Carregant...
Fitxers
Tipus de document
TesiVersió
Versió publicadaData de publicació
Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/197900
Nonlocal Lagrangian Formalism
Títol de la revista
Autors
ISSN de la revista
Títol del volum
Resum
[eng] This thesis aims to study nonlocal Lagrangians with a finite and 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 up a Hamiltonian formalism for them. Furthermore, we show that rth-order Lagrangians can be treated as a particular case, and the expected results are recovered. Finally, the method developed is applied to different examples: nonlocal harmonic oscillator, p-adic particle, p-adic open string field, and electrodynamics of dispersive media.
We divide this thesis into three blocks:
In the first block, we review the Lagrangian and Hamiltonian formalism for first-order and rth-order theories with a finite and an infinite number of degrees of freedom. In addition, we examine Noether’s first and second theorem for such Lagrangians, and we show that, by adding a total derivative to them, their equations of motion remain unchanged. For the particular case of rth-order Lagrangians with an infinite number of degrees of freedom, we focus on the Poincaré symmetry. We find the corresponding canonical energy-momentum tensor and the spin current. Furthermore, we indicate how to obtain the so-called Belinfante-Rosenfeld energy-momentum tensor. Finally, we discuss how Lagrangian and Hamiltonian formalisms are affected by extending them to theories of infinite order.
In the second block, we present the nonlocal Lagrangian formalism for a finite and an infinite number of degrees of freedom. We begin by introducing what we mean by time evolution. We then establish the principle of least action and obtain their equations of motion. Furthermore, we show that every nonlocal Lagrangian can be written as a nonlocal total derivative (or quadri-divergence for the case of an infinite number of degrees of freedom). Next, we demonstrate that its equations of motion are not identically zero and give an example to visualize this fact. We then find a sufficient asymptotic condition so that the equations of motion are unaffected by introducing a nonlocal total derivative (or quadri-divergence).
We extend Noether's first and second theorems for such Lagrangians. For the case of nonlocal Lagrangians with an infinite number of degrees of freedom, as in the first block, we focus on the Poincaré symmetry and obtain the canonical energy-momentum tensor and the spin current. Thanks to this extension, it allows us to politely infer (or propose) (based on the analogy of the local case) a Lagrange transformation to construct a Hamiltonian formalism for them.
Finally, in the third block, we exemplify the formalism developed. To do so, we choose four examples: nonlocal harmonic oscillator, p-adic particle, p-adic open string field, and electrodynamics of dispersive media. In the first two, we obtain the Hamiltonian and the symplectic form. As for the example of the p-adic open string field, we obtain the Hamiltonian, the symplectic form, and the components of the canonical and Belinfante-Rosenfeld energy-momentum tensors. For the latter, only the components of the canonical and Belinfante-Rosenfeld energy-momentum tensors.
Descripció
Matèries (anglès)
Citació
Citació
HEREDIA PIMIENTA, Carlos. Nonlocal Lagrangian Formalism. [consulta: 2 de desembre de 2025]. [Disponible a: https://hdl.handle.net/2445/197900]