Higher order normal modes

Giuseppe Gaeta; Sebastian Walcher

doi:10.3934/jgm.2020026

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September 2020, 12(3): 421-434. doi: 10.3934/jgm.2020026

Higher order normal modes

Giuseppe Gaeta ^1,2,, and Sebastian Walcher ^3,

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

^* Corresponding author: Giuseppe Gaeta

Dedicated to James Montaldi on his 25+something anniversary

Received September 2019 Revised July 2020 Published September 2020 Early access September 2020

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.

Keywords: Normal mode, Hamiltonian system, Eigenvectors of tensors.

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. Google Scholar
[2]	V. I. Arnold, Ordinary Differential Equations, Springer, 1992. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	J. F. Cariñena, J. Grabowski and G. Marmo, Lie-Scheffers Systems. A Geometric Approach, Bibliopolis, 2000. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	G. Gaeta and E. Virga, The symmetries of octupolar tensors, J. Elast., 135 (2019), 295-350. doi: 10.1007/s10659-018-09722-8. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	J. W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville, Va. 1965. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	L. Qi, Eigenvalues and invariants of tensors, J. Math. Anal. Appl., 325 (2007), 1363-1377. doi: 10.1016/j.jmaa.2006.02.071. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	H. Rohrl, Algebras and differential equations, Nagoya Math. J., 68 (1977), 59-122. doi: 10.1017/S0027763000017876. Google Scholar
[24]	H. Rohrl, On the zeros of polynomials over arbitrary finite-dimensional algebras, Manuscripta Math., 25 (1978), 359-390. doi: 10.1007/BF01168049. Google Scholar
[25]	H. Rohrl, Finite-dimensional algebras without nilpotents over algebraically closed fields, Arch. Math. (Basel), 32 (1979), 10-12. doi: 10.1007/BF01238461. Google Scholar
[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. Google Scholar
[27]	I. R. Shafarevich, Basic Algebraic Geometry, Springer, 1977. Google Scholar
[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. Google Scholar
[30]	S. Walcher, Algebras and Differential Equations, Hadronic Press, 1991. Google Scholar
[31]	S. Walcher, Eigenvectors of tensors – a primer, Acta Appl. Math., 162 (2019), 165-183. doi: 10.1007/s10440-018-0225-7. Google Scholar

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. Google Scholar
[2]	V. I. Arnold, Ordinary Differential Equations, Springer, 1992. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	J. F. Cariñena, J. Grabowski and G. Marmo, Lie-Scheffers Systems. A Geometric Approach, Bibliopolis, 2000. Google Scholar
[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. Google Scholar
[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. Google Scholar
[10]	G. Gaeta and E. Virga, The symmetries of octupolar tensors, J. Elast., 135 (2019), 295-350. doi: 10.1007/s10659-018-09722-8. Google Scholar
[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. Google Scholar
[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. Google Scholar
[16]	J. W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville, Va. 1965. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	L. Qi, Eigenvalues and invariants of tensors, J. Math. Anal. Appl., 325 (2007), 1363-1377. doi: 10.1016/j.jmaa.2006.02.071. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	H. Rohrl, Algebras and differential equations, Nagoya Math. J., 68 (1977), 59-122. doi: 10.1017/S0027763000017876. Google Scholar
[24]	H. Rohrl, On the zeros of polynomials over arbitrary finite-dimensional algebras, Manuscripta Math., 25 (1978), 359-390. doi: 10.1007/BF01168049. Google Scholar
[25]	H. Rohrl, Finite-dimensional algebras without nilpotents over algebraically closed fields, Arch. Math. (Basel), 32 (1979), 10-12. doi: 10.1007/BF01238461. Google Scholar
[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. Google Scholar
[27]	I. R. Shafarevich, Basic Algebraic Geometry, Springer, 1977. Google Scholar
[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. Google Scholar
[30]	S. Walcher, Algebras and Differential Equations, Hadronic Press, 1991. Google Scholar
[31]	S. Walcher, Eigenvectors of tensors – a primer, Acta Appl. Math., 162 (2019), 165-183. doi: 10.1007/s10440-018-0225-7. Google Scholar

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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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 $

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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 $

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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	$ S_1 $	$ S_2 $	$ S_3 $	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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Max	Min	$ S_1 $	$ S_2 $	$ S_3 $	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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