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An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps

Giorgio Gubbiotti, David McLaren, G. R. W. Quispel

Abstract

We show how to construct in an elementary way the invariant of the KHK discretisation of a cubic Hamiltonian system in two dimensions. That is, we show that this invariant is expressible as the product of the ratios of affine polynomials defining the prolongation of the three parallel sides of a hexagon. On the vertices of such a hexagon lie the indeterminacy points of the KHK map. This result is obtained analysing the structure of the singular fibres of the known invariant. We apply this construction to several examples, and we prove that a similar result holds true for a case outside the hypotheses of the main theorem, leading us to conjecture that further extensions are possible.

An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps

Abstract

We show how to construct in an elementary way the invariant of the KHK discretisation of a cubic Hamiltonian system in two dimensions. That is, we show that this invariant is expressible as the product of the ratios of affine polynomials defining the prolongation of the three parallel sides of a hexagon. On the vertices of such a hexagon lie the indeterminacy points of the KHK map. This result is obtained analysing the structure of the singular fibres of the known invariant. We apply this construction to several examples, and we prove that a similar result holds true for a case outside the hypotheses of the main theorem, leading us to conjecture that further extensions are possible.
Paper Structure (14 sections, 5 theorems, 84 equations, 7 figures)

This paper contains 14 sections, 5 theorems, 84 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Preliminaries
  4. Main theorem and its proof
  5. Examples
  6. Henon-Heiles potential
  7. The most general factorisable example
  8. A non-factorisable example
  9. Quadratic irreducible Hamiltonians and non-convex hexagons
  10. The degenerate case: the conic curve
  11. Conclusions
  12. Explicit form of the coefficients in eq.\ref{['eq:CDexpl']}
  13. Explicit form of coefficients in equation \ref{['eq:h3eq']}
  14. Explicit form of the polynomials in equation \ref{['eq:pprime']}

Key Result

Theorem 1.1

Consider a Hamiltonian vector field eq:firstordham with cubic Hamiltonian i.e. the function $H$ is a degree 3 polynomial. Then its KHK discretisation eq:kahan2 is birational and admits the following invariant: Moreover the KHK discretisation eq:kahan preserves the following measure on $\mathbb{R}^{n}$:

Figures (7)

  • Figure 1: An example of pencil \ref{['eq:pencilCD']} with $b_{1}=-b_{4}=2$, $b_{2}=b_{3}=-b_{5}=-b_{6}=1$$\mu_{1}=0$,$\mu_{1}=1$, and $\mu_{2}=-1$. In red and purple are shown the two singular curves factorised in three lines. The base points are highlighted in black.
  • Figure 2: The HH case \ref{['eq:hheqd']} with $h=1/2$: the lines $\overline{B_{1}B_{2}}$, $\overline{B_{3}B_{4}}$, and $\overline{B_{5}B_{6}}$ in red, the lines $\overline{B_{2}B_{3}}$, $\overline{B_{4}B_{5}}$, and $\overline{B_{6}B_{1}}$ in blue, and the $h$ independent lines $\overline{B_{1}B_{4}}$, $\overline{B_{2}B_{5}}$, and $\overline{B_{3}B_{6}}$ in green. The HH tringle HenonHeiles1964 is visualised in light green, while the circle \ref{['eq:hhcicle']} is drawn in purple.
  • Figure 3: The general factorisable case \ref{['eq:facteqd']} with $h=1/10$, $x_0=1$, $y_0=1/2$, $A=3$, $B=4$, and $C=5$: the lines $\overline{B_{1}B_{2}}$, $\overline{B_{3}B_{4}}$, and $\overline{B_{5}B_{6}}$ in red, the lines $\overline{B_{2}B_{3}}$, $\overline{B_{4}B_{5}}$, and $\overline{B_{6}B_{1}}$ in blue, and the $h$ independent lines $\overline{B_{1}B_{4}}$, $\overline{B_{2}B_{5}}$, and $\overline{B_{3}B_{6}}$ in green. The analog of the HH triangle is visualised in light green, while the ellipse \ref{['eq:factell']} is drawn in purple.
  • Figure 4: The level curves $H=\varepsilon$ with $H$ given by equation \ref{['eq:nonfact']} and 32 different values of $\varepsilon$. It it possible to note that there is only a linear factor (the line $y=0$) and that the base points are pushed to the line at infinity in $\mathbb{CP}^{2}$.
  • Figure 5: The non-factorisable case \ref{['eq:nonfacteqd']} with $h=1/5$: the lines $\overline{B_{1}B_{2}}$, $\overline{B_{3}B_{4}}$, and $\overline{B_{5}B_{6}}$ in red, the lines $\overline{B_{2}B_{3}}$, $\overline{B_{4}B_{5}}$, and $\overline{B_{6}B_{1}}$ in blue, and the singular pencil $p_{1,s}=0$ in green. Finally, the ellipse \ref{['eq:nonfactell']} is displayed in purple.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Theorem 1.1: CelledoniMcLachlanOwrenQuispel2013
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 2.7: PSS2019
  • Remark 2.8
  • ...and 7 more