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Higher-order mechanics: Variational principles and other topics
The Toda lattice, old and new
1. | Departamento de Matemática, PUC-Rio, R. Mq. S. Vicente 225, Rio de Janeiro 22451-900, Brazil |
  The text is mostly expositive and self contained, presenting alternative formulations of familiar results and applications to numerical analysis.
References:
[1] |
M. Adler, On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de-Vries type equations,, Invent. Math., 50 (): 219.
doi: 10.1007/BF01410079. |
[2] |
M. Atiyah, Convexity and commuting Hamiltonians,, Bull. London Math. Soc., 14 (1982), 1.
doi: 10.1112/blms/14.1.1. |
[3] |
M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems,, Stud. Appl. Math., 53 (1974), 249.
|
[4] |
R. Abraham and J. Marsden, Foundations of Mechanics,, Second edition, (1987). Google Scholar |
[5] |
R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line,, Math. Surveys and Monographs, (1988).
|
[6] |
A. M. Bloch, H. Flaschka and T. Ratiu, A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra,, Duke Math. J., 61 (1990), 41.
doi: 10.1215/S0012-7094-90-06103-4. |
[7] |
P. Deift, J. Demmel, L. C. Li and C. Tomei, The bidiagonal singular value decomposition and Hamiltonian mechanics,, SIAM J. Num. Anal., 28 (1991), 1463.
doi: 10.1137/0728076. |
[8] |
P. Deift, L. C. Li, T. Nanda and C. Tomei, The Toda flow on a generic orbit is integrable,, Comm. Pure Appl. Math., 39 (1986), 183.
doi: 10.1002/cpa.3160390203. |
[9] |
P. Deift, L. C. Li and C. Tomei, Matrix factorizations and integrable systems,, Comm. Pure Appl. Math., 42 (1989), 443.
doi: 10.1002/cpa.3160420405. |
[10] |
P. Deift, L. C. Li and C. Tomei, Loop groups, discrete versions of some classical integrable systems, and rank 2 extensions,, Memoirs of the Amer. Math. Soc., 100 (1992).
doi: 10.1090/memo/0479. |
[11] |
P. Deift, L. C. Li and C. Tomei, Toda flows with infinitely many variables,, J. Funct. Anal., 64 (1985), 358.
doi: 10.1016/0022-1236(85)90065-5. |
[12] |
P. Deift, T. Nanda and C. Tomei, Ordinary differential equations for the symmetric eigenvalue problem,, SIAM J. Num. Anal., 20 (1983), 1.
doi: 10.1137/0720001. |
[13] |
P. Deift, S. Rivera, C. Tomei and D. Watkins, A monotonicity property for Toda-type flows,, SIAM J. of Matrix Anal. and Appl., 12 (1991), 463.
doi: 10.1137/0612033. |
[14] |
J. W. Demmel, Applied Numerical Linear Algebra,, SIAM, (1997).
doi: 10.1137/1.9781611971446. |
[15] |
J. W. Demmel and W. Kahan, Accurate singular values of bidiagonal matrices,, SIAM J. Stat. Comput., 11 (1990), 873.
doi: 10.1137/0911052. |
[16] |
H. Flaschka, The Toda lattice. I. Existence of integrals,, Phys. Rev. B (3), 9 (1974), 1924.
doi: 10.1103/PhysRevB.9.1924. |
[17] |
D. Fried, The cohomology of an isospectral flow,, Proc. Amer. Math. Soc., 98 (1986), 363.
doi: 10.1090/S0002-9939-1986-0854048-6. |
[18] |
L. Feher and I. Tsutsui, Regularization of Toda lattices by Hamiltonian reduction,, Jour. Geom. Phys., 21 (1997), 97.
doi: 10.1016/S0393-0440(96)00010-1. |
[19] |
C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation,, Phys. Rev. Letter., 19 (1967), 1095. Google Scholar |
[20] |
V. Guillemin and S. Sternberg, Symplectic Techniques in Physics,, Cambridge University Press, (1984).
|
[21] |
T. Kapeller and J. Pöschel, KdV & KAM,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (2003).
|
[22] |
B. Kostant, Quantization and representation theory,, in Representation Theory of Lie Groups (ed. M. Atiyah), (1977), 287. Google Scholar |
[23] |
Y. Kodama and B. Shipman, The finite non-periodic toda lattice: A geometric and topological viewpoint,, , (2008). Google Scholar |
[24] |
I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations,, Russ. Math. Surv., 32 (1977), 185. Google Scholar |
[25] |
I. M. Krichever and S. P. Novikov, Holomorphich bundles over algebraic curves and nonlinear equations,, Russ. Math. Surv., 35 (1980), 53. Google Scholar |
[26] |
P. Lax, Integrals of nonlinear equations of evolution and solitary waves,, Comm. Pure Appl. Math., 21 (1968), 467.
doi: 10.1002/cpa.3160210503. |
[27] |
R. S. Leite, T. R. W. Richa and C. Tomei, Geometric proofs of some theorems of Schur-Horn type,, Lin. Alg. Appl., 286 (1999), 149.
doi: 10.1016/S0024-3795(98)10169-6. |
[28] |
R. S. Leite, N. C. Saldanha and C. Tomei, An atlas for tridiagonal isospectral manifolds,, Lin. Alg. Appl., 429 (2008), 387.
doi: 10.1016/j.laa.2008.03.001. |
[29] |
R. S. Leite, N. C. Saldanha and C. Tomei, The asymptotics of Wilkinson's shift: Loss of cubic convergence,, Found. Comp. Math., 10 (2010), 15.
doi: 10.1007/s10208-009-9047-3. |
[30] |
R. S. Leite, N. C. Saldanha and C. Tomei, Dynamics of the symmetric eigenvalue problem with shift strategies,, Int. Math. Res. Notices, 2013 (2013), 4382.
doi: 10.1093/imrn/rns186. |
[31] |
R. S. Leite and C. Tomei, Parametrization by polytopes of intersections of orbits by conjugation,, Lin. Alg. Appl., 361 (2003), 223.
doi: 10.1016/S0024-3795(02)00463-9. |
[32] |
J. Moser, Finitely many points on the line under the influence of an exponential potential-an integrable system,, in Dynamical Systems, (1974), 467.
|
[33] |
J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials,, Comm. Math. Phys., 139 (1991), 217.
doi: 10.1007/BF02352494. |
[34] |
B. Parlett, The Symmetric Eigenvalue Problem,, Prentice-Hall Series in Computational Mathematics, (1980).
|
[35] |
A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras. Vol. I,, Birkhäuser Verlag, (1990).
doi: 10.1007/978-3-0348-9257-5. |
[36] |
J. Pöschel and E. Trubowitz, Inverse Spectral Theory,, Pure and Applied Mathematics, (1987).
|
[37] |
S. N. M. Ruijsenaars, Relativistic Toda systems,, Comm. Math. Phys., 133 (1990), 217.
doi: 10.1007/BF02097366. |
[38] |
A. G. Reyman, M. A. Semenov-Tian-Shansky, Group-theoretical methods in the theory of finite-dimensional integrable-systems,, in Dynamical Systems VII (eds. V. I. Arnold and S. P. Novikov), (1994). Google Scholar |
[39] |
W. Symes, Hamiltonian group actions and integrable systems,, Physica D, 1 (1980), 339.
doi: 10.1016/0167-2789(80)90017-2. |
[40] |
W. Symes, The QR algorithm and scattering for the finite nonperiodic Toda lattice,, Physica D, 4 (): 275.
doi: 10.1016/0167-2789(82)90069-0. |
[41] |
N. C. Saldanha and C. Tomei, Manifolds of normal or symmetric matrices of given spectrum and envelope,, in preparation., (). Google Scholar |
[42] |
M. Toda, Wave propagation in anharmonic lattices,, J. Phys. Soc. Japan, 23 (1967), 501. Google Scholar |
[43] |
C. Tomei, The topology of isospectral manifolds of tridiagonal matrices,, Duke Math. J., 51 (1984), 981.
doi: 10.1215/S0012-7094-84-05144-5. |
[44] |
L. N.Trefethen and D. Bau, III, Numerical Linear Algebra,, SIAM, (1997).
doi: 10.1137/1.9780898719574. |
[45] |
P. van Moerbeke, The spectrum of Jacobi matrices,, Invent. Math., 37 (1976), 45.
doi: 10.1007/BF01418827. |
[46] |
D. S. Watkins and L. Elsner, On Rutishauser's approach to self-similar flows,, SIAM J. Matrix Anal. Appl., 11 (1990), 301.
doi: 10.1137/0611020. |
show all references
References:
[1] |
M. Adler, On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de-Vries type equations,, Invent. Math., 50 (): 219.
doi: 10.1007/BF01410079. |
[2] |
M. Atiyah, Convexity and commuting Hamiltonians,, Bull. London Math. Soc., 14 (1982), 1.
doi: 10.1112/blms/14.1.1. |
[3] |
M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems,, Stud. Appl. Math., 53 (1974), 249.
|
[4] |
R. Abraham and J. Marsden, Foundations of Mechanics,, Second edition, (1987). Google Scholar |
[5] |
R. Beals, P. Deift and C. Tomei, Direct and Inverse Scattering on the Line,, Math. Surveys and Monographs, (1988).
|
[6] |
A. M. Bloch, H. Flaschka and T. Ratiu, A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra,, Duke Math. J., 61 (1990), 41.
doi: 10.1215/S0012-7094-90-06103-4. |
[7] |
P. Deift, J. Demmel, L. C. Li and C. Tomei, The bidiagonal singular value decomposition and Hamiltonian mechanics,, SIAM J. Num. Anal., 28 (1991), 1463.
doi: 10.1137/0728076. |
[8] |
P. Deift, L. C. Li, T. Nanda and C. Tomei, The Toda flow on a generic orbit is integrable,, Comm. Pure Appl. Math., 39 (1986), 183.
doi: 10.1002/cpa.3160390203. |
[9] |
P. Deift, L. C. Li and C. Tomei, Matrix factorizations and integrable systems,, Comm. Pure Appl. Math., 42 (1989), 443.
doi: 10.1002/cpa.3160420405. |
[10] |
P. Deift, L. C. Li and C. Tomei, Loop groups, discrete versions of some classical integrable systems, and rank 2 extensions,, Memoirs of the Amer. Math. Soc., 100 (1992).
doi: 10.1090/memo/0479. |
[11] |
P. Deift, L. C. Li and C. Tomei, Toda flows with infinitely many variables,, J. Funct. Anal., 64 (1985), 358.
doi: 10.1016/0022-1236(85)90065-5. |
[12] |
P. Deift, T. Nanda and C. Tomei, Ordinary differential equations for the symmetric eigenvalue problem,, SIAM J. Num. Anal., 20 (1983), 1.
doi: 10.1137/0720001. |
[13] |
P. Deift, S. Rivera, C. Tomei and D. Watkins, A monotonicity property for Toda-type flows,, SIAM J. of Matrix Anal. and Appl., 12 (1991), 463.
doi: 10.1137/0612033. |
[14] |
J. W. Demmel, Applied Numerical Linear Algebra,, SIAM, (1997).
doi: 10.1137/1.9781611971446. |
[15] |
J. W. Demmel and W. Kahan, Accurate singular values of bidiagonal matrices,, SIAM J. Stat. Comput., 11 (1990), 873.
doi: 10.1137/0911052. |
[16] |
H. Flaschka, The Toda lattice. I. Existence of integrals,, Phys. Rev. B (3), 9 (1974), 1924.
doi: 10.1103/PhysRevB.9.1924. |
[17] |
D. Fried, The cohomology of an isospectral flow,, Proc. Amer. Math. Soc., 98 (1986), 363.
doi: 10.1090/S0002-9939-1986-0854048-6. |
[18] |
L. Feher and I. Tsutsui, Regularization of Toda lattices by Hamiltonian reduction,, Jour. Geom. Phys., 21 (1997), 97.
doi: 10.1016/S0393-0440(96)00010-1. |
[19] |
C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation,, Phys. Rev. Letter., 19 (1967), 1095. Google Scholar |
[20] |
V. Guillemin and S. Sternberg, Symplectic Techniques in Physics,, Cambridge University Press, (1984).
|
[21] |
T. Kapeller and J. Pöschel, KdV & KAM,, Ergebnisse der Mathematik und ihrer Grenzgebiete, (2003).
|
[22] |
B. Kostant, Quantization and representation theory,, in Representation Theory of Lie Groups (ed. M. Atiyah), (1977), 287. Google Scholar |
[23] |
Y. Kodama and B. Shipman, The finite non-periodic toda lattice: A geometric and topological viewpoint,, , (2008). Google Scholar |
[24] |
I. M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations,, Russ. Math. Surv., 32 (1977), 185. Google Scholar |
[25] |
I. M. Krichever and S. P. Novikov, Holomorphich bundles over algebraic curves and nonlinear equations,, Russ. Math. Surv., 35 (1980), 53. Google Scholar |
[26] |
P. Lax, Integrals of nonlinear equations of evolution and solitary waves,, Comm. Pure Appl. Math., 21 (1968), 467.
doi: 10.1002/cpa.3160210503. |
[27] |
R. S. Leite, T. R. W. Richa and C. Tomei, Geometric proofs of some theorems of Schur-Horn type,, Lin. Alg. Appl., 286 (1999), 149.
doi: 10.1016/S0024-3795(98)10169-6. |
[28] |
R. S. Leite, N. C. Saldanha and C. Tomei, An atlas for tridiagonal isospectral manifolds,, Lin. Alg. Appl., 429 (2008), 387.
doi: 10.1016/j.laa.2008.03.001. |
[29] |
R. S. Leite, N. C. Saldanha and C. Tomei, The asymptotics of Wilkinson's shift: Loss of cubic convergence,, Found. Comp. Math., 10 (2010), 15.
doi: 10.1007/s10208-009-9047-3. |
[30] |
R. S. Leite, N. C. Saldanha and C. Tomei, Dynamics of the symmetric eigenvalue problem with shift strategies,, Int. Math. Res. Notices, 2013 (2013), 4382.
doi: 10.1093/imrn/rns186. |
[31] |
R. S. Leite and C. Tomei, Parametrization by polytopes of intersections of orbits by conjugation,, Lin. Alg. Appl., 361 (2003), 223.
doi: 10.1016/S0024-3795(02)00463-9. |
[32] |
J. Moser, Finitely many points on the line under the influence of an exponential potential-an integrable system,, in Dynamical Systems, (1974), 467.
|
[33] |
J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials,, Comm. Math. Phys., 139 (1991), 217.
doi: 10.1007/BF02352494. |
[34] |
B. Parlett, The Symmetric Eigenvalue Problem,, Prentice-Hall Series in Computational Mathematics, (1980).
|
[35] |
A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras. Vol. I,, Birkhäuser Verlag, (1990).
doi: 10.1007/978-3-0348-9257-5. |
[36] |
J. Pöschel and E. Trubowitz, Inverse Spectral Theory,, Pure and Applied Mathematics, (1987).
|
[37] |
S. N. M. Ruijsenaars, Relativistic Toda systems,, Comm. Math. Phys., 133 (1990), 217.
doi: 10.1007/BF02097366. |
[38] |
A. G. Reyman, M. A. Semenov-Tian-Shansky, Group-theoretical methods in the theory of finite-dimensional integrable-systems,, in Dynamical Systems VII (eds. V. I. Arnold and S. P. Novikov), (1994). Google Scholar |
[39] |
W. Symes, Hamiltonian group actions and integrable systems,, Physica D, 1 (1980), 339.
doi: 10.1016/0167-2789(80)90017-2. |
[40] |
W. Symes, The QR algorithm and scattering for the finite nonperiodic Toda lattice,, Physica D, 4 (): 275.
doi: 10.1016/0167-2789(82)90069-0. |
[41] |
N. C. Saldanha and C. Tomei, Manifolds of normal or symmetric matrices of given spectrum and envelope,, in preparation., (). Google Scholar |
[42] |
M. Toda, Wave propagation in anharmonic lattices,, J. Phys. Soc. Japan, 23 (1967), 501. Google Scholar |
[43] |
C. Tomei, The topology of isospectral manifolds of tridiagonal matrices,, Duke Math. J., 51 (1984), 981.
doi: 10.1215/S0012-7094-84-05144-5. |
[44] |
L. N.Trefethen and D. Bau, III, Numerical Linear Algebra,, SIAM, (1997).
doi: 10.1137/1.9780898719574. |
[45] |
P. van Moerbeke, The spectrum of Jacobi matrices,, Invent. Math., 37 (1976), 45.
doi: 10.1007/BF01418827. |
[46] |
D. S. Watkins and L. Elsner, On Rutishauser's approach to self-similar flows,, SIAM J. Matrix Anal. Appl., 11 (1990), 301.
doi: 10.1137/0611020. |
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