Abstract
The study of the symmetry of Pais-Uhlenbeck oscillator initiated in [Nucl. Phys. B 885 (2014) 150] is continued with special emphasis put on the Hamiltonian formalism. The symmetry generators within the original Pais and Uhlenbeck Hamiltonian approach as well as the canonical transformation to the Ostrogradski Hamiltonian framework are derived. The resulting algebra of generators appears to be the central extension of the one obtained on the Lagrangian level; in particular, in the case of odd frequencies one obtains the centrally extended l-conformal Newton-Hooke algebra. In this important case the canonical transformation to an alternative Hamiltonian formalism (related to the free higher derivatives theory) is constructed. It is shown that all generators can be expressed in terms of the ones for the free theory and the result agrees with that obtained by the orbit method.
Highlights
The theories we are usually dealing with are Newtonian in the sense that the Lagrangian function depends on the first time derivatives only
There exists a few approaches to Hamiltonian formalism of the PU model: decomposition into the set of the independent harmonic oscillators proposed by Pais and Uhlenbeck in their original paper
On the Hamiltonian level, the form of generators and we show that they, form the algebra which is central extension the one appearing on the Lagrangian level
Summary
The theories we are usually dealing with are Newtonian in the sense that the Lagrangian function depends on the first time derivatives only. There exists a few approaches to Hamiltonian formalism of the PU model: decomposition into the set of the independent harmonic oscillators proposed by Pais and Uhlenbeck in their original paper [13], Ostrogradski approach based on the Ostrogradski method [29] of constructing Hamiltonian formalism for theories with higher time derivatives and the last one, applicable in the case of odd frequencies (mentioned above), which exhibits the l-conformal Newton–Hooke group structure of the model. The section is devoted to the case of odd frequencies where the additional natural approach can be constructed In this framework the Hamiltonian is the sum of the one for the free higher derivatives theory and the conformal generator. We derive there some relations and identities which are crucial for our work
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