A solvable N-body problem in the plane. I

Journal of Mathematical Physics

Volumn 37, Issue 4, 1996, Pages 1735-1759

  Calogero, F ^a,b  

^a UNIVERSITÀ DI ROMA LA SAPIENZA   (Italy)

^b INFN SEZIONE DI ROMA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030532332     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531666     Document Type: Article

References (17)

1. F. Calogero, "Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations, and related 'solvable' many-body problems," Nuovo Cimento 43B, 177-241 (1978).
2. M. Bruschi and F. Calogero, "The Lax representation for an integrable class of relativistic dynamical systems," Commun. Math. Phys. 109, 481-492 (1987).
3. S. Graffi (private communication).
4. F. Calogero, "Integrable many-body problems," in Nonlinear Equations in Physics and Mathematics, edited by A. O. Barut, Proceedings of the NATO Advanced Study Institute held in Istanbul in August 1977 (Reidel, Dordrecht, 1978), pp. 3-53; "Solvable many-body problems and related mathematical findings (and conjectures)," in Bifurcation Phenomena in Mathematical Physics and Related Topics, edited by C. Bardos and D. Bessis (Reidel, Dordrecht, 1980), pp. 371-384; "Integrable many-body problems and related mathematical results," in Proceedings of the 1980 Summer School on Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen (North Holland, Amsterdam, 1981), pp. 143-150; "Integrable dynamical systems and related mathematical results," in Nonlinear Phenomena, Proceedings of the CIFMO School and Workshop held at Oaxtepec, Mexico, November 29-December 17, 1982, Lecture Notes in Physics, Vol. 189 (Springer, Heidelberg, 1983), pp. 47-109; "Solvable many-body problems," in Statics and Dynamics of Nonlinear Systems, edited by G. Benedek, H. Bilz, and R. Zeyher, Springer Series in Solid-State Science, Vol. 47 (Springer, Berlin, 1983), pp. 35-36; "Some solvable dynamical systems," in Geometry Today, Giornate di Geometria, Roma 1984, edited by E. Arbarello, C. Procesi, and E. Strickland (Birkhauser, Boston, 1985), pp. 39-45; "Some applications of a convenient finite-dimensional matrix representation of the differential operator," Proceedings of the International Conference on Special Functions: Theory and Computation, Rend. Sem. Mat., Univ. & Polit. Torino, October 1984 (special issue), pp. 23-61; "Integrable dynamical systems and some other mathematical results (remarkable matrices, identities, basic hypergeometric functions)," Notes of lectures presented at the summer school held at the Université de Montréal, July 29-August 16, 1985, in Systèmes Dynamiques Non Linéaires: Integrabilité et Comportement Qualitatif, edited by P. Winternitz, SMS 102 (Presses de l'Université de Montréal, Montréal, 1986), pp. 40-70.
5. F. Calogero, "Integrable many-body problems," in Nonlinear Equations in Physics and Mathematics, edited by A. O. Barut, Proceedings of the NATO Advanced Study Institute held in Istanbul in August 1977 (Reidel, Dordrecht, 1978), pp. 3-53; "Solvable many-body problems and related mathematical findings (and conjectures)," in Bifurcation Phenomena in Mathematical Physics and Related Topics, edited by C. Bardos and D. Bessis (Reidel, Dordrecht, 1980), pp. 371-384; "Integrable many-body problems and related mathematical results," in Proceedings of the 1980 Summer School on Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen (North Holland, Amsterdam, 1981), pp. 143-150; "Integrable dynamical systems and related mathematical results," in Nonlinear Phenomena, Proceedings of the CIFMO School and Workshop held at Oaxtepec, Mexico, November 29-December 17, 1982, Lecture Notes in Physics, Vol. 189 (Springer, Heidelberg, 1983), pp. 47-109; "Solvable many-body problems," in Statics and Dynamics of Nonlinear Systems, edited by G. Benedek, H. Bilz, and R. Zeyher, Springer Series in Solid-State Science, Vol. 47 (Springer, Berlin, 1983), pp. 35-36; "Some solvable dynamical systems," in Geometry Today, Giornate di Geometria, Roma 1984, edited by E. Arbarello, C. Procesi, and E. Strickland (Birkhauser, Boston, 1985), pp. 39-45; "Some applications of a convenient finite-dimensional matrix representation of the differential operator," Proceedings of the International Conference on Special Functions: Theory and Computation, Rend. Sem. Mat., Univ. & Polit. Torino, October 1984 (special issue), pp. 23-61; "Integrable dynamical systems and some other mathematical results (remarkable matrices, identities, basic hypergeometric functions)," Notes of lectures presented at the summer school held at the Université de Montréal, July 29-August 16, 1985, in Systèmes Dynamiques Non Linéaires: Integrabilité et Comportement Qualitatif, edited by P. Winternitz, SMS 102 (Presses de l'Université de Montréal, Montréal, 1986), pp. 40-70.
6. F. Calogero, "Integrable many-body problems," in Nonlinear Equations in Physics and Mathematics, edited by A. O. Barut, Proceedings of the NATO Advanced Study Institute held in Istanbul in August 1977 (Reidel, Dordrecht, 1978), pp. 3-53; "Solvable many-body problems and related mathematical findings (and conjectures)," in Bifurcation Phenomena in Mathematical Physics and Related Topics, edited by C. Bardos and D. Bessis (Reidel, Dordrecht, 1980), pp. 371-384; "Integrable many-body problems and related mathematical results," in Proceedings of the 1980 Summer School on Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen (North Holland, Amsterdam, 1981), pp. 143-150; "Integrable dynamical systems and related mathematical results," in Nonlinear Phenomena, Proceedings of the CIFMO School and Workshop held at Oaxtepec, Mexico, November 29-December 17, 1982, Lecture Notes in Physics, Vol. 189 (Springer, Heidelberg, 1983), pp. 47-109; "Solvable many-body problems," in Statics and Dynamics of Nonlinear Systems, edited by G. Benedek, H. Bilz, and R. Zeyher, Springer Series in Solid-State Science, Vol. 47 (Springer, Berlin, 1983), pp. 35-36; "Some solvable dynamical systems," in Geometry Today, Giornate di Geometria, Roma 1984, edited by E. Arbarello, C. Procesi, and E. Strickland (Birkhauser, Boston, 1985), pp. 39-45; "Some applications of a convenient finite-dimensional matrix representation of the differential operator," Proceedings of the International Conference on Special Functions: Theory and Computation, Rend. Sem. Mat., Univ. & Polit. Torino, October 1984 (special issue), pp. 23-61; "Integrable dynamical systems and some other mathematical results (remarkable matrices, identities, basic hypergeometric functions)," Notes of lectures presented at the summer school held at the Université de Montréal, July 29-August 16, 1985, in Systèmes Dynamiques Non Linéaires: Integrabilité et Comportement Qualitatif, edited by P. Winternitz, SMS 102 (Presses de l'Université de Montréal, Montréal, 1986), pp. 40-70.
7. F. Calogero, "Integrable many-body problems," in Nonlinear Equations in Physics and Mathematics, edited by A. O. Barut, Proceedings of the NATO Advanced Study Institute held in Istanbul in August 1977 (Reidel, Dordrecht, 1978), pp. 3-53; "Solvable many-body problems and related mathematical findings (and conjectures)," in Bifurcation Phenomena in Mathematical Physics and Related Topics, edited by C. Bardos and D. Bessis (Reidel, Dordrecht, 1980), pp. 371-384; "Integrable many-body problems and related mathematical results," in Proceedings of the 1980 Summer School on Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen (North Holland, Amsterdam, 1981), pp. 143-150; "Integrable dynamical systems and related mathematical results," in Nonlinear Phenomena, Proceedings of the CIFMO School and Workshop held at Oaxtepec, Mexico, November 29-December 17, 1982, Lecture Notes in Physics, Vol. 189 (Springer, Heidelberg, 1983), pp. 47-109; "Solvable many-body problems," in Statics and Dynamics of Nonlinear Systems, edited by G. Benedek, H. Bilz, and R. Zeyher, Springer Series in Solid-State Science, Vol. 47 (Springer, Berlin, 1983), pp. 35-36; "Some solvable dynamical systems," in Geometry Today, Giornate di Geometria, Roma 1984, edited by E. Arbarello, C. Procesi, and E. Strickland (Birkhauser, Boston, 1985), pp. 39-45; "Some applications of a convenient finite-dimensional matrix representation of the differential operator," Proceedings of the International Conference on Special Functions: Theory and Computation, Rend. Sem. Mat., Univ. & Polit. Torino, October 1984 (special issue), pp. 23-61; "Integrable dynamical systems and some other mathematical results (remarkable matrices, identities, basic hypergeometric functions)," Notes of lectures presented at the summer school held at the Université de Montréal, July 29-August 16, 1985, in Systèmes Dynamiques Non Linéaires: Integrabilité et Comportement Qualitatif, edited by P. Winternitz, SMS 102 (Presses de l'Université de Montréal, Montréal, 1986), pp. 40-70.
8. F. Calogero, "Integrable many-body problems," in Nonlinear Equations in Physics and Mathematics, edited by A. O. Barut, Proceedings of the NATO Advanced Study Institute held in Istanbul in August 1977 (Reidel, Dordrecht, 1978), pp. 3-53; "Solvable many-body problems and related mathematical findings (and conjectures)," in Bifurcation Phenomena in Mathematical Physics and Related Topics, edited by C. Bardos and D. Bessis (Reidel, Dordrecht, 1980), pp. 371-384; "Integrable many-body problems and related mathematical results," in Proceedings of the 1980 Summer School on Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen (North Holland, Amsterdam, 1981), pp. 143-150; "Integrable dynamical systems and related mathematical results," in Nonlinear Phenomena, Proceedings of the CIFMO School and Workshop held at Oaxtepec, Mexico, November 29-December 17, 1982, Lecture Notes in Physics, Vol. 189 (Springer, Heidelberg, 1983), pp. 47-109; "Solvable many-body problems," in Statics and Dynamics of Nonlinear Systems, edited by G. Benedek, H. Bilz, and R. Zeyher, Springer Series in Solid-State Science, Vol. 47 (Springer, Berlin, 1983), pp. 35-36; "Some solvable dynamical systems," in Geometry Today, Giornate di Geometria, Roma 1984, edited by E. Arbarello, C. Procesi, and E. Strickland (Birkhauser, Boston, 1985), pp. 39-45; "Some applications of a convenient finite-dimensional matrix representation of the differential operator," Proceedings of the International Conference on Special Functions: Theory and Computation, Rend. Sem. Mat., Univ. & Polit. Torino, October 1984 (special issue), pp. 23-61; "Integrable dynamical systems and some other mathematical results (remarkable matrices, identities, basic hypergeometric functions)," Notes of lectures presented at the summer school held at the Université de Montréal, July 29-August 16, 1985, in Systèmes Dynamiques Non Linéaires: Integrabilité et Comportement Qualitatif, edited by P. Winternitz, SMS 102 (Presses de l'Université de Montréal, Montréal, 1986), pp. 40-70.
9. F. Calogero, "Integrable many-body problems," in Nonlinear Equations in Physics and Mathematics, edited by A. O. Barut, Proceedings of the NATO Advanced Study Institute held in Istanbul in August 1977 (Reidel, Dordrecht, 1978), pp. 3-53; "Solvable many-body problems and related mathematical findings (and conjectures)," in Bifurcation Phenomena in Mathematical Physics and Related Topics, edited by C. Bardos and D. Bessis (Reidel, Dordrecht, 1980), pp. 371-384; "Integrable many-body problems and related mathematical results," in Proceedings of the 1980 Summer School on Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen (North Holland, Amsterdam, 1981), pp. 143-150; "Integrable dynamical systems and related mathematical results," in Nonlinear Phenomena, Proceedings of the CIFMO School and Workshop held at Oaxtepec, Mexico, November 29-December 17, 1982, Lecture Notes in Physics, Vol. 189 (Springer, Heidelberg, 1983), pp. 47-109; "Solvable many-body problems," in Statics and Dynamics of Nonlinear Systems, edited by G. Benedek, H. Bilz, and R. Zeyher, Springer Series in Solid-State Science, Vol. 47 (Springer, Berlin, 1983), pp. 35-36; "Some solvable dynamical systems," in Geometry Today, Giornate di Geometria, Roma 1984, edited by E. Arbarello, C. Procesi, and E. Strickland (Birkhauser, Boston, 1985), pp. 39-45; "Some applications of a convenient finite-dimensional matrix representation of the differential operator," Proceedings of the International Conference on Special Functions: Theory and Computation, Rend. Sem. Mat., Univ. & Polit. Torino, October 1984 (special issue), pp. 23-61; "Integrable dynamical systems and some other mathematical results (remarkable matrices, identities, basic hypergeometric functions)," Notes of lectures presented at the summer school held at the Université de Montréal, July 29-August 16, 1985, in Systèmes Dynamiques Non Linéaires: Integrabilité et Comportement Qualitatif, edited by P. Winternitz, SMS 102 (Presses de l'Université de Montréal, Montréal, 1986), pp. 40-70.
10. F. Calogero, "Integrable many-body problems," in Nonlinear Equations in Physics and Mathematics, edited by A. O. Barut, Proceedings of the NATO Advanced Study Institute held in Istanbul in August 1977 (Reidel, Dordrecht, 1978), pp. 3-53; "Solvable many-body problems and related mathematical findings (and conjectures)," in Bifurcation Phenomena in Mathematical Physics and Related Topics, edited by C. Bardos and D. Bessis (Reidel, Dordrecht, 1980), pp. 371-384; "Integrable many-body problems and related mathematical results," in Proceedings of the 1980 Summer School on Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen (North Holland, Amsterdam, 1981), pp. 143-150; "Integrable dynamical systems and related mathematical results," in Nonlinear Phenomena, Proceedings of the CIFMO School and Workshop held at Oaxtepec, Mexico, November 29-December 17, 1982, Lecture Notes in Physics, Vol. 189 (Springer, Heidelberg, 1983), pp. 47-109; "Solvable many-body problems," in Statics and Dynamics of Nonlinear Systems, edited by G. Benedek, H. Bilz, and R. Zeyher, Springer Series in Solid-State Science, Vol. 47 (Springer, Berlin, 1983), pp. 35-36; "Some solvable dynamical systems," in Geometry Today, Giornate di Geometria, Roma 1984, edited by E. Arbarello, C. Procesi, and E. Strickland (Birkhauser, Boston, 1985), pp. 39-45; "Some applications of a convenient finite-dimensional matrix representation of the differential operator," Proceedings of the International Conference on Special Functions: Theory and Computation, Rend. Sem. Mat., Univ. & Polit. Torino, October 1984 (special issue), pp. 23-61; "Integrable dynamical systems and some other mathematical results (remarkable matrices, identities, basic hypergeometric functions)," Notes of lectures presented at the summer school held at the Université de Montréal, July 29-August 16, 1985, in Systèmes Dynamiques Non Linéaires: Integrabilité et Comportement Qualitatif, edited by P. Winternitz, SMS 102 (Presses de l'Université de Montréal, Montréal, 1986), pp. 40-70.
11. F. Calogero, "Integrable many-body problems," in Nonlinear Equations in Physics and Mathematics, edited by A. O. Barut, Proceedings of the NATO Advanced Study Institute held in Istanbul in August 1977 (Reidel, Dordrecht, 1978), pp. 3-53; "Solvable many-body problems and related mathematical findings (and conjectures)," in Bifurcation Phenomena in Mathematical Physics and Related Topics, edited by C. Bardos and D. Bessis (Reidel, Dordrecht, 1980), pp. 371-384; "Integrable many-body problems and related mathematical results," in Proceedings of the 1980 Summer School on Fundamental Problems in Statistical Mechanics, edited by E. G. D. Cohen (North Holland, Amsterdam, 1981), pp. 143-150; "Integrable dynamical systems and related mathematical results," in Nonlinear Phenomena, Proceedings of the CIFMO School and Workshop held at Oaxtepec, Mexico, November 29-December 17, 1982, Lecture Notes in Physics, Vol. 189 (Springer, Heidelberg, 1983), pp. 47-109; "Solvable many-body problems," in Statics and Dynamics of Nonlinear Systems, edited by G. Benedek, H. Bilz, and R. Zeyher, Springer Series in Solid-State Science, Vol. 47 (Springer, Berlin, 1983), pp. 35-36; "Some solvable dynamical systems," in Geometry Today, Giornate di Geometria, Roma 1984, edited by E. Arbarello, C. Procesi, and E. Strickland (Birkhauser, Boston, 1985), pp. 39-45; "Some applications of a convenient finite-dimensional matrix representation of the differential operator," Proceedings of the International Conference on Special Functions: Theory and Computation, Rend. Sem. Mat., Univ. & Polit. Torino, October 1984 (special issue), pp. 23-61; "Integrable dynamical systems and some other mathematical results (remarkable matrices, identities, basic hypergeometric functions)," Notes of lectures presented at the summer school held at the Université de Montréal, July 29-August 16, 1985, in Systèmes Dynamiques Non Linéaires: Integrabilité et Comportement Qualitatif, edited by P. Winternitz, SMS 102 (Presses de l'Université de Montréal, Montréal, 1986), pp. 40-70.
12. S. N. M. Ruijsenaars and H. Schneider, "A new class of integrable systems and its relation to solitons," Ann. Phys. 170, 370-405 (1986); S. N. M. Ruijsenaars, "Complete integrability of relativistic Calogero-Moser systems and elliptic function identities," Commun. Math. Phys. 110, 191-213 (1987); S. N. M. Ruijsenaars, "Finite-dimensional soliton systems," in Integrable and Superintegrable Systems, edited by B. Kupershmidt (World Scientific, Singapore, 1990), pp. 165-206.
13. S. N. M. Ruijsenaars and H. Schneider, "A new class of integrable systems and its relation to solitons," Ann. Phys. 170, 370-405 (1986); S. N. M. Ruijsenaars, "Complete integrability of relativistic Calogero-Moser systems and elliptic function identities," Commun. Math. Phys. 110, 191-213 (1987); S. N. M. Ruijsenaars, "Finite-dimensional soliton systems," in Integrable and Superintegrable Systems, edited by B. Kupershmidt (World Scientific, Singapore, 1990), pp. 165-206.
14. S. N. M. Ruijsenaars and H. Schneider, "A new class of integrable systems and its relation to solitons," Ann. Phys. 170, 370-405 (1986); S. N. M. Ruijsenaars, "Complete integrability of relativistic Calogero-Moser systems and elliptic function identities," Commun. Math. Phys. 110, 191-213 (1987); S. N. M. Ruijsenaars, "Finite-dimensional soliton systems," in Integrable and Superintegrable Systems, edited by B. Kupershmidt (World Scientific, Singapore, 1990), pp. 165-206.
15. P. D. Lax, "Integrals of nonlinear equations of evolution and solitary waves," Commun. Pure Appl. Math 21, 467-490 (1968).
16. F. Calogero and J.-P. Françoise, "Integrable dynamical systems obtained by duplications," Ann. Inst. H. Poincaré 57, 167-181 (1992).
17. F. Calogero and J.-P. Françoise, "Hamiltonian character of the motion of the zeros of a polynomial whose coefficients oscillate over time" (submitted to J. Phys. A).