数理解析研究所講究録 1203 巻 2001 年 71-88

71

Painlev\’e-Calogero correspondence
Kanehisa Takasaki*

Department of Fundamental Science
Faculty of Integrate Human Studies, Kyoto University

Abstract
The so called “Painleve-Calogero correspondence” relates the sixth Painleve
equation with an integrable system of the Calogero type. This relation was recently
generaized to the other Painleve equations and a“multi-component” analogue. This

paper reviews these results.

1Historical background
It was at the beginning of the twentieth century that Painleve’ discovered what are
nowadays called the “Painleve’ equations” [1], Painleve’ obtained those equations in thecourse of classification of second order nonlinear algebraic ordinary differential equations“without movable critical point”. The classification was eventually completed by his student Gambier[2], who supplemented several cases (in particular, the sixth equation) that

Painleve’ overlooked.
The property that the differential equation be free of movable critical point, whichlies in the heart of Painleve’s work, is now called the “Painleve’ property”. This kind ofanalysis is generally referred to as “Painleve’ analysis”. Actually, aprototype of Painlev\’e’
method can be found in Kowalevskaya’s work on integrability of the motion of arigid body[3]. In this respect, this method should be rather called “Kowalevskaya-Painleve analysis”.
At the time when Painleve’s classification was being completed, R. Fuchs (son of L.
Ruchs, who’s name is coined in the notion of “Fuchsian differential equations” , “Fuchsiangroups”, etc.) proposed two new approaches to the sixth Painleve’ equations [4]:$\mathrm{s}$” 高崎金久, 京都大学総合人間学部基礎科学科

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1. isomonodromic deformations

2. elliptic integrals
The first approach was soon generalized by Schlesinger[5] and Garnier[6] to alarge
extent, and (after along break) revived in the seventies along with an unexpected application in mathematical physics, such as the Ising model, soliton theory, etc.
The second approach, meanwhile, had been almost forgotten after Painleve improved
it slightly [7]. It should be mentioned that Okamoto [8] touched upon this work of Painleve
in his study on symmetries of the sixth Painleve equation. Manin [9] reexamined the workof Fuchs and Painleve after ninety years, and presented avery remarkable result.
Manin discovered that the sixth Painleve equation can be converted to anon-autonomous
Hamiltonian system with an elliptic potential. This is achieved by change of variables in
two steps. The first step was done by Painleve, who interpreted Fuchs’ elliptic integral
as anew dependent variable. Manin proposed to use the modulus of the elliptic curve as
anew independent variable, The outcome is the aforementioned non-autonomous Hamiltonian system. Furthermore, Manin noticed that this Hamiltonian system is reminiscentof aproblem of integrable systems studied by Treibich and Verdier [10].
Levin and Olshanetsky [11] pointed out that Manin’s Hamiltonian coincides with the
Hamiltonian of an integrable system called the “Inozemtsev system” $[13, 14]$ . Since theInozemtsev system is an integrable generalization of the Calogero system [12], Levin andOlshanetsky called this connection between the Painleve equations and the integrable
systems of the Calogero type the “Painleve- Calogero corresponden c\"e.
We shall briefly review this correspondence in the next section, then turn to recent
results on generalizations of the Painleve-Calogero correspondenc to the other Painleve
equations [15].

2Painlev\’e-Calogero correspondence for PVI
The sixth Painleve equation (PVI) is asecond order nonlinear differential equation of thefollowing form:$\frac{d^{2}\lambda}{dt^{2}}=$$\frac{1}{2}(\frac{1}{\lambda}+\frac{1}{\lambda-1}+\frac{1}{\lambda-t})(\frac{d\lambda}{dt})^{2}-(\frac{1}{t}+\frac{1}{t-1}+\frac{1}{\lambda-t})\frac{d\lambda}{dt}$$+ \frac{\lambda((\lambda-1)(\lambda-t)}{t^{2}(t-1)^{2}}(\alpha+\frac{\beta t}{\lambda^{2}}+\frac{\gamma(t-1)}{(\lambda-1)^{2}}+\frac{\delta t(t-1)}{(\lambda-t)^{2}})$.

We now follow the work of Fuchs, Painleve and Manin in the historical order

(1)

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What Fuchs did

2.1

Fuchs presents the following expression for this equation:$t(1-t) \mathcal{L}_{t}\int_{\infty}^{\lambda}\frac{dz}{\sqrt{z(z-1)(z-t)}}$

$=$

Here

$\mathcal{L}_{t}$$\sqrt{\lambda(\lambda-1)(\lambda-t)}[\alpha+\frac{\beta t}{\lambda^{2}}+\frac{\gamma(t-1)}{(\lambda-1)^{2}}+(\delta-\frac{1}{2})\frac{t(t-1)}{(\lambda-t)^{2}}]$.

(2)is the second order linear differential operator(3)$\mathcal{L}_{t}=t(1-t)\frac{d^{2}}{dt^{2}}+(1-2t)\frac{d}{dt}-\frac{1}{4}$called the “Picard-Fuchs operator”, whichs originates in the differential equation (the)
Picard-Fuchs$\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}^{1}$(4)$\mathcal{L}_{t}\oint_{\gamma}\frac{dz}{\sqrt{z(z-1)(z-t)}}--0$

for complete elliptic integrals. Fuchs thus applied it to an incomplete elliptic integral,and discovered the miraculous phenomena that all derivative terms in the sixth Painleve
equation are absorbed therein. We shall see that the same phenomena takes place for theother Painleve’ equations, too.

What Painlev\’e did

2.2

Painleve uses the Weierstrass function to rewrite Fuchs’ equation in terms of anew
dependent variable. Following Painleve, we now change the dependent variable from Ato
$\wp$

.

$q= \frac{1}{2(e_{2}-e_{1})^{1/2}}\int_{\infty}^{\lambda}\frac{dz}{\sqrt{z(z-1)(z-\lambda)}}$(5)(Actually, this is already modified \‘a la Manin.) This equation can be solved for Aas

A
where

$\mathrm{p}(\mathrm{u})$is the$\wp$$= \frac{\wp(q)-e_{1}}{e_{2}-e_{1}}$,(6)function$\wp(u)=\wp(u|1, \tau)=\frac{1}{u^{2}}+$$\sum$$(m,n)\neq(0,0)$

named after L. Fuch$( \frac{1}{(u+m+n\tau)^{2}}-\frac{1}{(m+n\tau)^{2}})$

(7)

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with fundamental periods 1, , and$\tau$$e_{1}$,$e_{2}$,$e_{3}$

denotes the values(8)$e_{n}=\wp(\omega_{n})$

of

$\wp$

at the three half-periods$\omega_{1}=\frac{1}{2}$,$\omega_{2}=-\frac{1}{2}-\frac{\tau}{2}$,$\omega_{3}=\frac{\tau}{2}$.

This change of dependent variables stems from the parametrization$z= \frac{\wp(u)-e_{1}}{e_{2}-e_{1}}$$,\mathrm{o}\mathrm{f}$,$y= \frac{\wp’(u)}{2(e_{3}-e_{1})^{3/2}}$,(9)

the elliptic curve$y^{2}=z(z-1)(z-t)$(10)that lies behind Puchs’ elliptic integral. The three half-periods corresponds to the threebranch points $z=0,1,t$ of the covering map from the elliptic curve onto the z-plane:–$z=0$ ,

$u=\omega_{2}$$rightarrow$$z=1$ ,

$u=\omega_{3}$

$-$

$z=t$

$u=\omega_{1}$.

$u=0$ correspond to the fourth branch point at $z=\infty$This change of dependent variable
turns into$\lambdaarrow q$.

transforms (2) as follows. The left hand side$t(1-t)\mathcal{L}_{t}(2(e_{2}-e_{1})^{12}q)$.

Let us examine the right hand side term by term by expanding the parentheses. Recallingthe differential equation$\wp’(u)^{2}=4(\wp(u)-e_{1})(\wp(u)-e_{2})(\wp(u)-e_{3})$

,

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one can rewrite the first term as$\sqrt{\lambda(\lambda-1)(\lambda-t)}=$$=$$(e_{2}-e_{1})^{-3/2}\sqrt{(\wp(q)-e_{1})(\wp(q)-e_{2})(\wp(q)-e_{3})}$$\frac{1}{2}(e_{2}-e_{1})^{-3/2}\wp’(q)$.

Furthermore, by the functional identit$\wp(u+\omega_{j})=e_{j}+\frac{(e_{j}-e_{k})(e_{j}-e\ell)}{\wp(u)-e_{j})}$

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( ,
$j$

$k,\ell$being acyclic permutation of 1, 2, 3), the other terms give alinear combination of$(n=1,2,3)$ .

$\frac{(e_{2}-e_{1})^{3/2}}{2}\wp’(q+\omega_{n})$Thus Fuchs’ equation can be eventually rewritten$t(1-t) \mathcal{L}_{t}(2(e_{2}-e_{1})^{1/2}q)=\frac{1}{2}(e_{2}-e_{1})^{-3/2}\sum_{n=0}^{3}\alpha_{n}\wp(q+\omega_{n})$

,

(13)

where
$\alpha_{0}=\alpha$,$\alpha_{1}=-\beta$,$\alpha_{2}=\gamma$,$\alpha_{3}=-\delta+\frac{1}{2}$

.

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This is essentially what Painleve did.

2.3

What Manin did

Manin further changes the dependent variable from tofunctional relation$t$$t= \frac{e_{3}-e_{1}}{e_{2}-e_{1}}$$\tau$. and are connected by the$t$$\tau$.

(15)(Note that are special values of at half-periods, thereby depends on ) Geometrically,
they are both amodulus of afamily of elliptic curves $-t$ is the modulus in Jacobi’s elliptic
function theory, and is its counterpart in the modular upper half plane. Manin shows
the beautiful formula$\tau.$$\wp$$e_{n}$$\tau$$\frac{d\tau}{dt}=\frac{\pi i}{t(t-1)(e_{2}-e_{1})}$

,

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for the Jacobian of this change of variable, and, with the aid of this formula, derives therelation$(e_{2}-e_{1})^{3/2}t(1-t)0 \mathcal{L}_{t}\mathrm{o}(e_{2}-e_{1})^{1/2}=(\pi i)^{2}\frac{d^{2}}{d\tau^{2}}$(17)that connects the Picard-Fuchs operator with adifferential operator in the new dependent variable . ( stands for composition of operators.) Equation (13) can be thereby
rewritten$\tau$$0$$(2 \pi i)^{2}\frac{d^{2}q}{d\tau^{2}}=\sum_{n=0}^{3}\alpha_{n}\wp’(q+\omega_{n})$

.

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This is the equation that Manin discovered.
This equation has another expression as anon-autonomous Hamiltonian system$2 \pi\dot{\iota}\frac{dq}{d\tau}=\frac{\partial \mathcal{H}}{\partial p}$

,

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$2 \pi\dot{\iota}\frac{dp}{d\tau}=-\frac{\partial \mathcal{H}}{\partial q}$

with the Hamiltonian$\mathcal{H}=\frac{p^{2}}{2}-\sum_{n=0}^{3}\alpha_{n}\wp(q+\omega_{n})$.

Note that the Hamiltonian depends on explicitly via the
Hamiltonian system is non-autonomous.
$\tau$

2.4(20)$\tau$

dependence of , so that the$\wp$

Relation to elliptic Inozemtsev system

The elliptic Inozemtsev system is amany-body particle system on aline. The equationsof motion take the Hamiltonian form$\frac{dqj}{dt}=\frac{\partial \mathcal{H}}{\partial p_{j}}$

,

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$\frac{dp_{j}}{dt}=-\frac{m}{\partial q_{j}}$with the Hamiltonian ( being the number of particles)$\ell$$\mathcal{H}$$= \sum_{j=1}^{\ell}(\frac{p_{j}^{2}}{2}+\sum_{n=0}^{3}g_{n}^{2}\wp(q_{j}+\omega_{n}))+g_{4}^{2}\sum_{j\neq k}(\wp(q_{j}-q_{k})+\wp(q_{j}+q_{k}))$

,

(22)

, are coupling constants. The modulus of the functions, too, is treated
where ,as an independent constant. Naturally, this is an autonomous system. Inozemtsev [13]discovered this system (along with analogues with hyperbolic and rational potentials) as
ageneralization of the Calogero system.
Levin and Olshanetsky noticed that Manin’s Hamiltonian is nothing but aspecial
case $(\ell=1)$ of Inozemtsev’s Hamiltonian. (The tw0-body potential is absent therein.)
An essential difference is that plays the role of time variable in Manin’s equation. One
may similarly consider the non-autonomous system$g_{0}$$\ldots$$\tau$$g_{4}$$\wp$$\tau$$2 \pi i\frac{dq_{j}}{d\tau}=\frac{\partial \mathcal{H}}{\partial p_{j}}$,$2 \pi i\frac{dp_{j}}{d\tau}=-\frac{\partial \mathcal{H}}{\partial q_{j}}$(23)

for any value of . This generalization of Manin’s equation turns out to give asystem ofisomonodromic deformations on the torus (Takasaki[16]).
$\ell$

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3Other Painleve equations
We now show how this correspondence was extended to the other Painleve’ equations [15].
Acrucial idea can be obtained from the degeneration relation among the six Painleve
equations (OkamOtO[17]). This relation can be schematically displayed by the following
diagram:$arrow$$\mathrm{P}_{\mathrm{V}\mathrm{I}}$$\mathrm{P}_{\mathrm{V}}$$arrow$$\mathrm{P}_{\mathrm{I}\mathrm{V}}$$\downarrow$$\downarrow$$\mathrm{p}_{\Pi \mathrm{I}}$$arrow$$\mathrm{P}_{\mathrm{I}\mathrm{I}}$$arrow$$\mathrm{P}_{\mathrm{I}}$The arrows stand for adegeneration process between two equations. One will notice, inparticular, that the five equations other than PVI can be reached from PVI. Remarkably,asimilar diagram of degeneration was known (without any connection with the Painleve
equations) for the Inozemtsev systems (van Diejen [18]). After all, the “Painlev\’e-Calogero
correspondence” for PVI turns out to be inherited by the other Painleve equations inaccordance with the degeneration relations on both the Painleve and Calogero sides.
The correspondence can be formulated in both the second order formalism and Hamil’tonian formalism. We here present the second order formalism, which is arather straightforward generalization of the work of Fuchs, Painleve’ and Manin.

3.1

Analogue of Fuchs’ equation for PV

The fifth Painleve
the following form:$\mathrm{V}$$\frac{d^{2}\lambda}{dt^{2}}$equation (PV) is asecond order nonlinear differential equation of$=$$( \frac{1}{2\lambda}+\frac{1}{\lambda-1})(\frac{d\lambda}{dt})^{2}-\frac{1}{t}\frac{d\lambda}{dt}$$+ \frac{\lambda(\lambda-1)^{2}}{t^{2}}(\alpha+\frac{\beta}{t^{2}}+\frac{\gamma t}{(\lambda-1)^{2}}+\frac{\delta t^{2}(\lambda+1)}{(\lambda-1)^{3}})$

.

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This equation can be derived from PVI by degeneration. This process is achieved by
setting$t=1+\epsilon\tilde{t}$,$\alpha=\tilde{\alpha}$,$\beta=\tilde{\beta}$,$\gamma=\frac{\tilde{\gamma}}{\epsilon}-\frac{\tilde{\delta}}{\epsilon^{2}}$,$\delta=\frac{\tilde{\delta}}{\epsilon^{2}}$(25), finite. This is akind of “scaling
while leaving ,and taking the limit aslimit” that is frequently used in physics.
In this scaling limit, the building blocks of Fuchs’ equation (2) behave as follows$\tilde{\delta}$$\epsilonarrow 0$$\tilde{\alpha}$$\ldots$

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1. Picard-Fuchs operator:$t(1-t) \mathcal{L}_{t}arrow\tilde{t}^{2}\frac{d^{2}}{d\tilde{t}^{2}}+\tilde{t}\frac{d}{d\tilde{t}}=(\tilde{t}\frac{d}{d\tilde{t}})^{2}$

2. The part$\alpha+\cdots$on the right hand side:.

$\alpha+\frac{\beta t}{\lambda^{2}}+\frac{\gamma(t-1)}{(\lambda-1)^{2}}+(\delta-\frac{1}{2})\frac{t(t-1)}{(\lambda-t)^{2}}arrow\tilde{\alpha}+\frac{\tilde{\beta}}{\lambda^{2}}+\frac{\tilde{\gamma}\tilde{t}}{(\lambda-1)^{2}}+\frac{\tilde{\delta}\tilde{t}^{2}(\lambda+1)}{(\lambda-1)^{3}}$3. The square root on the right hand side:$\sqrt{\lambda(\lambda-1)(\lambda-t)}arrow\sqrt{\lambda}(\lambda-1)$.

4. The incomplete elliptic integral:.

$\int_{\infty}^{\lambda}\frac{dz}{\sqrt{z(z-1)(z-t)}}arrow\int_{\infty}^{\lambda}\frac{dz}{\sqrt{z}(z-1)}$

Renaming ,, and as ,$\tilde{\delta}$$\tilde{\alpha}$$\ldots$$\tilde{t}$$\alpha$$\ldots$,$\delta kt$, we eventually obtains the equation$(t \frac{d}{dt})^{2}\int_{\infty}^{\lambda}\frac{dz}{\sqrt{z}(z-1)}=\sqrt{\lambda}(\lambda-1)(\alpha+\frac{\beta}{\lambda^{2}}+\frac{\gamma t}{(\lambda-1)^{2}}+\frac{\delta t^{2}(\lambda+1)}{(\lambda-1)^{3}})$(26)

as an analogue of Fuchs’ equation.

3.2

Analogue of Manin’s equation for PV

We now change the dependent variable from

$q$

to

$q= \int_{\infty}^{\lambda}\frac{dz}{\sqrt{z}(z-1)}$

.

(27)

in (26). A remark is in order: To be faithful to the construction for PVI, one should
rather define$q= \frac{1}{2\pi i}\int_{\infty}^{\lambda}\frac{dz}{\sqrt{z}(z-1)}$.

In fact, the degeneration process is associated with the degeneration$y^{2}=z(z-1)(z-t)$$arrow y^{2}=z(z-1)^{2}$

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of the elliptic curve to a singular rational curve as $t-arrow 1$ , and the constant factor $2(e_{2}-$on the right hand side of the definition of q behaves as$e_{1})^{1/2}$$2(e_{2}-e_{1})^{1/2}arrow 2\pi i$

causes no substantial difference,in this limit. Since omitting the numerical factor
we adopt the simpler definition. If $1/2\pi i$ is inserted, one will obtain a trigonometricfunction rather than the hyperbolic function that arises in the following calculation.
The integral defining can be calculated by elementary calculus:$1/2\mathrm{i}\mathrm{r}\mathrm{i}$

$q$

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$q= \log(\frac{\sqrt{\lambda}-1}{\sqrt{\lambda}+1})$

Solving this relation for$\lambda$

yields$\sqrt{\lambda}=-\coth(q/2)$

.

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Thus the function is now replaced by the hyperbolic function. Geometrically, thismeans that the torus turns into a cylinder as. It should be mentioned that asimilar change of variable for PV is already known in the literature (Iwasaki et al. [19]).
Let us rewrite (26) in terms of the new dependent variable . Each term on the righthand side of the equation can be written$\coth^{2}$$\wp$${\rm Im}\tauarrow\infty$$q$$\sqrt{\lambda}(\lambda-1)$$=$$- \frac{\cosh(q/2)}{\sinh^{3}(q/2)}$,$\sqrt{\lambda}(\lambda-1)\frac{1}{\lambda^{2}}$$=$$- \frac{\sinh(q/2)}{\cosh^{3}(q/2)}$,$\sqrt{\lambda}(\lambda-1)\frac{1}{(\lambda-1)^{2}}$$=$$- \frac{1}{2}\sinh(q)$$\sqrt{\lambda}(\lambda-1)\frac{(\lambda+1)}{(\lambda-1)^{3}}$$=$$- \frac{\lambda^{3/2}+\lambda^{1/2}}{(\lambda-1)^{2}}=-\frac{1}{4}\sinh(2q)$,,

so that (26) turns into the equation$(t \frac{d}{dt})^{2}q=-\frac{\partial V(q)}{\partial q}$

,

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for . $V(q)$ is the “potential”$q$$V(q)=- \frac{\alpha}{\sinh^{2}(q/2)}-\frac{\beta}{\cosh^{2}(q/2)}+\frac{\gamma t}{2}\cosh(q)+\frac{\delta t^{2}}{8}\cosh(2q)$This gives an analogue of Manin’s equation for PV

.

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This second order equation can be readily converted to a Hamiltonian system of thefom$t \frac{dq}{dt}=\frac{\partial \mathcal{H}}{\partial p}$

,

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$t \frac{dp}{dt}=-\frac{\theta \mathcal{H}}{\partial q}$

with the Hamiltonian $H$ $=p^{2}/2+V(q)$ and the time variable
non-autonomous system.
Remarkably, this Hamiltonian$\log t$. Note that this is a$\mathcal{H}=\frac{p^{2}}{2}-\frac{\alpha}{\sinh^{2}(q/2)}-\frac{\beta}{\cosh^{2}(q/2)}+\frac{\gamma t}{2}\cosh(q)+\frac{\delta t^{2}}{8}\cosh(2q)$

,

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too, is a special case of the Hamiltonian
$?t$

$=$

$\sum_{j=1}^{\ell}(\frac{p_{j}^{2}}{2}+\frac{g_{0}^{2}}{\sinh^{2}(q_{j}/2)}+\frac{g_{1}^{2}}{\cosh^{2}(q_{j}/2)}+g_{2}^{2}\cosh(qj)+g_{3}^{2}\cosh(2q_{j}))$(34)$+g_{4}^{2} \sum_{j\neq k}(\frac{1}{\sinh^{2}((q_{j}-q_{k})/2)}+\frac{1}{\sinh^{2}((qj+q_{k})/2)})$that Inozemtsev [13] considered as a generalization of the Calogero system. More precisely,
this Hamiltonian was first discovered by Levi and Wojciechowski [20], and Inozemtsev
rediscovered it in the course of his classification of generalized Calogero systems. The
foregoing Hamiltonian amounts to the case with $\ell=1$ , for which the tw0-body potential isabsent. Furthermore, as opposed to the autonomous case, some of the coupling constants
now depends on .
$t$

3.3

Direct method

Although the same method, in principle, works for the other Painleve equations, complexity of calculations soon increases as one proceeds deep into the lower equations. Fortunately, a simpler and direct method is available. This method, which works for the four
Painleve equations other than PII and , tells us how to find an appropriate definitionof the variable by simply inspecting the second order nonlinear differential equations.
, for which we already know the
To illustrate the idea, let us return to PVI andterm of the
explicit form of the variable. A clue lies in the coefficient of theequation. This coefficient is connected with the integrand of the definition of by thefollowing simple relation:$\mathrm{P}\mathrm{I}$$q$$\mathrm{P}\mathrm{V}$$(d\lambda/dt)^{2}$

$q$

$q$

$\frac{1}{\sqrt{z(z-1)(z-t)}}=\exp[-\int\frac{1}{2}(\frac{1}{z}+\frac{1}{z-1}+\frac{1}{z-t})dz]$$\sqrt{z}(z-1)1$$= \exp[-\int\frac{1}{2}(\frac{1}{2z}+\frac{1}{z-1})dz]$

.

,
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One will be naturally expect the same relation for PIV(36)$\frac{d^{2}\lambda}{dt^{2}}=\frac{1}{2\lambda}(\frac{d\lambda}{dt})^{2}+\frac{3}{2}\lambda^{3}+4t\lambda^{2}+2(t^{2}-\alpha)\lambda+\frac{\beta}{\lambda}$

and$\mathrm{P}$II.

$\frac{d^{2}\lambda}{dt^{2}}=\frac{1}{\lambda}(\frac{d\lambda}{dt})^{2}-\frac{1}{t}\frac{d\lambda}{dt}+\frac{\lambda^{2}}{4t^{2}}(\alpha+\frac{\beta t}{\lambda^{2}}+\gamma\lambda+\frac{\delta t^{2}}{4\lambda^{3}})$(37)

This is indeed the case, as we shall show below.
Let us note that this idea obviously does not apply to PII(38)$\frac{d^{2}\lambda}{dt^{2}}=2\lambda^{3}+t\lambda+\alpha$and PI(39)$\frac{d^{2}\lambda}{dt^{2}}=6\lambda^{2}+t$(simply because they have nodifferent way.

3.4$(d\lambda/dt)^{2}$term). These equations have to be treated in a

Case of PIVApplying the foregoing idea to PIV suggests to use the function$\exp(-\int\frac{dz}{2z})=\frac{1}{\sqrt{z}}$(40)

for defining . We thus define$q$$q= \int^{\lambda}\frac{dz}{\sqrt{z}}=2\sqrt{\lambda}$

This can be solved for$\lambda$

.

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as
$\lambda=(\frac{q}{2})^{2}$(42)

By direct calculation, one finds that PIV turns into the equation$\frac{d^{2}q}{dt^{2}}=-\frac{\partial V(q)}{\partial q}$(43)

with the potentia$V(q)=- \frac{1}{2}(\frac{q}{2})^{6}-2t$ $( \frac{q}{2})^{4}-2(t^{2}-\alpha)(\frac{q}{2})^{2}+\beta(\frac{q}{2})^{-2}$

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This equation, too, can be converted to a Hamiltonian system:$\frac{dq}{dt}=\frac{\partial \mathcal{H}}{\partial p}$

,

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$\frac{dp}{dt}=-\frac{\partial \mathcal{H}}{\partial q}$

The Hamiltonian takes the form$\mathcal{H}$(46)$= \frac{p^{2}}{2}-\frac{1}{2}(\frac{q}{2})^{6}-2t(\frac{q}{2})^{4}-2(t^{2}-\alpha)(\frac{q}{2})^{2}+\beta(\frac{q}{2})^{-2}$

A similar Hamiltonian can be found in the work of Levi, Wojciechowski [20] and Inozemtsev [13]:$\mathcal{H}=\sum_{j=1}^{\ell}(\frac{p_{j}^{2}}{2}+g_{0}^{2}q_{j}^{6}+g_{1}^{2}q_{j}^{4}+g_{2}^{2}q_{j}^{2}+g_{3}^{2}q_{j}^{-2)}+g_{4}^{2}\sum_{j\neq k}(\frac{1}{(q_{j}-q_{k})^{2}}-\frac{1}{(q_{j}+q_{k})^{2}}).$(47)

The foregoing Hamiltonian amounts to the case where $\ell=1$ . Some of the coupling
constants, too, are time-dependent.

3.5

Case of PIIIThe integrand for PIII reads$\exp(-\int\frac{dz}{z})=\frac{1}{z}$

The

$q$

.

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variable is given by$q= \int^{\lambda}\frac{dz}{z}=\log\lambda$

and solved for$\lambda$

,

(49)

as
$\lambda=e^{q}$

.

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Now PIII is converted to the equation(51)$(t \frac{d}{dt})^{2}q=-\frac{\partial V(q)}{\partial q}$

with potential$V(q)=- \frac{\alpha}{4}e^{q}+\frac{\beta t}{4}e^{-q}-\frac{\gamma}{8}e^{2q}+\frac{\delta t^{2}}{8}e^{-2q}$.

The associated Hamiltonian system is somewhat similar to the case of$t \frac{dq}{dt}=\frac{\partial \mathcal{H}}{\partial p}$,$t \frac{dp}{dt}=-\frac{\partial \mathcal{H}}{\partial q}$(52)$\mathrm{P}\mathrm{V}$

:
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The Hamiltonian$\mathcal{H}$(54)$= \frac{p^{2}}{2}-\frac{\alpha}{4}e^{q}+\frac{\beta t}{4}e^{-q}-\frac{\gamma}{8}e^{2q}+\frac{\delta t^{2}}{8}e^{-2q}$

has no analogue in Inozemtsev’s classification. One can, however, derive this HamiltOnian (and its generalization with degrees of freedom) from the hyperbolic nozemtsev
Hamiltonian by a degeneration process; this is exactly what van Diejen considered [18].
$\ell$

Correspondence in Hamiltonian formalism

4

The foregoing correspondence in the second order formalism can be reformulated as atime-dependent canonical transformation between Hamiltonian systems. The status of
PII and PI is also clarified in this Hamiltonian formalism.

Hamiltonian structure of Painlev\’e equations

4.1

As first pointed out by Malmquist[21], the six Painleve equations can be expressed as a
Hamiltonian system of the form$\frac{d\lambda}{dt}=\frac{\partial H}{\partial\mu}$,$\frac{d\lambda}{dt}=-\frac{\partial H}{\partial\lambda}$.

This expression is not unique; we now consider the one with the following “polynomial
Hamiltonians” (OkamOtO[17]):

PVI
PV
PIV
$\mathrm{P}$

II

PII
PI

$H= \frac{\lambda(\lambda-1)(\lambda-t)}{t(t-1)}[\mu^{2}-(\frac{\kappa_{0}}{\lambda}+\frac{\kappa_{1}}{\lambda-1}+\frac{\theta-1}{\lambda-t})\mu+\frac{\kappa}{\lambda(\lambda-1)}]$$H= \frac{\lambda(\lambda-1)^{2}}{t}[\mu^{2}-(\frac{\kappa_{0}}{\lambda}+\frac{\theta_{1}}{\lambda-1}-\frac{\eta_{1}t}{(\lambda-1)^{2}})\mu+\frac{\kappa}{\lambda(\lambda-1)}]$$H=2 \lambda[\mu^{2}-(\frac{\lambda}{2}+t+\frac{\kappa_{0}}{\lambda})\mu+\frac{\theta_{\infty}}{2}]$

.

.

.

$H= \frac{\lambda^{2}}{t}[\mu^{2}-(\eta_{\infty}+\frac{\theta_{0}}{\lambda}-\frac{\eta_{0}t}{\lambda^{2}})\mu+\frac{\eta_{\infty}(\theta_{0}+\theta_{\infty})}{2\lambda}]$.

.

$H= \frac{\mu^{2}}{2}-(\lambda^{2}+\frac{t}{2})\mu-(\alpha+\frac{1}{2})\lambda$$H= \frac{\mu^{2}}{2}-2\lambda^{3}-t\lambda$.

, are constants that are connected with the parameters of the Painleve equations
by simple algebraic relations$\kappa_{0}$,$\theta$$\kappa_{1}$

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4.2

How to find canonical transformation

We now show the outline of the construction of a canonical transformation that connects
the Hamiltonian form of the Painleve equations with the Hamiltonian system of the
Inozemtsev type.
A clue lies in the equation for in the foregoing Hamiltonian system. For illustration,
let us consider PVI. The equation for reads:$\lambda$$\lambda$$\frac{d\lambda}{dt}=\frac{\lambda(\lambda-1)(\lambda-t)}{t(t-1)}(2\mu-\frac{\kappa_{0}}{\lambda}-\frac{\kappa_{1}}{\lambda-1}-\frac{\theta-1}{\lambda-t})$This equation can be solvled for

.

(55)

as follows:$\mu$$\mu=\frac{t(t-1)}{2\lambda(\lambda-1)(\lambda-t)}\frac{d\lambda}{dt}+\frac{1}{2}(\frac{\kappa_{0}}{\lambda}+\frac{\kappa_{1}}{\lambda-1}+\frac{\theta-1}{\lambda-t})$

.

(56)

Our goal is to rewrite this expression of to a function of , (and ). To this end, let
us recall (29). Differentiating (29) against results in the equation
$p$

$\mu$

$q$

$\tau$$t$$\frac{d\lambda}{dt}=(\frac{\wp’(q)}{e_{2}-e_{1}}\frac{dq}{d\tau}+f_{\tau}(q))\frac{d\tau}{dt}$

,

(57)

where
$f(u)= \frac{\wp(u)-e_{1}}{e_{2}-e_{1}}$The equation of motion for$q$,$f_{\tau}(u)= \frac{\partial f(u)}{\partial\tau}$

.

gives
(58)

$\frac{dq}{d\tau}=\frac{1}{2\pi i}\frac{\partial \mathcal{H}}{\partial p}=\frac{p}{2\pi i}$

Furthermore, one can use (16) to rewrite
$\mu$

$=$

$\mathrm{d}\mathrm{r}/\mathrm{d}\mathrm{t}$. Thus one eventually obtain the expression$\frac{e_{2}-e_{1}}{\wp’(q)}p+\frac{2\pi i(e_{2}-e_{1})^{2}}{\wp(q)^{2}},f_{\tau}(q)$$+ \frac{e_{2}-e_{1}}{2}(\frac{\kappa_{0}}{\wp(q)-e_{1}}+\frac{\kappa_{1}}{\wp(q)-e_{2}}+\frac{\theta-1}{\wp(q)-e_{3}})$(59)

in terms of , and .
Having derived (59), we now change the point of view: We now interpret (59) and (6)as defining a time-dependent map. By somewhat lengthy calculations, one
can prove that this map satisfies the equation

for

$\mu$

$p$

$q$

$\tau$$(q,p)arrow(\mathrm{q},\mathrm{p})$$\mu d\lambda-Hdt$$=pdq- \mathcal{H}\frac{d\tau}{2\pi i}+\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{c}\mathrm{t}$

form,

(60)

which means that this map is a time-dependent canonical transformation between the
two Hamiltonian systems on the Painleve’ and Calogero sides.
In much the same way, the correspondence for, PIV and PIII can be reformulated
as a time-dependent canonical transformation$\mathrm{P}\mathrm{V}$85
PII and PI

4.3

PII and PI have to be treated separately. In view of the result for PIV-Pill, one shouldseek for a transformation to a Hamiltonian system with a Hamiltonian of the “standard
form”$\mathcal{H}$$= \frac{p^{2}}{2}+V(q)$.

Since PI is already of that form, what is left is PII only.
One can readily find such a canonical transformation:$\lambda=q$

,

(61)

$\mu=p+\lambda^{2}+\frac{t}{2}$This converts PII to a Hamiltonian system with the Hamiltonian$\mathcal{H}=\frac{p^{2}}{2}-\frac{1}{2}(q^{2}+\frac{t}{2})^{2}-\alpha q$

.

(62)

Besides this rather ad hoc way, one can show that this Hamiltonian can be derived fromthe others (for PIV and PHI) by a degeneration process. Thus the six Hamiltonians on
the Calogero side, like those on the Painleve side, are connected by degeneration relation.

Multi-component analogues

5

All the Hamiltonian systems in the Painlev\’e-Calogero correspondence have just one degreeof freedom. Since the Inozemtsev systems themselves are generalized to a many-bodysystem, one will naturally ask if the Painlev\’e equations have a many-body analogue.
“
This question, too, is answered afirmatively [15]. Namely, an -body” generalizationof the Painleve equations can be constructed. This generalization has pairs $(\lambda j,\mu j)(j=$, ) of canonically conjugate variables, each of which is connected with a canonical,pair (Aj, ) on the Calogero side by the same functional relation as in the $\ell=1$ case. The
Hamiltonian $H$ is a rational function of the canonical variables and the time variable, andtake the form$\ell$$\ell$$\ell$$1$$\ldots$$p_{j}$-body interaction.

$H= \sum_{j=1}^{\ell}H_{j}+\mathrm{t}\mathrm{w}\mathrm{o}$(63)

is the “one-body” Hamiltonian in (Aj, ) with the same functional form as the corresponding Painleve’ equation. The tw0-body interaction tersm have singularities of the. In other words, this is a “perturbation) along
CalogerO-type (i.e.,$H_{j}$$\mu_{j}$$(\lambda_{j}-\lambda_{k})^{-2}$$\lambda_{j}=\lambda_{k}$86of independent Painleve equations by tw0-body interaction terms. For instance, the-body Hamiltonian for the PVI type reads:$\ell$$\ell$$H$$=$$\sum_{j=1}^{\ell}\frac{\lambda_{j}(\lambda_{j}-1)(\lambda_{j}-t)}{t(t-1)}[\mu_{j}^{2}-(\frac{\kappa_{0}}{\lambda_{j}}+\frac{\kappa_{1}}{\lambda_{j}-1}+\frac{\theta-1}{\lambda_{j}-t})\mu_{j}+\frac{\kappa}{\lambda_{j}(\lambda_{j}-1)}]$$+ \frac{g_{4}^{2}}{2t(t-1)}\sum_{j\neq k}[\frac{\lambda_{j}(\lambda_{j}-1)(\lambda_{j}-t)+\lambda_{k}(\lambda_{k}-1)(\lambda_{k}-t)}{8(\lambda_{j}-\lambda_{k})^{2}}-2(\lambda_{j}+\lambda_{k})]$. (64)

This is a kind of “multi-component” version of the Painleve equations, similar but
obviously different from another family of multi-dimensional generalizations called the“Ga mier systems” (Garnier[6], OkamOtO[17]). Presumably, these multi-component generalizations, too, will describe isomonodromic deformations on the Riemann sphere.

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