§32.6 Hamiltonian Structure

Keywords:: Painlevé equations, Painlevé transcendents
Permalink:: http://dlmf.nist.gov/32.6

§32.6(i) Introduction
§32.6(ii) First Painlevé Equation
§32.6(iii) Second Painlevé Equation
§32.6(iv) Third Painlevé Equation
§32.6(v) Other Painlevé Equations

§32.6(i) Introduction

Notes:: See Okamoto (1981).
Permalink:: http://dlmf.nist.gov/32.6.i

$\mbox{P}_{{\mbox{\scriptsize I}}}$ – $\mbox{P}_{{\mbox{\scriptsize VI}}}$ can be written as a Hamiltonian system

32.6.1

$\frac{dq}{dz}=\frac{\partial\mathrm{H}}{\partial p},$

$\frac{dp}{dz}=-\frac{\partial\mathrm{H}}{\partial q},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $z$ : real, $q$ : coordinate, $p$ : momentum and $\mathrm{H}(q,p,z)$ : Hamiltonian
Permalink:: http://dlmf.nist.gov/32.6.E1
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for suitable (non-autonomous) Hamiltonian functions $\mathrm{H}(q,p,z)$ .

§32.6(ii) First Painlevé Equation

Notes:: See Jimbo and Miwa (1981), Okamoto (1981).
Permalink:: http://dlmf.nist.gov/32.6.ii

The Hamiltonian for $\mbox{P}_{{\mbox{\scriptsize I}}}$ is

32.6.2 $\mathrm{H}_{{\mbox{\scriptsize I}}}(q,p,z)=\tfrac{1}{2}p^{2}-2q^{3}-zq,$

Symbols:: $z$ : real, $q$ : coordinate, $p$ : momentum and $\mathrm{H}(q,p,z)$ : Hamiltonian
Referenced by:: §32.6(ii)
Permalink:: http://dlmf.nist.gov/32.6.E2
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and so

32.6.3 $q^{{\prime}}=p,$

Symbols:: $q$ : coordinate and $p$ : momentum
Referenced by:: §32.6(ii)
Permalink:: http://dlmf.nist.gov/32.6.E3
Encodings:: TeX, pMML, png

32.6.4 $p^{{\prime}}=6q^{2}+z.$

Symbols:: $z$ : real, $q$ : coordinate and $p$ : momentum
Referenced by:: §32.6(ii)
Permalink:: http://dlmf.nist.gov/32.6.E4
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Then $q=w$ satisfies $\mbox{P}_{{\mbox{\scriptsize I}}}$ . The function

32.6.5 $\sigma=\mathrm{H}_{{\mbox{\scriptsize I}}}(q,p,z),$

Symbols:: $z$ : real, $q$ : coordinate, $p$ : momentum, $\mathrm{H}(q,p,z)$ : Hamiltonian and $\sigma$ : function
Permalink:: http://dlmf.nist.gov/32.6.E5
Encodings:: TeX, pMML, png

defined by (32.6.2) satisfies

32.6.6 $\left(\sigma^{{\prime\prime}}\right)^{2}+4\left(\sigma^{{\prime}}\right)^{3}+2z\sigma^{{\prime}}-2\sigma=0.$

Symbols:: $z$ : real and $\sigma$ : function
Referenced by:: §32.6(ii)
Permalink:: http://dlmf.nist.gov/32.6.E6
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Conversely, if $\sigma$ is a solution of (32.6.6), then

32.6.7 $q=-\sigma^{{\prime}},$

Symbols:: $q$ : coordinate and $\sigma$ : function
Permalink:: http://dlmf.nist.gov/32.6.E7
Encodings:: TeX, pMML, png

32.6.8 $p=-\sigma^{{\prime\prime}},$

Symbols:: $p$ : momentum and $\sigma$ : function
Permalink:: http://dlmf.nist.gov/32.6.E8
Encodings:: TeX, pMML, png

are solutions of (32.6.3) and (32.6.4).

§32.6(iii) Second Painlevé Equation

Notes:: See Jimbo and Miwa (1981), Okamoto (1986).
Permalink:: http://dlmf.nist.gov/32.6.iii

The Hamiltonian for $\mbox{P}_{{\mbox{\scriptsize II}}}$ is

32.6.9 $\mathrm{H}_{{\mbox{\scriptsize II}}}(q,p,z)=\tfrac{1}{2}p^{2}-(q^{2}+\tfrac{1}{2}z)p-(\alpha+\tfrac{1}{2})q,$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant, $q$ : coordinate, $p$ : momentum and $\mathrm{H}(q,p,z)$ : Hamiltonian
Referenced by:: §32.6(iii)
Permalink:: http://dlmf.nist.gov/32.6.E9
Encodings:: TeX, pMML, png

and so

32.6.10 $q^{{\prime}}=p-q^{2}-\tfrac{1}{2}z,$

Symbols:: $z$ : real, $q$ : coordinate and $p$ : momentum
Referenced by:: §32.6(iii)
Permalink:: http://dlmf.nist.gov/32.6.E10
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32.6.11 $p^{{\prime}}=2qp+\alpha+\tfrac{1}{2}.$

Symbols:: $\alpha$ : arbitrary constant, $q$ : coordinate and $p$ : momentum
Referenced by:: §32.6(iii)
Permalink:: http://dlmf.nist.gov/32.6.E11
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Then $q=w$ satisfies $\mbox{P}_{{\mbox{\scriptsize II}}}$ and $p$ satisfies

32.6.12 $pp^{{\prime\prime}}=\tfrac{1}{2}(p^{{\prime}})^{2}+2p^{3}-zp^{2}-\tfrac{1}{2}(\alpha+\tfrac{1}{2})^{2}.$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant and $p$ : momentum
Permalink:: http://dlmf.nist.gov/32.6.E12
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The function $\sigma(z)=\mathrm{H}_{{\mbox{\scriptsize II}}}(q,p,z)$ defined by (32.6.9) satisfies

32.6.13 $\left(\sigma^{{\prime\prime}}\right)^{2}+4\left(\sigma^{{\prime}}\right)^{3}+2\sigma^{{\prime}}\left(z\sigma^{{\prime}}-\sigma\right)=\tfrac{1}{4}(\alpha+\tfrac{1}{2})^{2}.$

Symbols:: $z$ : real, $\alpha$ : arbitrary constant and $\sigma(z)$ : function
Referenced by:: §32.6(iii)
Permalink:: http://dlmf.nist.gov/32.6.E13
Encodings:: TeX, pMML, png

Conversely, if $\sigma(z)$ is a solution of (32.6.13), then

32.6.14 $q=\ifrac{(4\sigma^{{\prime\prime}}+2\alpha+1)}{(8\sigma^{{\prime}})},$

Symbols:: $\alpha$ : arbitrary constant, $q$ : coordinate and $\sigma(z)$ : function
Permalink:: http://dlmf.nist.gov/32.6.E14
Encodings:: TeX, pMML, png

32.6.15 $p=-2\sigma^{{\prime}},$

Symbols:: $p$ : momentum and $\sigma(z)$ : function
Permalink:: http://dlmf.nist.gov/32.6.E15
Encodings:: TeX, pMML, png

are solutions of (32.6.10) and (32.6.11).

§32.6(iv) Third Painlevé Equation

Notes:: See Jimbo and Miwa (1981), Okamoto (1987c); also Forrester and Witte (2002).
Permalink:: http://dlmf.nist.gov/32.6.iv

The Hamiltonian for $\mbox{P}_{{\mbox{\scriptsize III}}}$ is

32.6.16 $z\mathrm{H}_{{\mbox{\scriptsize III}}}(q,p,z)=q^{2}p^{2}-{\left(\kappa _{{\infty}}zq^{2}+(2\theta _{0}+1)q-\kappa _{0}z\right)p}+\kappa _{{\infty}}(\theta _{0}+\theta _{{\infty}})zq,$

Symbols:: $z$ : real, $q$ : coordinate, $\mathrm{H}(q,p,z)$ : Hamiltonian, $\theta$ : parameter, $p$ : momentum and $\kappa _{j}$ : parameters
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E16
Encodings:: TeX, pMML, png

and so

32.6.17 $zq^{{\prime}}=2q^{2}p-\kappa _{{\infty}}zq^{2}-(2\theta _{0}+1)q+\kappa _{0}z,$

Symbols:: $z$ : real, $q$ : coordinate, $\theta$ : parameter, $p$ : momentum and $\kappa _{j}$ : parameters
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E17
Encodings:: TeX, pMML, png

32.6.18 $zp^{{\prime}}=-2qp^{2}+2\kappa _{{\infty}}zqp+(2\theta _{0}+1)p-\kappa _{{\infty}}(\theta _{0}+\theta _{\infty})z.$

Symbols:: $z$ : real, $q$ : coordinate, $\theta$ : parameter, $p$ : momentum and $\kappa _{j}$ : parameters
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E18
Encodings:: TeX, pMML, png

Then $q=w$ satisfies $\mbox{P}_{{\mbox{\scriptsize III}}}$ with

32.6.19 $(\alpha,\beta,\gamma,\delta)=\left(-2\kappa _{{\infty}}\theta _{{\infty}},2\kappa _{0}(\theta _{0}+1),\kappa _{{\infty}}^{2},-\kappa _{0}^{2}\right).$

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant, $\theta$ : parameter and $\kappa _{j}$ : parameters
Permalink:: http://dlmf.nist.gov/32.6.E19
Encodings:: TeX, pMML, png

The function

32.6.20 $\sigma=z\mathrm{H}_{{\mbox{\scriptsize III}}}(q,p,z)+pq+\theta _{0}^{2}-\tfrac{1}{2}\kappa _{0}\kappa _{{\infty}}z^{2}$

Symbols:: $z$ : real, $q$ : coordinate, $\sigma$ : function, $\mathrm{H}(q,p,z)$ : Hamiltonian, $\theta$ : parameter, $p$ : momentum and $\kappa _{j}$ : parameters
Permalink:: http://dlmf.nist.gov/32.6.E20
Encodings:: TeX, pMML, png

defined by (32.6.16) satisfies

32.6.21 $(z\sigma^{{\prime\prime}}-\sigma^{{\prime}})^{2}+2\left((\sigma^{{\prime}})^{2}-\kappa _{0}^{2}\kappa _{{\infty}}^{2}z^{2}\right)(z\sigma^{{\prime}}-2\sigma)+8\kappa _{0}\kappa _{{\infty}}\theta _{0}\theta _{{\infty}}z\sigma^{{\prime}}=4\kappa _{0}^{2}\kappa _{{\infty}}^{2}(\theta _{0}^{2}+\theta _{{\infty}}^{2})z^{2}.$

Symbols:: $z$ : real, $\sigma$ : function, $\theta$ : parameter and $\kappa _{j}$ : parameters
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E21
Encodings:: TeX, pMML, png

Conversely, if $\sigma$ is a solution of (32.6.21), then

32.6.22 $q=\frac{\kappa _{0}\left(z\sigma^{{\prime\prime}}-(2\theta _{0}+1)\sigma^{{\prime}}+2\kappa _{0}\kappa _{{\infty}}\theta _{{\infty}}z\right)}{\kappa _{0}^{2}\kappa _{{\infty}}^{2}z^{2}-(\sigma^{{\prime}})^{2}},$

Symbols:: $z$ : real, $q$ : coordinate, $\sigma$ : function, $\theta$ : parameter and $\kappa _{j}$ : parameters
Permalink:: http://dlmf.nist.gov/32.6.E22
Encodings:: TeX, pMML, png

32.6.23 $p=\ifrac{(\sigma^{{\prime}}+\kappa _{0}\kappa _{{\infty}}z)}{(2\kappa _{0})},$

Symbols:: $z$ : real, $\sigma$ : function, $p$ : momentum and $\kappa _{j}$ : parameters
Permalink:: http://dlmf.nist.gov/32.6.E23
Encodings:: TeX, pMML, png

are solutions of (32.6.17) and (32.6.18).

The Hamiltonian for $\mbox{P}^{{\prime}}_{{\mbox{\scriptsize III}}}$ (§32.2(iii)) is

32.6.24 $\zeta\mathrm{H}_{{\mbox{\scriptsize III}}}(q,p,\zeta)=q^{2}p^{2}-\left(\eta _{{\infty}}q^{2}+\theta _{0}q-\eta _{0}\zeta\right)p+\tfrac{1}{2}\eta _{{\infty}}(\theta _{0}+\theta _{{\infty}})q,$

Symbols:: $q$ : coordinate, $\mathrm{H}(q,p,z)$ : Hamiltonian, $\theta$ : parameter and $p$ : momentum
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E24
Encodings:: TeX, pMML, png

and so

32.6.25 $\zeta q^{{\prime}}=2q^{2}p-\eta _{{\infty}}q^{2}-\theta _{0}q+\eta _{0}\zeta,$

Symbols:: $q$ : coordinate, $\theta$ : parameter and $p$ : momentum
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E25
Encodings:: TeX, pMML, png

32.6.26 $\zeta p^{{\prime}}=-2qp^{2}+2\eta _{{\infty}}qp+\theta _{0}p-\tfrac{1}{2}\eta _{{\infty}}(\theta _{0}+\theta _{1}).$

Symbols:: $q$ : coordinate, $\theta$ : parameter and $p$ : momentum
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E26
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Then $q=u$ satisfies $\mbox{P}^{{\prime}}_{{\mbox{\scriptsize III}}}$ with

32.6.27 $(\alpha,\beta,\gamma,\delta)=\left(-4\eta _{{\infty}}\theta _{{\infty}},4\eta _{0}(\theta _{0}+1),4\eta _{{\infty}}^{2},-4\eta _{0}^{2}\right).$

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant and $\theta$ : parameter
Permalink:: http://dlmf.nist.gov/32.6.E27
Encodings:: TeX, pMML, png

The function

32.6.28 $\sigma=\zeta\mathrm{H}_{{\mbox{\scriptsize III}}}(q,p,\zeta)+\tfrac{1}{4}\theta _{0}^{2}-\tfrac{1}{2}\eta _{0}\eta _{{\infty}}\zeta$

Symbols:: $q$ : coordinate, $\sigma$ : function, $\mathrm{H}(q,p,z)$ : Hamiltonian, $\theta$ : parameter and $p$ : momentum
Permalink:: http://dlmf.nist.gov/32.6.E28
Encodings:: TeX, pMML, png

defined by (32.6.24) satisfies

32.6.29 $\zeta^{2}(\sigma^{{\prime\prime}})^{2}+\left(4(\sigma^{{\prime}})^{2}-\eta _{0}^{2}\eta _{{\infty}}^{2}\right)(\zeta\sigma^{{\prime}}-\sigma)+\eta _{0}\eta _{{\infty}}\theta _{0}\theta _{{\infty}}\sigma^{{\prime}}=\tfrac{1}{4}\eta _{0}^{2}\eta _{{\infty}}^{2}(\theta _{0}^{2}+\theta _{{\infty}}^{2}).$

Symbols:: $\sigma$ : function and $\theta$ : parameter
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E29
Encodings:: TeX, pMML, png

Conversely, if $\sigma$ is a solution of (32.6.29), then

32.6.30 $q=\frac{\eta _{0}\left(\zeta\sigma^{{\prime\prime}}-2\theta _{0}\sigma^{{\prime}}+\eta _{0}\eta _{{\infty}}\theta _{{\infty}}\right)}{\eta _{0}^{2}\eta _{{\infty}}^{2}-4(\sigma^{{\prime}})^{2}},$

Symbols:: $q$ : coordinate, $\sigma$ : function and $\theta$ : parameter
Permalink:: http://dlmf.nist.gov/32.6.E30
Encodings:: TeX, pMML, png

32.6.31 $p=\ifrac{(2\sigma^{{\prime}}+\eta _{0}\eta _{{\infty}}\zeta)}{(2\eta _{0})},$

Symbols:: $\sigma$ : function and $p$ : momentum
Permalink:: http://dlmf.nist.gov/32.6.E31
Encodings:: TeX, pMML, png

are solutions of (32.6.25) and (32.6.26).

The Hamiltonian for $\mbox{P}_{{\mbox{\scriptsize III}}}$ with $\gamma=0$ is

32.6.32 $z\mathrm{H}_{{\mbox{\scriptsize III}}}(q,p,z)=q^{2}p^{2}+(\theta q-\kappa _{0}z)p-\kappa _{{\infty}}zq,$

Symbols:: $z$ : real, $q$ : coordinate, $\mathrm{H}(q,p,z)$ : Hamiltonian, $\theta$ : parameter, $p$ : momentum and $\kappa _{j}$ : parameters
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E32
Encodings:: TeX, pMML, png

and so

32.6.33 $zq^{{\prime}}=2q^{2}p+\theta q-\kappa _{0}z,$

Symbols:: $z$ : real, $q$ : coordinate, $\theta$ : parameter, $p$ : momentum and $\kappa _{j}$ : parameters
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E33
Encodings:: TeX, pMML, png

32.6.34 $zp^{{\prime}}=-2qp^{2}-\theta p+\kappa _{{\infty}}z.$

Symbols:: $z$ : real, $q$ : coordinate, $\theta$ : parameter, $p$ : momentum and $\kappa _{j}$ : parameters
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E34
Encodings:: TeX, pMML, png

Then $q=w$ satisfies $\mbox{P}_{{\mbox{\scriptsize III}}}$ with

32.6.35 $(\alpha,\beta,\gamma,\delta)=\left(2\kappa _{{\infty}},\kappa _{0}(\theta-1),0,-\kappa _{0}^{2}\right).$

Symbols:: $\alpha$ : arbitrary constant, $\beta$ : arbitrary constant, $\gamma$ : arbitrary constant, $\delta$ : arbitrary constant, $\theta$ : parameter and $\kappa _{j}$ : parameters
Permalink:: http://dlmf.nist.gov/32.6.E35
Encodings:: TeX, pMML, png

The function

32.6.36 $\sigma=z\mathrm{H}_{{\mbox{\scriptsize III}}}(q,p,z)+pq+\tfrac{1}{4}(\theta+1)^{2}$

Symbols:: $z$ : real, $q$ : coordinate, $\sigma$ : function, $\mathrm{H}(q,p,z)$ : Hamiltonian, $\theta$ : parameter and $p$ : momentum
Permalink:: http://dlmf.nist.gov/32.6.E36
Encodings:: TeX, pMML, png

defined by (32.6.32) satisfies

32.6.37 $(z\sigma^{{\prime\prime}}-\sigma^{{\prime}})^{2}+2(\sigma^{{\prime}})^{2}(z\sigma^{{\prime}}-2\sigma)-4\kappa _{0}\kappa _{{\infty}}(\theta+1)\theta _{{\infty}}z\sigma^{{\prime}}=4\kappa _{0}^{2}\kappa _{{\infty}}^{2}z^{2}.$

Symbols:: $z$ : real, $\sigma$ : function, $\theta$ : parameter and $\kappa _{j}$ : parameters
Referenced by:: §32.6(iv)
Permalink:: http://dlmf.nist.gov/32.6.E37
Encodings:: TeX, pMML, png

Conversely, if $\sigma$ is a solution of (32.6.37), then

32.6.38 $q=\ifrac{\kappa _{0}\left(z\sigma^{{\prime\prime}}-\theta\sigma^{{\prime}}+2\kappa _{0}\kappa _{{\infty}}z\right)}{(\sigma^{{\prime}})^{2}},$