§32.6 Hamiltonian Structure

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

32.6.1		${\displaystyle \frac{dq}{dz}}$	$={\displaystyle \frac{\partial \mathrm{H}}{\partial p}},$
		${\displaystyle \frac{dp}{dz}}$	$=-{\displaystyle \frac{\partial \mathrm{H}}{\partial q}},$
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $z$: real, $q$: coordinate, $p$: momentum and $\mathrm{H}(q,p,z)$: Hamiltonian Permalink: http://dlmf.nist.gov/32.6.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §32.6(i), §32.6 and Ch.32

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

The Hamiltonian for ${\text{P}}_{\text{I}}$ is

32.6.2		$${\mathrm{H}}_{\text{I}}(q,p,z)=\frac{1}{2}{p}^2-2q^3-zq,$$
ⓘ 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 ${\text{P}}_{\text{I}}$. The function

32.6.5		$$\sigma ={\mathrm{H}}_{\text{I}}(q,p,z),$$
ⓘ 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).

The Hamiltonian for ${\text{P}}_{\text{II}}$ is

32.6.9		$${\mathrm{H}}_{\text{II}}(q,p,z)=\frac{1}{2}{p}^2-(q^2+\frac{1}{2}z)p-(\alpha +\frac{1}{2})q,$$
ⓘ 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		$$q^{\prime }=p-q^2-\frac{1}{2}z,$$
ⓘ 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		$$p^{\prime }=2qp+\alpha +\frac{1}{2}.$$
ⓘ 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 ${\text{P}}_{\text{II}}$ and $p$ satisfies

32.6.12		$$p{p}^{\prime \prime }=\frac{1}{2}{(p^{\prime })}^2+2p^3-z{p}^2-\frac{1}{2}{(\alpha +\frac{1}{2})}^2.$$
ⓘ 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 $\sigma (z)={\mathrm{H}}_{\text{II}}(q,p,z)$ defined by (32.6.9) satisfies

32.6.13		$${\left({\sigma }^{\prime \prime }\right)}^2+4{\left({\sigma }^{\prime }\right)}^3+2{\sigma }^{\prime }\left(z{\sigma }^{\prime }-\sigma \right)=\frac{1}{4}{(\alpha +\frac{1}{2})}^2.$$
ⓘ 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		$$q=(4{\sigma }^{\prime \prime }+2\alpha +1)/(8{\sigma }^{\prime }),$$
ⓘ 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).

The Hamiltonian for ${\text{P}}_{\text{III}}$ is

32.6.16		$$z{\mathrm{H}}_{\text{III}}(q,p,z)=q^2{p}^2-\left({\kappa }_{\mathrm{\infty }}z{q}^2+(2{\theta }_0+1)q-{\kappa }_{0}z\right)p+{\kappa }_{\mathrm{\infty }}({\theta }_0+{\theta }_{\mathrm{\infty }})zq,$$
ⓘ 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		$z{q}^{\prime }$	$=2q^{2}p-{\kappa }_{\mathrm{\infty }}z{q}^2-(2{\theta }_0+1)q+{\kappa }_{0}z,$
ⓘ 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		$z{p}^{\prime }$	$=-2q{p}^2+2{\kappa }_{\mathrm{\infty }}zqp+(2{\theta }_0+1)p-{\kappa }_{\mathrm{\infty }}({\theta }_0+{\theta }_{\mathrm{\infty }})z.$
ⓘ 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 ${\text{P}}_{\text{III}}$ with

32.6.19		$$(\alpha ,\beta ,\gamma ,\delta )=(-2{\kappa }_{\mathrm{\infty }}{\theta }_{\mathrm{\infty }},2{\kappa }_0({\theta }_0+1),{\kappa }_{\mathrm{\infty }}^2,-{\kappa }_0^2).$$
ⓘ 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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

The function

32.6.20		$$\sigma =z{\mathrm{H}}_{\text{III}}(q,p,z)+pq+{\theta }_0^2-\frac{1}{2}{\kappa }_0{\kappa }_{\mathrm{\infty }}{z}^2$$
ⓘ 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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

defined by (32.6.16) satisfies

32.6.21		$$\begin{array}{l}{(z{\sigma }^{\prime \prime }-{\sigma }^{\prime })}^2+2\left({({\sigma }^{\prime })}^2-{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2{z}^2\right)(z{\sigma }^{\prime }-2\sigma )+8{\kappa }_0{\kappa }_{\mathrm{\infty }}{\theta }_0{\theta }_{\mathrm{\infty }}z{\sigma }^{\prime }\\ =4{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2({\theta }_0^2+{\theta }_{\mathrm{\infty }}^2)z^2.\end{array}$$
ⓘ 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		$$q=\frac{{\kappa }_0\left(z{\sigma }^{\prime \prime }-(2{\theta }_0+1){\sigma }^{\prime }+2{\kappa }_0{\kappa }_{\mathrm{\infty }}{\theta }_{\mathrm{\infty }}z\right)}{{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2{z}^2-{({\sigma }^{\prime })}^2},$$
ⓘ 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		$$p=({\sigma }^{\prime }+{\kappa }_0{\kappa }_{\mathrm{\infty }}z)/(2{\kappa }_0),$$
ⓘ 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 ${\text{P}}_{\text{III}}^{\prime }$ (§32.2(iii)) is

32.6.24		$$\zeta {\mathrm{H}}_{\text{III}}(q,p,\zeta )=q^2{p}^2-\left({\eta }_{\mathrm{\infty }}{q}^2+{\theta }_{0}q-{\eta }_0\zeta \right)p+\frac{1}{2}{\eta }_{\mathrm{\infty }}({\theta }_0+{\theta }_{\mathrm{\infty }})q,$$
ⓘ 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		$$\zeta q^{\prime }=2q^{2}p-{\eta }_{\mathrm{\infty }}{q}^2-{\theta }_{0}q+{\eta }_0\zeta ,$$
ⓘ 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		$$\zeta p^{\prime }=-2q{p}^2+2{\eta }_{\mathrm{\infty }}qp+{\theta }_{0}p-\frac{1}{2}{\eta }_{\mathrm{\infty }}({\theta }_0+{\theta }_1).$$
ⓘ 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 ${\text{P}}_{\text{III}}^{\prime }$ with

32.6.27		$$(\alpha ,\beta ,\gamma ,\delta )=(-4{\eta }_{\mathrm{\infty }}{\theta }_{\mathrm{\infty }},4{\eta }_0({\theta }_0+1),4{\eta }_{\mathrm{\infty }}^2,-4{\eta }_0^2).$$
ⓘ 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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

The function

32.6.28		$$\sigma =\zeta {\mathrm{H}}_{\text{III}}(q,p,\zeta )+\frac{1}{4}{\theta }_0^2-\frac{1}{2}{\eta }_0{\eta }_{\mathrm{\infty }}\zeta$$
ⓘ 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		$${\zeta }^2{({\sigma }^{\prime \prime })}^2+\left(4{({\sigma }^{\prime })}^2-{\eta }_0^2{\eta }_{\mathrm{\infty }}^2\right)(\zeta {\sigma }^{\prime }-\sigma )+{\eta }_0{\eta }_{\mathrm{\infty }}{\theta }_0{\theta }_{\mathrm{\infty }}{\sigma }^{\prime }=\frac{1}{4}{\eta }_0^2{\eta }_{\mathrm{\infty }}^2({\theta }_0^2+{\theta }_{\mathrm{\infty }}^2).$$
ⓘ 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		$$q=\frac{{\eta }_0\left(\zeta {\sigma }^{\prime \prime }-2{\theta }_0{\sigma }^{\prime }+{\eta }_0{\eta }_{\mathrm{\infty }}{\theta }_{\mathrm{\infty }}\right)}{{\eta }_0^2{\eta }_{\mathrm{\infty }}^2-4{({\sigma }^{\prime })}^2},$$
ⓘ 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		$$p=(2{\sigma }^{\prime }+{\eta }_0{\eta }_{\mathrm{\infty }}\zeta )/(2{\eta }_0),$$
ⓘ 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 ${\text{P}}_{\text{III}}$ with $\gamma =0$ is

32.6.32		$$z{\mathrm{H}}_{\text{III}}(q,p,z)=q^2{p}^2+(\theta q-{\kappa }_{0}z)p-{\kappa }_{\mathrm{\infty }}zq,$$
ⓘ 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		$$z{q}^{\prime }=2q^{2}p+\theta q-{\kappa }_{0}z,$$
ⓘ 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		$$z{p}^{\prime }=-2q{p}^2-\theta p+{\kappa }_{\mathrm{\infty }}z.$$
ⓘ 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 ${\text{P}}_{\text{III}}$ with

32.6.35		$$(\alpha ,\beta ,\gamma ,\delta )=(2{\kappa }_{\mathrm{\infty }},{\kappa }_0(\theta -1),0,-{\kappa }_0^2).$$
ⓘ 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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

The function

32.6.36		$$\sigma =z{\mathrm{H}}_{\text{III}}(q,p,z)+pq+\frac{1}{4}{(\theta +1)}^2$$
ⓘ 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 See also: Annotations for §32.6(iv), §32.6 and Ch.32

defined by (32.6.32) satisfies

32.6.37		$${(z{\sigma }^{\prime \prime }-{\sigma }^{\prime })}^2+2{({\sigma }^{\prime })}^2(z{\sigma }^{\prime }-2\sigma )-4{\kappa }_0{\kappa }_{\mathrm{\infty }}(\theta +1){\theta }_{\mathrm{\infty }}z{\sigma }^{\prime }=4{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2{z}^2.$$
ⓘ 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		$$q={\kappa }_0\left(z{\sigma }^{\prime \prime }-\theta {\sigma }^{\prime }+2{\kappa }_0{\kappa }_{\mathrm{\infty }}z\right)/{({\sigma }^{\prime })}^2,$$
ⓘ 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		$$p={\sigma }^{\prime }/(2{\kappa }_0),$$
ⓘ 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).

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

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

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