Figure 1.
The potential $ W (\theta) $ as in eq.(11) for different values of $ {\beta} $ ; here $ {\beta} = -1, -0.5,0.5,1 $ . The exchange of stability takes place at $ {\beta} = 0 $
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September
2020, 12(3): 421-434.
doi: 10.3934/jgm.2020026
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.
Mathematics Subject Classification:
Primary: 37J06, 34A05; Secondary: 15A69.
Citation:
Giuseppe Gaeta, Sebastian Walcher. Higher order normal modes. Journal of Geometric Mechanics,
2020, 12
(3)
: 421-434.
doi: 10.3934/jgm.2020026
![]()
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. |
| [4] |
J. P. Bornsen and A. E. M. van de Ven, Tangent developable orbit space of an octupole, preprint, arXiv: 1807.04817, 2018. |
| [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. |
| [12] |
G. Gaeta and E. G. Virga, Octupolar order in three dimensions, Eur. Phys. J. E, 39 (2016), 113pp. |
| [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. |
| [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. |
| [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
Dedicated to James Montaldi on his 25+something anniversary
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. |
| [4] |
J. P. Bornsen and A. E. M. van de Ven, Tangent developable orbit space of an octupole, preprint, arXiv: 1807.04817, 2018. |
| [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. |
| [12] |
G. Gaeta and E. G. Virga, Octupolar order in three dimensions, Eur. Phys. J. E, 39 (2016), 113pp. |
| [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. |
| [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. |
| [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. |
Figure 2.
Numerical integration of the motion generated by the potential (10) with the choice $ {\beta} = 1 $ for initial conditions near to normal modes. In all cases, initial data correspond to zero speed and position at $ r = 1 $ along eigenvectors, with an offset of 0.001 from the latter. The simulation show the outcome, for $ t \in (0,100) $ , for initial data: (a) near the eigenvector $ \theta = 0 $ , (b) near the eigenvector $ \theta = \pi $ , (c) near the eigenvector $ \theta = \pi/4 $ , (d) near the eigenvector $ \theta = - \pi/4 $
Figure 3.
The potential $ W(\theta) $ , see (13), for $ {\alpha} = 1/4 $ and various choices of $ {\beta} $ . Here $ \theta $ is measured in units of $ \pi $
Table 1.
Different possibilities for the number and type of critical points in the case of a cubic potential in three dimensions; here "Max"and "Min" represent the number of maxima and minima, while "$ S_k $ " represents the number of saddle points of index $ - k $ . Finally, "NCP" is the total number of critical points
| 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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