# Non-local Lagrangian mechanics: Noether’s theorem and Hamiltonian formalism

@article{Heredia2021NonlocalLM, title={Non-local Lagrangian mechanics: Noether’s theorem and Hamiltonian formalism}, author={Carlos Heredia and Josep Llosa}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2021}, volume={54} }

Lagrangian systems with a finite number of degrees of freedom that are non-local in time are studied. We obtain an extension of Noether’s theorem and Noether identities to this kind of Lagrangians. A Hamiltonian formalism has then been set up for these systems. n-order local Lagrangians can be treated as a particular case of non-local ones and standard results are recovered. The method is then applied to several other cases, namely two examples of non-local oscillators and the p-adic particle.

## 6 Citations

### Nonlocal Lagrangian fields: Noether’s theorem and Hamiltonian formalism

- Mathematics, PhysicsPhysical Review D
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This article aims to study non-local Lagrangians with an inﬁnite number of degrees of freedom. We obtain an extension of Noether’s theorem and Noether’s identities for such Lagrangians. We then set…

### Infinite-derivative linearized gravity in convolutional form

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We solve the non-local equation −e−` 2 φ = J for i) static scalar fields in static spacetimes and ii) time-dependent scalar fields in ultrastatic spacetimes. Corresponding equations are rewritten as…

### Ghost-free infinite-derivative dilaton gravity in two dimensions

- MathematicsPhysical Review D
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We present the ghost-free inﬁnite-derivative extensions of the Spherically-Reduced Gravity (SRG) and Callan-Giddings-Harvey-Strominger (CGHS) theories in two space-time dimensions. For the case of…

### Topological defects in nonlocal field theories

- MathematicsJournal of High Energy Physics
- 2022

In this paper we study for the first time topological defects in the context of nonlocal field theories in which Lagrangians contain infinite-order differential operators. In particular, we analyze…

## References

SHOWING 1-10 OF 27 REFERENCES

### Functional Lagrange formalism for time-non-local Lagrangians

- Physics, Mathematics
- 2012

We develop a time-non-local (TNL) formalism based on variational calculus, which allows for the analysis of TNL Lagrangians. We derive the generalized Euler-Lagrange equations starting from the…

### Hamiltonian formalism for nonlocal Lagrangians

- Physics
- 1994

A Hamiltonian formalism is set up for nonlocal Lagrangian systems. The method is based on obtaining an equivalent singular first order Lagrangian, which is processed according to the standard…

### Canonical formalism for Lagrangians with nonlocality of finite extent

- Physics, Mathematics
- 2000

I consider Lagrangians which depend nonlocally in time but in such a way that there is no mixing between times differing by more than some finite value {Delta}t. By considering these systems as the…

### Hamiltonian formalism for nonlocal

- Physics
- 1999

A Hamiltonian formalism is set up for nonlocal Lagrangian systems. The method is based on obtaining an equivalent singular first order Lagrangian, which is processed according to the standard…

### Nonlinear Dynamics Equation in p-Adic String Theory

- Mathematics
- 2003

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation…

### Towards understanding the ultraviolet behavior of quantum loops in infinite-derivative theories of gravity

- Physics
- 2014

In this paper we will consider quantum aspects of a non-local, infinite-derivative scalar field theory—a toy model depiction of a covariant infinite-derivative, non-local extension of Einstein’s…

### Ostrogradsky's theorem on Hamiltonian instability

- PhysicsScholarpedia
- 2015

Ostrogradsky's construction of a Hamiltonian formalism for nondegenerate higher derivative Lagrangians is reviewed and a discussion is given of the problems raised by attempts to evade this restriction.

### Dynamics with infinitely many derivatives: The Initial value problem

- Mathematics
- 2008

Differential equations of infinite order are an increasingly important class of equations in theoretical physics. Such equations are ubiquitous in string field theory and have recently attracted…

### Dynamics with Infinitely Many Time Derivatives and Rolling Tachyons

- Physics
- 2002

Both in string field theory and in p-adic string theory the equations of motion involve infinite number of time derivatives. We argue that the initial value problem is qualitatively different from…