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Spinning top in quadratic potential and matrix dressing chain

V. E. Adler, A. P. Veselov

TL;DR

This work recasts the classical problem of a spinning top in a Newtonian field with a quadratic potential as a period-one closure of a matrix Darboux dressing chain for matrix Schrödinger operators. The authors derive a GL(d) extension of the top dynamics via the matrix system $CF'+F'C=[C,F^2+B]+2α C$, $B'=[B,F]$, relate it to finite-gap spectral theory, and show that for α=0 the reduction yields a matrix Dubrovin-type integrable system; in the 2×2 case they obtain explicit solutions in terms of elliptic functions (α=0) or Painlevé II/IV transcendents (α≠0) and analyze the corresponding spectral problems. They further connect the dressing chain to the matrix KdV hierarchy and Novikov equations, highlighting a rich noncommutative integrable structure and spectral-characterization of the associated matrix Schrödinger operators.

Abstract

We show that the equations of motion of the rigid body about a fixed point in the Newtonian field with a quadratic potential are special reduction of period-one closure of the Darboux dressing chain for the Schrödinger operators with matrix potentials. Some new explicit solutions of the corresponding matrix system and the spectral properties of the related Schrödinger operators are discussed.

Spinning top in quadratic potential and matrix dressing chain

TL;DR

This work recasts the classical problem of a spinning top in a Newtonian field with a quadratic potential as a period-one closure of a matrix Darboux dressing chain for matrix Schrödinger operators. The authors derive a GL(d) extension of the top dynamics via the matrix system , , relate it to finite-gap spectral theory, and show that for α=0 the reduction yields a matrix Dubrovin-type integrable system; in the 2×2 case they obtain explicit solutions in terms of elliptic functions (α=0) or Painlevé II/IV transcendents (α≠0) and analyze the corresponding spectral problems. They further connect the dressing chain to the matrix KdV hierarchy and Novikov equations, highlighting a rich noncommutative integrable structure and spectral-characterization of the associated matrix Schrödinger operators.

Abstract

We show that the equations of motion of the rigid body about a fixed point in the Newtonian field with a quadratic potential are special reduction of period-one closure of the Darboux dressing chain for the Schrödinger operators with matrix potentials. Some new explicit solutions of the corresponding matrix system and the spectral properties of the related Schrödinger operators are discussed.

Paper Structure

This paper contains 6 sections, 3 theorems, 107 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motion of the top in Newtonian field with quadratic potential
  3. Matrix Darboux transformations and dressing chain
  4. Periodic closure of matrix dressing chain and spinning top
  5. Solutions of 1-periodic dressing chain in 2x2 matrix case
  6. Matrix KdV hierarchy and Novikov equation

Key Result

Theorem 1

The matrix dressing chain (fbtop) with $\alpha=0$ and generic $c$ is integrable in terms of $\theta$-functions of the Jacobi variety $J(\mathcal{C}).$ The corresponding Schrödinger operator $L=-D^2_x+U(x)$ is finite-gap with the spectral curve $\mathcal{C}.$

Figures (3)

  • Figure 1: Characteristic curves and spectral bands
  • Figure 2: Components $U_{11}$ and $U_{12}$ of the matrix potential (\ref{['pot2']}), for $\alpha=1$, $k=0.5$, $\gamma=0.05$ and $\omega_1-\omega_0=0$.
  • Figure 3: Real parts of solutions of equation (\ref{['phieq']}), for $k=0.5$ and $\gamma=0.05$, with initial data $(\varphi(0),\varphi'(0))$ equal to $(1,0)$ and $(0,1)$ (top), $(i,0)$ and $(0,i)$ (bottom).

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof