Motion in a symmetric potential on the hyperbolic plane. (English) Zbl 1373.37134
The motion of a particle on the hyperbolic plane under the influence of a central potential \(U\) is an interesting and old problem. By equipping \(\mathbb R^3\) with the Lorenz inner product \(\langle, \rangle_L\), the authors obtain on \(\mathbb R^{2, 1}\) a pseudo-Riemannian metric and pull-back the canonical symplectic 2-form on \(T^* \mathbb R^{2,1}\).
They study the motion of a particle on:
\[ \mathbb{H}^2= \{ x \in \mathbb{R}^{2,1}: \langle x,x \rangle_L+1=0, \, x_3> 0\} \]
with the Hamiltonian
\[ H(x, y)= \frac{1}{2}\langle y, y\rangle_L + U(x). \]
By the principal axis theorem for the Lorenz group, the authors reduce the problem and discuss the situation for three subgroups: \(G_e \) (elliptic rotations), \(G_h\) (hyperbolic rotations), and \( G_p\) (parabolic rotations), which leave respectively the \(x_3\)-axis, \(x_1\)-axis and \(x_1=0, x_2= x_3\) invariant.
They study the motion of a particle on:
\[ \mathbb{H}^2= \{ x \in \mathbb{R}^{2,1}: \langle x,x \rangle_L+1=0, \, x_3> 0\} \]
with the Hamiltonian
\[ H(x, y)= \frac{1}{2}\langle y, y\rangle_L + U(x). \]
By the principal axis theorem for the Lorenz group, the authors reduce the problem and discuss the situation for three subgroups: \(G_e \) (elliptic rotations), \(G_h\) (hyperbolic rotations), and \( G_p\) (parabolic rotations), which leave respectively the \(x_3\)-axis, \(x_1\)-axis and \(x_1=0, x_2= x_3\) invariant.
Reviewer: Mohamed Selmi (Sousse-Riadh)
MSC:
37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |
70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |
37C80 | Symmetries, equivariant dynamical systems (MSC2010) |
34C14 | Symmetries, invariants of ordinary differential equations |
20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |