F. Calogero, On a technique to identify solvable/integrable many-body problems / Milan, 30.09.2011 / page 1 / 28 On a technique to identify solvable/integrable many-body problems Francesco Calogero Physics Department, University of Rome “La Sapienza” Istituto Nazionale di Fisica Nucleare, Sezione di Roma francesco.calogero@roma1.infn.it, francesco.calogero@uniroma1.it Summary ( ) U ≡ U t The starting point is a square matrix , of rank N , evolving in the independent variable t (“time”) according to a solvable (or perhaps just integrable ) matrix evolution equation. One then ( ) n ≡ z z t focuses on the evolution of its N eigenvalues . This evolution generally also involves n N(N−1) additional variables. In some cases via a compatible ansatz these additional variables can be ( ) n ≡ z z t expressed in terms of the N variables . Thereby one obtains a system of evolution equations n ( ) n ≡ z z t involving only the N dependent variables , which is generally interpretable as a many-body n problem (characterized by Newtonian equations of motion). This approach is tersely reviewed, and several new solvable (and one integrable ) many-body problems identified in this manner are reported.
F. Calogero, On a technique to identify solvable/integrable many-body problems / Milan, 30.09.2011 / page 2 / 28 Recent papers containing findings including those reported below: - F. Calogero, “Two new solvable dynamical systems of goldfish type”, J. Nonlinear Math. Phys. 17 , 397-414 (2010). DOI: 10.1142/S1402925110000970. - F. Calogero, “A new goldfish model”, Theor. Math. Phys. 167 , 714-724 (2011). - F. Calogero, “Another new goldfish model”, Theor. Math. Phys. (in press). - F. Calogero, “A new integrable many-body problem”, J. Math. Phys. (in press). - F. Calogero, “New solvable many-body model of goldfish type”, J. Nonlinear Math. Phys. (submitted to, 4 September 2011). - F. Calogero, “Two quite similar matrix ODEs and the many-body problems related to them”, Proceedings of the Beppe Marmo 65 th Birthdate Meeting, Int. J. Geom. Met. Mod. Phys. (in press).
F. Calogero, On a technique to identify solvable/integrable many-body problems / Milan, 30.09.2011 / page 3 / 28 A few decades ago the many-body problem of N equal particles interacting pairwise in one-dimensional space via a force proportional to the inverse cube of their separation was shown to be solvable by algebraic operations. This many-body model is generally identified by the names of the two authors who demonstrated its solvable character, firstly in a quantum context ["Schrödinger equation": F. Calogero, "Solution of the one-dimensional N- body problem with quadratic and/or inversely quadratic pair potentials", J. Math. Phys. 12, 419-436 (1971); "Erratum", ibidem 37, 3646 (1996)] and then in a classical context ["Newtonian equations of motion": J. Moser, "Three integrable Hamiltonian systems connected with isospectral deformations", Adv. Math. 16, 197-220 (1975)]. Accordingly, it will be hereafter referred to as the CM model . The Newtonian equations of motion of this prototypical many- body problem read as follows: N ( ) ∑ 3 − 2 = − z g z z � � n n m 1 , = ≠ m m n
F. Calogero, On a technique to identify solvable/integrable many-body problems / Milan, 30.09.2011 / page 4 / 28 Notation . Here the N coordinates z n ≡ z n (t) are the dependent variables (and below we generally assume they are complex ), superimposed dots denote differentiations with respect to the ( real ) independent variable t ("time"), N is an arbitrary positive integer, and g 2 is an arbitrary "coupling constant". Indices such as n , m , k , generally run from 1 to N , unless otherwise specified. This model was the prototype of many subsequent developments in the theory of integrable dynamical systems: the relevant literature is vast, amounting to several hundred, possibly a few thousand, papers and tens of books, which can nowadays be easily traced via internet. An early, landmark contribution due to Olshanetsky and Perelomov [M. A. Olshanetsky and A. M. Perelomov, "Explicit solution of the Calogero model in the classical case and geodesic flows on symmetric spaces of zero curvature", Lett. Nuovo Cimento 16, 333-339 (1976)] explained the solvable character of the CM model in the classical context via the possibility to identify the N coordinates of its moving particles with the N eigenvalues of a square matrix of rank N itself evolving in time extremely simply (linearly). The time-evolution of that matrix could therefore be explicitly ascertained, and the solution of the CM many-body problem was thereby reduced to the purely algebraic task of evaluating the N eigenvalues z n ≡ z n (t) of an explicitly known matrix of rank N , or equivalently to finding the N zeros z n ≡ z n (t) of an explicitly known time-dependent polynomial of degree N in its argument z .
F. Calogero, On a technique to identify solvable/integrable many-body problems / Milan, 30.09.2011 / page 5 / 28 A bit later another prototypical class of solvable many-body problems was identified [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)], and it was eventually suggested that the simplest model belonging to this class be considered a "goldfish" [F. Calogero, "The "neatest" many-body problem amenable to exact treatments (a "goldfish"?)", Physica D 152-153 , 78-84 (2001)]. The original "goldfish" dynamical system is characterized by the two neat formulas N N [ ] [ ] ( ) { ( ) ( ) } ∑ ∑ 2 / , 1 ,..., ; 0 / 0 1 / , = − = − = z z z z z n N z z z t � � � � � n n m n m k k = 1 , ≠ = 1 m m n k the first of which provides the N Newtonian equations of motion of this dynamical system, while the second provides the solution of the corresponding initial-value problem: the N values of the dependent variables z n ≡ z n (t) at time t are the N roots of the polynomial equation of degree N in the variable z that obtains by multiplying this [ ] ( ) ∏ = N 0 . − z z formula by the polynomial n 1 n
F. Calogero, On a technique to identify solvable/integrable many-body problems / Milan, 30.09.2011 / page 6 / 28 The origin of the terms “goldfish”, “goldfishing” The following quotation is from my book (F. Calogero, Isochronous systems , Oxford University Press, 2008), in particular from its Section 4.2.2 entitled “Goldfishing”. “This terminology originated from the following description of the search for integrable systems given by V. E. Zakharov: "A mathematician, using the dressing method to find a new integrable system, could be compared with a fisherman, plunging his net into the sea. He does not know what a fish he will pull out. He hopes to catch a goldfish, of course. But too often his catch is something that could not be used for any known to him purpose. He invents more and more sophisticated nets and equipments and plunges all that deeper and deeper. As a result he pulls on the shore after a hard work more and more strange creatures. He should not despair, nevertheless. The strange creatures may be interesting enough if you are not too pragmatic. And who knows how deep in the sea do goldfishes live?" [ V. E. Zakharov, “On the dressing method”, in: P. C. Sabatier (editor), Inverse Methods in Action , Springer, Heidelberg, 1990, pp. 602-623].” Motivated by this poetic description of the search for integrable systems I - somewhat presumptuously (albeit with a question mark) - used the term “goldfish” in a contribution to Zakharov‘s 60 th birthday meeting: F. Calogero, "The ‘neatest’ many-body problem amenable to exact treatments (a ‘goldfish’?)", Physica D 152-153 , 78-84 (2001). Actually the solvable character of the prototypical goldfish model had been identified several years earlier as the simplest specimen of a solvable class of dynamical systems: 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).
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