Ostrogradsky's theorem on Hamiltonian instability

By Richard P. Woodard. Scholarpedia, 2015

563 citations recorded by the source corpus.

Ostrogradsky's construction of a Hamiltonian formalism for nondegenerate higher derivative Lagrangians is reviewed. The resulting instability imposes by far the most powerful restriction on fundamental, interacting, continuum Lagrangian field theories. A discussion is given of the problems raised by attempts to evade this restriction.

Subfields: Astronomy and Astrophysics, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/256475a2c244bcaad6d8bcc05651c25b5cd28234

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In Astronomy and Astrophysics

In Atomic and Molecular Physics, and Optics

In Statistical and Nonlinear Physics

Citations in the corpus

Dark Energy in Light of Multi-Messenger Gravitational-Wave Astronomy (2018, background): “The classification is motivated by Ostrogradski’s theorem, which states that theories with second and higher (time) derivatives in the action generically introduce unstable degrees of freedom (Ostrogradski, 1850; Woodard, 2015).”
Dark energy and modified gravity in degenerate higher-order scalar–tensor (DHOST) theories: A review (2018, background): “This is the characteristic signature of Ostrogradski’s instability [2].”
Horndeski theory and beyond: a review (2019, background): “One of the guiding principles we follow when we seek for a ‘healthy’ extension of general relativity is to avoid what is called the Ostrogradsky instability [9, 10].”
Born–Infeld inspired modifications of gravity (2017, background): “ghosts owing to the Ostrogradski instability associated to higher order equations of motion [366, 367].”
Effective description of higher-order scalar-tensor theories (2017, background): “1), which depends on second-order derivatives of φ, contain two tensor modes and two scalar modes, one of which is associated with a so-called Ostrogradsky instability [55, 56].”
Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order (2016, unknown): “Horndenski [7] analysed the most general actions for scalar-tensor theories which lead to second order equations of motion (EOMs), and avoid Ostrogradsky instabilities [8, 9].”
An Asymptotically Safe Guide to Quantum Gravity and Matter (2018, background): “Higher-order derivatives in QFTs on a flat background generically imply an instability in the kinetic term (Ostrogradsky, 1985; Woodard, 2015), translating into a violation of reflection positivity for the Euclidean propagator (Arici et al., 2017).”
Quadratic Gravity (2018, background): “However, this leads to non-normalizable wave functions [56, 57].”
A systematic approach to generalisations of General Relativity and their cosmological implications (2018, background): “As can clearly be seen, the equations of motion are at most second order in the derivatives, avoiding the Ostrogradski instability [94, 95].”