# §32.6 Hamiltonian Structure

## §32.6(i) Introduction

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

 32.6.1 $\displaystyle\frac{\mathrm{d}q}{\mathrm{d}z}$ $\displaystyle=\frac{\partial\mathrm{H}}{\partial p},$ $\displaystyle\frac{\mathrm{d}p}{\mathrm{d}z}$ $\displaystyle=-\frac{\partial\mathrm{H}}{\partial q},$

for suitable (non-autonomous) Hamiltonian functions $\mathrm{H}(q,p,z)$.

## §32.6(ii) First Painlevé Equation

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 Encodings: TeX, pMML, png See also: Annotations for §32.6(ii), §32.6 and Ch.32

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 See also: Annotations for §32.6(ii), §32.6 and Ch.32
 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 Encodings: TeX, pMML, png See also: Annotations for §32.6(ii), §32.6 and Ch.32

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 See also: Annotations for §32.6(ii), §32.6 and Ch.32

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 Encodings: TeX, pMML, png See also: Annotations for §32.6(ii), §32.6 and Ch.32

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 See also: Annotations for §32.6(ii), §32.6 and Ch.32
 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 See also: Annotations for §32.6(ii), §32.6 and Ch.32

are solutions of (32.6.3) and (32.6.4).

## §32.6(iii) Second Painlevé Equation

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 See also: Annotations for §32.6(iii), §32.6 and Ch.32

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 Encodings: TeX, pMML, png See also: Annotations for §32.6(iii), §32.6 and Ch.32
 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 Encodings: TeX, pMML, png See also: Annotations for §32.6(iii), §32.6 and Ch.32

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 Encodings: TeX, pMML, png See also: Annotations for §32.6(iii), §32.6 and Ch.32

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 See also: Annotations for §32.6(iii), §32.6 and Ch.32

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 See also: Annotations for §32.6(iii), §32.6 and Ch.32
 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 See also: Annotations for §32.6(iii), §32.6 and Ch.32

are solutions of (32.6.10) and (32.6.11).

## §32.6(iv) Third Painlevé Equation

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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

and so

 32.6.17 $\displaystyle zq^{\prime}$ $\displaystyle=2q^{2}p-\kappa_{\infty}zq^{2}-(2\theta_{0}+1)q+\kappa_{0}z,$ ⓘ 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 See also: Annotations for §32.6(iv), §32.6 and Ch.32 32.6.18 $\displaystyle zp^{\prime}$ $\displaystyle=-2qp^{2}+2\kappa_{\infty}zqp+(2\theta_{0}+1)p-\kappa_{\infty}(% \theta_{0}+\theta_{\infty})z.$ ⓘ 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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

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

 32.6.19 $(\alpha,\beta,\gamma,\delta)=\left(-2\kappa_{\infty}\theta_{\infty},2\kappa_{0% }(\theta_{0}+1),\kappa_{\infty}^{2},-\kappa_{0}^{2}\right).$

The function

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

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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

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 See also: Annotations for §32.6(iv), §32.6 and Ch.32
 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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

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 See also: Annotations for §32.6(iv), §32.6 and Ch.32
 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 Encodings: TeX, pMML, png See also: Annotations for §32.6(iv), §32.6 and Ch.32

Then $q=u$ satisfies $\mbox{P}^{\prime}_{\mbox{\scriptsize III}}$ with

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

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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

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 See also: Annotations for §32.6(iv), §32.6 and Ch.32
 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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

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 See also: Annotations for §32.6(iv), §32.6 and Ch.32
 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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

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

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

The function

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

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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

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

 32.6.38 $q=\ifrac{\kappa_{0}\left(z\sigma^{\prime\prime}-\theta\sigma^{\prime}+2\kappa_% {0}\kappa_{\infty}z\right)}{(\sigma^{\prime})^{2}},$ ⓘ Symbols: $z$: real, $q$: coordinate, $\sigma$: function, $\theta$: parameter and $\kappa_{j}$: parameters Permalink: http://dlmf.nist.gov/32.6.E38 Encodings: TeX, pMML, png See also: Annotations for §32.6(iv), §32.6 and Ch.32
 32.6.39 $p=\ifrac{\sigma^{\prime}}{(2\kappa_{0})},$ ⓘ Symbols: $\sigma$: function, $p$: momentum and $\kappa_{j}$: parameters Permalink: http://dlmf.nist.gov/32.6.E39 Encodings: TeX, pMML, png See also: Annotations for §32.6(iv), §32.6 and Ch.32

are solutions of (32.6.33) and (32.6.34).

## §32.6(v) Other Painlevé Equations

For Hamiltonian structure for $\mbox{P}_{\mbox{\scriptsize IV}}$ see Jimbo and Miwa (1981), Okamoto (1986); also Forrester and Witte (2001).

For Hamiltonian structure for $\mbox{P}_{\mbox{\scriptsize V}}$ see Jimbo and Miwa (1981), Okamoto (1987b); also Forrester and Witte (2002).

For Hamiltonian structure for $\mbox{P}_{\mbox{\scriptsize VI}}$ see Jimbo and Miwa (1981) and Okamoto (1987a); also Forrester and Witte (2004).