### Abstract

The Schwarzian derivative is invariant under SL(2,R)-transformations and, as thus, any function of it can be used to determine the equation of motion or the Lagrangian density of a higher derivative SL(2,R)-invariant 1d mechanics or the Schwarzian mechanics for short. In this note, we consider the simplest variant which results from setting the Schwarzian derivative to be equal to a dimensionful coupling constant. It is shown that the corresponding dynamical system in general undergoes stable evolution but for one fixed point solution which is only locally stable. Conserved charges associated with the SL(2,R)-symmetry transformations are constructed and a Hamiltonian formulation reproducing them is proposed. An embedding of the Schwarzian mechanics into a larger dynamical system associated with the geodesics of a Brinkmann-like metric obeying the Einstein equations is constructed.

Original language | English |
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Pages (from-to) | 661-667 |

Number of pages | 7 |

Journal | Nuclear Physics B |

Volume | 936 |

DOIs | |

Publication status | Published - 1 Nov 2018 |

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### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

**A variant of Schwarzian mechanics.** / Galajinsky, Anton.

Research output: Contribution to journal › Article

*Nuclear Physics B*, vol. 936, pp. 661-667. https://doi.org/10.1016/j.nuclphysb.2018.10.004

}

TY - JOUR

T1 - A variant of Schwarzian mechanics

AU - Galajinsky, Anton

PY - 2018/11/1

Y1 - 2018/11/1

N2 - The Schwarzian derivative is invariant under SL(2,R)-transformations and, as thus, any function of it can be used to determine the equation of motion or the Lagrangian density of a higher derivative SL(2,R)-invariant 1d mechanics or the Schwarzian mechanics for short. In this note, we consider the simplest variant which results from setting the Schwarzian derivative to be equal to a dimensionful coupling constant. It is shown that the corresponding dynamical system in general undergoes stable evolution but for one fixed point solution which is only locally stable. Conserved charges associated with the SL(2,R)-symmetry transformations are constructed and a Hamiltonian formulation reproducing them is proposed. An embedding of the Schwarzian mechanics into a larger dynamical system associated with the geodesics of a Brinkmann-like metric obeying the Einstein equations is constructed.

AB - The Schwarzian derivative is invariant under SL(2,R)-transformations and, as thus, any function of it can be used to determine the equation of motion or the Lagrangian density of a higher derivative SL(2,R)-invariant 1d mechanics or the Schwarzian mechanics for short. In this note, we consider the simplest variant which results from setting the Schwarzian derivative to be equal to a dimensionful coupling constant. It is shown that the corresponding dynamical system in general undergoes stable evolution but for one fixed point solution which is only locally stable. Conserved charges associated with the SL(2,R)-symmetry transformations are constructed and a Hamiltonian formulation reproducing them is proposed. An embedding of the Schwarzian mechanics into a larger dynamical system associated with the geodesics of a Brinkmann-like metric obeying the Einstein equations is constructed.

UR - http://www.scopus.com/inward/record.url?scp=85054905133&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054905133&partnerID=8YFLogxK

U2 - 10.1016/j.nuclphysb.2018.10.004

DO - 10.1016/j.nuclphysb.2018.10.004

M3 - Article

VL - 936

SP - 661

EP - 667

JO - Nuclear Physics B

JF - Nuclear Physics B

SN - 0550-3213

ER -