We observe that a wide class of higher-derivative systems admits a bounded integral of motion that ensures the classical stability of dynamics, while the canonical energy is unbounded.We use the concept of a Lagrange anchor to demonstrate that the bounded integral of motion is connected with the time-translation invariance. A procedure is suggested for switching on interactions in free higher-derivative systems without breaking their stability. We also demonstrate the quantization technique that keeps the higher-derivative dynamics stable at quantum level.The general construction is illustrated by the examples of the Pais-Uhlenbeck oscillator, higherderivative scalar field model, and the Podolsky electrodynamics. For all these models, the positive integrals of motion are explicitly constructed and the interactions are included such that keep the system stable.

Making use of the Lagrange anchor construction introduced earlier to quantize non-Lagrangian field theories, we extend the Noether theorem beyond the class of variational dynamics.

Abstract. Starting from the concept of involution of field equations, a universal method is proposed for constructing consistent interactions between the fields. The method equally well applies to the Lagrangian and non-Lagrangian equations and it is explicitly covariant.No auxiliary fields are introduced. The equations may have (or have no) gauge symmetry and/or second class constraints in Hamiltonian formalism, providing the theory admits a Hamiltonian description. In every case the method identifies all the consistent interactions.

We consider the class of higher derivative 3d vector field models with the field equation operator being a polynomial of the Chern-Simons operator. For the nth-order theory of this type, we provide a general recipe for constructing n-parameter family of conserved second rank tensors. The family includes the canonical energy-momentum tensor, which is unbounded, while there are bounded conserved tensors that provide classical stability of the system for certain combinations of the parameters in the Lagrangian. We also demonstrate the examples of consistent interactions which are compatible with the requirement of stability.

We consider the general higher derivative field theories of derived type. At free level, the wave operator of derived-type theory is a polynomial of the order n ≥ 2 of another operator W which is of the lower order. Every symmetry of W gives rise to the series of independent higher order symmetries of the field equations of derived system. In its turn, these symmetries give rise to the series of independent conserved quantities. In particular, the translation invariance of operator W results in the series of conserved tensors of the derived theory. The series involves n independent conserved tensors including canonical energymomentum. Even if the canonical energy is unbounded, the other conserved tensors in the series can be bounded, that will make the dynamics stable. The general procedure is worked out to switch on the interactions such that the stability persists beyond the free level. The stable interaction vertices are inevitably non-Lagrangian. The stable theory, however, can admit consistent quantization. The general construction is exemplified by the order N extension of Chern-Simons coupled to the Pais-Uhlenbeck-type higher derivative complex scalar field.