Md Fazlul HOQUE
Integrable and superintegrable classical systems in magnetic fields and Poisson algebras of their integrals of motion
The talk presents the construction of all nonstandard integrable systems in magnetic fields whose integrals have leading order structure that are elements of the universal enveloping algebra of the three-dimensional Euclidean algebra. We consider the natural Hamiltonian for a particle moving in the three dimensional Euclidean space
\begin{eqnarray}
H(\vec{x},\vec{p})=\frac{1}{2}\left(\vec{p}+\vec{A}(\vec{x})\right)^2+W(\vec{x}),\label{chm1}\nonumber
\end{eqnarray}
where $\vec{p}=(p_1,p_2,p_3)$ are components of the linear momentum and $\vec{x}=(x_1,x_2,x_3)\equiv (x,y,z)$ the Cartesian spatial coordinates, $\vec{A}(\vec{x}) =(A_1(\vec{x}), A_2(\vec{x}), A_3(\vec{x}))$ is the vector potential depending on the position vector $\vec{x}$ and $W(\vec{x})$ is the electrostatic potential function involving only the coordinates $\vec{x}$. We choose the units in which the mass of the particle has the numerical value 1 and the charge of the particle is $-1$.
The system assume to be integrable with a pair of integrals of motion $X_1$, $X_2$ which are quadratic polynomials in the momenta. The general form of the quadratic integrals of motion as
\begin{eqnarray}
&&X_1= l^A_3p_3^A+a(p^A_1)^2+bp^A_1p_2^A+\sum_{j=1}^3 s_{j}(\vec{x}) p_j^A+m(\vec{x}),\label{int1}\nonumber
\\&&
X_2=(p^A_3)^2+\sum_{j=1}^3 S_{j}(\vec{x})p^A_j+M(\vec{x}),\quad a,b\in \mathbb{R}, \label{int2}\nonumber
\end{eqnarray}
where the gauge covariant expressions were used, namely
\begin{eqnarray}
p_i^A=p_i+A_i(\vec{x}),\quad l_i^A=\sum_{j,k=1}^{3}\varepsilon_{ijk}x_jp^A_k, \quad {i=1,2,3}.\nonumber
\end{eqnarray}
Here $\varepsilon_{ijk}$ is the completely antisymmetric tensor with $\varepsilon_{123}=1$. {The functions $s_j(\vec x)$, $S_j(\vec x)$, $m(\vec x)$ and $M(\vec x)$ are to be determined by requiring that $X_i$, $i=1,2$ {are} commute with the Hamiltonian $H$ as well as with each other. We show how these pairs of commuting elements lead to distinct independent integrals of motion in several nonvanishing magnetic fields. We also search for additional first- and second-order integrals of motion of these systems to arrive at superintegrable systems. We construct the corresponding Poisson algebras of integrals of motion. The separability of the systems are verified in various coordinates by the Levi-Civita condition.
The talk is based on joint work with Libor \v{S}nobl and Antonella Marchesiello, Czech Technical University in Prague, Czech Republic.
