§32.6 Hamiltonian Structure

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

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

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

and so

Then $q=w$ satisfies ${\text{P}}_{\text{I}}$. The function

defined by (32.6.2) satisfies

Conversely, if $\sigma$ is a solution of (32.6.6), then

are solutions of (32.6.3) and (32.6.4).

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

and so

Then $q=w$ satisfies ${\text{P}}_{\text{II}}$ and $p$ satisfies

The function $\sigma (z)={\mathrm{H}}_{\text{II}}(q,p,z)$ defined by (32.6.9) satisfies

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

are solutions of (32.6.10) and (32.6.11).

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

and so

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

The function

defined by (32.6.16) satisfies

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

are solutions of (32.6.17) and (32.6.18).

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

and so

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

The function

defined by (32.6.24) satisfies

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

are solutions of (32.6.25) and (32.6.26).

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

and so

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

The function

defined by (32.6.32) satisfies

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

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).