# Healthy degenerate theories with higher derivatives

## Abstract

In the context of classical mechanics, we study the conditions under which higher-order derivative theories can evade the so-called Ostrogradsky instability. More precisely, we consider general Lagrangians with second order time derivatives, of the form L(ϕ{sup ¨a}, ϕ-dot {sup a},ϕ{sup a}; q-dot {sup i},q{sup i}) with a=1,⋯,n and i=1,⋯,m. For n=1, assuming that the q{sup i}’s form a nondegenerate subsystem, we confirm that the degeneracy of the kinetic matrix eliminates the Ostrogradsky instability. The degeneracy implies, in the Hamiltonian formulation of the theory, the existence of a primary constraint, which generates a secondary constraint, thus eliminating the Ostrogradsky ghost. For n>1, we show that, in addition to the degeneracy of the kinetic matrix, one needs to impose extra conditions to ensure the presence of a sufficient number of secondary constraints that can eliminate all the Ostrogradsky ghosts. When these conditions that ensure the disappearance of the Ostrogradsky instability are satisfied, we show that the Euler-Lagrange equations, which involve a priori higher order derivatives, can be reduced to a second order system.

- Authors:

- Kavli Institute for Cosmological Physics, The University of Chicago,Chicago, Illinois 60637 (United States)
- Laboratoire de Mathématiques et Physique Théorique,Université François Rabelais,Parc de Grandmont, 37200 Tours (France)
- (France)
- Research Center for the Early Universe (RESCEU), Graduate School of Science,The University of Tokyo, Tokyo 113-0033 (Japan)
- Department of Physics, Tokyo Institute of Technology,2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551 (Japan)
- Laboratoire APC - Astroparticule et Cosmologie,Université Paris Diderot Paris 7,75013 Paris (France)

- Publication Date:

- Sponsoring Org.:
- SCOAP3, CERN, Geneva (Switzerland)

- OSTI Identifier:
- 22572120

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2016; Journal Issue: 07; Other Information: PUBLISHER-ID: JCAP07(2016)033; OAI: oai:repo.scoap3.org:16499; cc-by Article funded by SCOAP3. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CLASSICAL MECHANICS; COSMOLOGICAL INFLATION; HAMILTONIANS; INFLATIONARY UNIVERSE; INSTABILITY; LAGRANGE EQUATIONS; LAGRANGIAN FUNCTION; LIMITING VALUES; MATRICES; NONLUMINOUS MATTER

### Citation Formats

```
Motohashi, Hayato, Noui, Karim, Laboratoire APC - Astroparticule et Cosmologie,Université Paris Diderot Paris 7,75013 Paris, Suyama, Teruaki, Yamaguchi, Masahide, and Langlois, David.
```*Healthy degenerate theories with higher derivatives*. United States: N. p., 2016.
Web. doi:10.1088/1475-7516/2016/07/033.

```
Motohashi, Hayato, Noui, Karim, Laboratoire APC - Astroparticule et Cosmologie,Université Paris Diderot Paris 7,75013 Paris, Suyama, Teruaki, Yamaguchi, Masahide, & Langlois, David.
```*Healthy degenerate theories with higher derivatives*. United States. doi:10.1088/1475-7516/2016/07/033.

```
Motohashi, Hayato, Noui, Karim, Laboratoire APC - Astroparticule et Cosmologie,Université Paris Diderot Paris 7,75013 Paris, Suyama, Teruaki, Yamaguchi, Masahide, and Langlois, David. Wed .
"Healthy degenerate theories with higher derivatives". United States.
doi:10.1088/1475-7516/2016/07/033.
```

```
@article{osti_22572120,
```

title = {Healthy degenerate theories with higher derivatives},

author = {Motohashi, Hayato and Noui, Karim and Laboratoire APC - Astroparticule et Cosmologie,Université Paris Diderot Paris 7,75013 Paris and Suyama, Teruaki and Yamaguchi, Masahide and Langlois, David},

abstractNote = {In the context of classical mechanics, we study the conditions under which higher-order derivative theories can evade the so-called Ostrogradsky instability. More precisely, we consider general Lagrangians with second order time derivatives, of the form L(ϕ{sup ¨a}, ϕ-dot {sup a},ϕ{sup a}; q-dot {sup i},q{sup i}) with a=1,⋯,n and i=1,⋯,m. For n=1, assuming that the q{sup i}’s form a nondegenerate subsystem, we confirm that the degeneracy of the kinetic matrix eliminates the Ostrogradsky instability. The degeneracy implies, in the Hamiltonian formulation of the theory, the existence of a primary constraint, which generates a secondary constraint, thus eliminating the Ostrogradsky ghost. For n>1, we show that, in addition to the degeneracy of the kinetic matrix, one needs to impose extra conditions to ensure the presence of a sufficient number of secondary constraints that can eliminate all the Ostrogradsky ghosts. When these conditions that ensure the disappearance of the Ostrogradsky instability are satisfied, we show that the Euler-Lagrange equations, which involve a priori higher order derivatives, can be reduced to a second order system.},

doi = {10.1088/1475-7516/2016/07/033},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 07,

volume = 2016,

place = {United States},

year = {Wed Jul 20 00:00:00 EDT 2016},

month = {Wed Jul 20 00:00:00 EDT 2016}

}