRI, I-00058 Santa Marinella, Italy |
3. | Mathematik A, RWTH Aachen, D-52056 Aachen, Germany |
Normal modes are intimately related to the quadratic approximation of a potential at its hyperbolic equilibria. Here we extend the notion to the case where the Taylor expansion for the potential at a critical point starts with higher order terms, and show that such an extension shares some of the properties of standard normal modes. Some symmetric examples are considered in detail.
References:
[1] |
V. I. Arnold, Mathematical Methods of Classical Mechanics, Third edition. "Nauka", Moscow, 1989. |
[2] |
V. I. Arnold, Ordinary Differential Equations, Springer, 1992. |
[3] |
V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, 1983. Google Scholar |
[4] |
J. P. Bornsen and A. E. M. van de Ven, Tangent developable orbit space of an octupole, preprint, arXiv: 1807.04817, 2018. Google Scholar |
[5] |
D. Cartwright and B. Sturmfels,
The number of eigenvalues of a tensor, Linear Algebra Appl., 438 (2013), 942-952.
doi: 10.1016/j.laa.2011.05.040. |
[6] |
J. F. Cariñena and A. Ramos,
A new geometric approach to Lie systems and physical applications, Acta Appl. Math., 70 (2002), 43-69.
doi: 10.1023/A:1013913930134. |
[7] |
J. F. Cariñena, J. Grabowski and G. Marmo, Lie-Scheffers Systems. A Geometric Approach, Bibliopolis, 2000. |
[8] |
Y. Chen, L. Qi and E. G. Virga, Octupolar tensors for liquid crystals, J. Phys. A, 51 (2018), 025206, 20pp.
doi: 10.1088/1751-8121/aa98a8. |
[9] |
C. Elphick, E. Tirapegui, M. E. Brachet, P. Coullet and G. Iooss,
A simple global characterization for normal forms of singular vector fields, Physica D, 29 (1987), 95-127.
doi: 10.1016/0167-2789(87)90049-2. |
[10] |
G. Gaeta and E. Virga,
The symmetries of octupolar tensors, J. Elast., 135 (2019), 295-350.
doi: 10.1007/s10659-018-09722-8. |
[11] |
F. Gantmacher, Lectures in Analytical Mechanics, MIR, 1970. Google Scholar |
[12] |
G. Gaeta and E. G. Virga, Octupolar order in three dimensions, Eur. Phys. J. E, 39 (2016), 113pp. Google Scholar |
[13] |
H. Goldstein, Classical Mechanics, Addison-Wesley, 1980. |
[14] |
N. Kruff, J. Llibre, C. Pantazi and S. Walcher, Invariant algebraic surfaces of polynomial vector fields in dimension three, preprint, arXiv: 1907.12536, 2019. Google Scholar |
[15] |
L. D. Landau and E. M. Lifhsitz, Mechanics, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960. |
[16] |
J. W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville, Va. 1965. |
[17] |
J. Moser,
Regularization of Kepler's problem and the averaging method on a manifold, Comm.Pure Appl. Math., 23 (1970), 609-636.
doi: 10.1002/cpa.3160230406. |
[18] |
J. Montaldi, M. Roberts and I. Stewart,
Existence of nonlinear normal modes in symmetric Hamiltonian systems, Nonlinearity, 3 (1990), 695-730.
doi: 10.1088/0951-7715/3/3/009. |
[19] |
J. Montaldi, M. Roberts and I. Stewart,
Stability of nonlinear normal modes in symmetric Hamiltonian systems, Nonlinearity, 3 (1990), 731-772.
doi: 10.1088/0951-7715/3/3/010. |
[20] |
L. Qi,
Eigenvalues and invariants of tensors, J. Math. Anal. Appl., 325 (2007), 1363-1377.
doi: 10.1016/j.jmaa.2006.02.071. |
[21] |
L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, (Advances in Mechanics and Mathematics, vol. 39), Springer, 2018.
doi: 10.1007/978-981-10-8058-6. |
[22] |
H. Rohrl,
A theorem on non-associative algebras and its application to differential equations, Manus. Math., 21 (1977), 181-187.
doi: 10.1007/BF01168018. |
[23] |
H. Rohrl,
Algebras and differential equations, Nagoya Math. J., 68 (1977), 59-122.
doi: 10.1017/S0027763000017876. |
[24] |
H. Rohrl,
On the zeros of polynomials over arbitrary finite-dimensional algebras, Manuscripta Math., 25 (1978), 359-390.
doi: 10.1007/BF01168049. |
[25] |
H. Rohrl,
Finite-dimensional algebras without nilpotents over algebraically closed fields, Arch. Math. (Basel), 32 (1979), 10-12.
doi: 10.1007/BF01238461. |
[26] |
H. Rohrl and S. Walcher,
Projections of polynomial vector fields and the Poincaré sphere, J. Diff. Eqs., 139 (1997), 22-40.
doi: 10.1006/jdeq.1997.3298. |
[27] |
I. R. Shafarevich, Basic Algebraic Geometry, Springer, 1977. |
[28] |
E. Virga, Octupolar order in two dimensions, Eur.Phys. J. E, 38 (2015), 63pp. Google Scholar |
[29] |
A. Weinstein,
Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57.
doi: 10.1007/BF01405263. |
[30] |
S. Walcher, Algebras and Differential Equations, Hadronic Press, 1991.
![]() |
[31] |
S. Walcher,
Eigenvectors of tensors – a primer, Acta Appl. Math., 162 (2019), 165-183.
doi: 10.1007/s10440-018-0225-7. |
show all references
References:
[1] |
V. I. Arnold, Mathematical Methods of Classical Mechanics, Third edition. "Nauka", Moscow, 1989. |
[2] |
V. I. Arnold, Ordinary Differential Equations, Springer, 1992. |
[3] |
V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, 1983. Google Scholar |
[4] |
J. P. Bornsen and A. E. M. van de Ven, Tangent developable orbit space of an octupole, preprint, arXiv: 1807.04817, 2018. Google Scholar |
[5] |
D. Cartwright and B. Sturmfels,
The number of eigenvalues of a tensor, Linear Algebra Appl., 438 (2013), 942-952.
doi: 10.1016/j.laa.2011.05.040. |
[6] |
J. F. Cariñena and A. Ramos,
A new geometric approach to Lie systems and physical applications, Acta Appl. Math., 70 (2002), 43-69.
doi: 10.1023/A:1013913930134. |
[7] |
J. F. Cariñena, J. Grabowski and G. Marmo, Lie-Scheffers Systems. A Geometric Approach, Bibliopolis, 2000. |
[8] |
Y. Chen, L. Qi and E. G. Virga, Octupolar tensors for liquid crystals, J. Phys. A, 51 (2018), 025206, 20pp.
doi: 10.1088/1751-8121/aa98a8. |
[9] |
C. Elphick, E. Tirapegui, M. E. Brachet, P. Coullet and G. Iooss,
A simple global characterization for normal forms of singular vector fields, Physica D, 29 (1987), 95-127.
doi: 10.1016/0167-2789(87)90049-2. |
[10] |
G. Gaeta and E. Virga,
The symmetries of octupolar tensors, J. Elast., 135 (2019), 295-350.
doi: 10.1007/s10659-018-09722-8. |
[11] |
F. Gantmacher, Lectures in Analytical Mechanics, MIR, 1970. Google Scholar |
[12] |
G. Gaeta and E. G. Virga, Octupolar order in three dimensions, Eur. Phys. J. E, 39 (2016), 113pp. Google Scholar |
[13] |
H. Goldstein, Classical Mechanics, Addison-Wesley, 1980. |
[14] |
N. Kruff, J. Llibre, C. Pantazi and S. Walcher, Invariant algebraic surfaces of polynomial vector fields in dimension three, preprint, arXiv: 1907.12536, 2019. Google Scholar |
[15] |
L. D. Landau and E. M. Lifhsitz, Mechanics, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960. |
[16] |
J. W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville, Va. 1965. |
[17] |
J. Moser,
Regularization of Kepler's problem and the averaging method on a manifold, Comm.Pure Appl. Math., 23 (1970), 609-636.
doi: 10.1002/cpa.3160230406. |
[18] |
J. Montaldi, M. Roberts and I. Stewart,
Existence of nonlinear normal modes in symmetric Hamiltonian systems, Nonlinearity, 3 (1990), 695-730.
doi: 10.1088/0951-7715/3/3/009. |
[19] |
J. Montaldi, M. Roberts and I. Stewart,
Stability of nonlinear normal modes in symmetric Hamiltonian systems, Nonlinearity, 3 (1990), 731-772.
doi: 10.1088/0951-7715/3/3/010. |
[20] |
L. Qi,
Eigenvalues and invariants of tensors, J. Math. Anal. Appl., 325 (2007), 1363-1377.
doi: 10.1016/j.jmaa.2006.02.071. |
[21] |
L. Qi, H. Chen and Y. Chen, Tensor Eigenvalues and Their Applications, (Advances in Mechanics and Mathematics, vol. 39), Springer, 2018.
doi: 10.1007/978-981-10-8058-6. |
[22] |
H. Rohrl,
A theorem on non-associative algebras and its application to differential equations, Manus. Math., 21 (1977), 181-187.
doi: 10.1007/BF01168018. |
[23] |
H. Rohrl,
Algebras and differential equations, Nagoya Math. J., 68 (1977), 59-122.
doi: 10.1017/S0027763000017876. |
[24] |
H. Rohrl,
On the zeros of polynomials over arbitrary finite-dimensional algebras, Manuscripta Math., 25 (1978), 359-390.
doi: 10.1007/BF01168049. |
[25] |
H. Rohrl,
Finite-dimensional algebras without nilpotents over algebraically closed fields, Arch. Math. (Basel), 32 (1979), 10-12.
doi: 10.1007/BF01238461. |
[26] |
H. Rohrl and S. Walcher,
Projections of polynomial vector fields and the Poincaré sphere, J. Diff. Eqs., 139 (1997), 22-40.
doi: 10.1006/jdeq.1997.3298. |
[27] |
I. R. Shafarevich, Basic Algebraic Geometry, Springer, 1977. |
[28] |
E. Virga, Octupolar order in two dimensions, Eur.Phys. J. E, 38 (2015), 63pp. Google Scholar |
[29] |
A. Weinstein,
Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57.
doi: 10.1007/BF01405263. |
[30] |
S. Walcher, Algebras and Differential Equations, Hadronic Press, 1991.
![]() |
[31] |
S. Walcher,
Eigenvectors of tensors – a primer, Acta Appl. Math., 162 (2019), 165-183.
doi: 10.1007/s10440-018-0225-7. |



Max | Min | NCP | |||
1 | 1 | 0 | 0 | 0 | 2 |
2 | 2 | 2 | 0 | 0 | 6 |
3 | 3 | 4 | 0 | 0 | 10 |
3 | 3 | 0 | 2 | 0 | 8 |
4 | 4 | 6 | 0 | 0 | 14 |
4 | 4 | 2 | 2 | 0 | 12 |
4 | 4 | 0 | 0 | 2 | 10 |
Max | Min | NCP | |||
1 | 1 | 0 | 0 | 0 | 2 |
2 | 2 | 2 | 0 | 0 | 6 |
3 | 3 | 4 | 0 | 0 | 10 |
3 | 3 | 0 | 2 | 0 | 8 |
4 | 4 | 6 | 0 | 0 | 14 |
4 | 4 | 2 | 2 | 0 | 12 |
4 | 4 | 0 | 0 | 2 | 10 |
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