Dynamical realizations of l-conformal Newton–Hooke group

Abstract

The method of nonlinear realizations and the technique previously developed in [Nucl. Phys. B 866 (2013) 212] are used to construct a dynamical system without higher derivative terms, which holds invariant under the l-conformal Newton-Hooke group. A configuration space of the model involves coordinates, which parametrize a particle moving in d spatial dimensions and a conformal mode, which gives rise to an effective external field. The dynamical system describes a generalized multi-dimensional oscillator, which undergoes accelerated/decelerated motion in an ellipse in accord with evolution of the conformal mode. Higher derivative formulations are discussed as well. It is demonstrated that the multi-dimensional Pais-Uhlenbeck oscillator enjoys the l = 3 2 -conformal Newton-Hooke symmetry for a particular choice of its frequencies.

ℓ-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras

“…In a series of papers [9,10,11,12,13] it was shown that the Pais-Uhlenbeck oscillators are invariant under the ℓ = 1 2 +N 0 CGAs. These systems, however, unlike the ℓ-oscillator (1), are defined by higher-derivatives equations.…”

Higher-derivative generalization of conformal mechanics

“…To obtain NH counterpart of the model (16), it follows to apply Niederer's coordinate transformation of the form…”

“…Recently, symmetries of these higher-derivative models have been extensively studied 13,14,[16][17][18][19][20][21][22][23] . In particular it has been shown that the l-conformal Galilei group is the maximal symmetry group of the free (2l + 1)-order particle 14 .…”

“…In particular it has been shown that the l-conformal Galilei group is the maximal symmetry group of the free (2l + 1)-order particle 14 . Similarly, the PU oscillator accommodates the l-conformal NH symmetry for a special choice of its oscillation frequencies 16,18,21 .…”

On dynamical realizations of l-conformal Galilei and Newton–Hooke algebras

“…2 First, dynamical realizations of these algebras constructed so far did not assign any clear physical meaning to the parameter l. Second, apart from the oscillator-like models coupled to external field [4][5][6], no interacting theory which exhibits such symmetries is known. Third, because a number of functionally independent integrals of motion needed to integrate a differential equation correlates with its order, dynamical realizations of the l-conformal Galilei/Newton-Hooke algebra in general involve higher derivative terms (see, e.g., [7][8][9][10][11][12][13] and the references therein).…”