Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)4999310.22363/2658-4670-2026-34-1-113-124UPXGCSPhysics and AstronomyФизикаResearch ArticleInteraction of relativistic electrons with intense electromagnetic fields: ponderomotive\,effect, acceleration,\,refraction,\,reflection, dependence\,on\,initial\,conditionsВзаимодействие релятивистских электронов с интенсивными электромагнитными полями: пондеромоторные эффекты, ускорение, преломление, отражение и зависимость от начальных условийhttps://orcid.org/0000-0003-0001-8764CastilloAlejandro J.КастильоА. Х.

PhD Student of Institute of Physical Research and Technology, RUDN University; Junior Research Fellow of Р.N. Lebedev Physical Institute of the Russian Academy of Sciences (LPI); Lecturer of Department of Higher Mathematics of State Autonomous Educational Institution of Higher Education “N.I. Pirogov Russian National Research Medical University” of Ministry of Health of Russian Federation

114222068@rudn.ruhttps://orcid.org/0000-0002-7130-4859RudoyYuriy Gr.РудойЮ. Г.

Professor of Institute of Physical Research and Technology

rudikar@mail.ruRUDN UniversityРоссийский университет дружбы народовР. N. Lebedev Physical Institute of the Russian Academy of SciencesФизический институт имени П. Н. Лебедева РАНN. I. Pirogov Russian National Research Medical UniversityРНИМУ имени Н. И. Пирогова30042026341Vol 34, No 1 (2026)ТОМ 34, № 1 (2026)11312429042026Copyright ©; 2026, Castillo A.J., Rudoy Y.G.Copyright ©; 2026, Кастильо А.Х., Рудой Ю.Г.2026Castillo A.J., Rudoy Y.G.Кастильо А.Х., Рудой Ю.Г.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/49993

The rigorous theory and characterization of charged-particle dynamics in high-intensity electromagnetic fields are fundamental for the development of advanced plasma-based applications. Accurate analytical models must bridge the gap between smoothed trajectories and exact particle motion to predetermine injection and energy gain. The main objective of this review is to establish a rigorous framework for the averaged relativistic motion of electrons, focusing on the strict derivation of ponderomotive forces and the impact of fast-oscillating periodic additions on dynamical variables. By making use of the Krylov--Bogoliubov--Mitropolsky averaging method to obtain the equations of motion, the study analyzes relativistic effects in laser beams and waveguides. These theoretical predictions are substantiated through numerical validation, including test-particle simulations and three-dimensional particle-in-cell simulations (PIC) of relativistic self-trapping regimes such as “laser bullet” and “bubble” structures. The review details the independence of the results on the formulation framework, the strict dependence on wave polarization, and the non-strict potential character of the relativistic ponderomotive force. The analysis demonstrates that periodic fast-oscillating additions are essential for a complete description, accurately setting initial conditions in averaged equations and enabling precise predictions of electron reflection and refraction. Simulations confirm that these fast-oscillating corrections determine electron injection and beam charge in realistic laser–plasma acceleration scenarios. The present review clearly shows that the dual framework of test-particle and PIC models is vital for probing the limits of averaged motion theory. The findings are of direct practical relevance for the optimization of radiation sources and guide the development of future theories incorporating non-adiabatic and field topology dependent effects.

Строгая теория и описание динамики заряженных частиц в высокоинтенсивных электромагнитных полях имеют фундаментальное значение для разработки перспективных плазменных приложений. Точные аналитические модели должны устранять разрыв между усредненными траекториями и истинным движением частиц для предварительного определения параметров инжекции и набора энергии. Основная цель данного обзора заключается в создании строгого аналитического описания усредненного релятивистского движения электронов с упором на строгий вывод пондеромоторных сил и влияние быстро осциллирующих периодических добавок на динамические переменные. С помощью метода усреднения Крылова--Боголюбова--Митропольского для получения уравнений движения в работе анализируются релятивистские эффекты в лазерных пучках и волноводах. Теоретические результаты подтверждаются в ходе численной валидации, включающей моделирование тестовых частиц в заданных полях и трехмерное моделирование плазмы в ячейках (PIC) для режимов релятивистского самозахвата, таких как структуры «лазерная пуля» и «пузырь». В обзоре подробно описаны независимость результатов от способа описания (приближение), строгая зависимость от поляризации волны и нестрого потенциальный характер релятивистской пондеромоторной силы. Анализ показывает, что периодические быстро осциллирующие добавки необходимы для полного описания, точного задания начальных условий в усреднённых уравнениях и обеспечения достоверного прогнозирования явлений отражения и преломления электронов. Моделирование подтверждает, что быстро осциллирующие добавки определяют инжекцию электронов и заряд пучка в реалистичных сценариях лазерно-плазменного ускорения. Данный обзор демонстрирует, что комбинированное использование моделей тестовых частиц и PIC-моделирования является крайне важным для исследования пределов теории усреднённого движения. Полученные результаты имеют прямую практическую значимость для оптимизации источников излучения и служат ориентиром для развития будущих теорий, учитывающих неадиабатические эффекты и эффекты, зависящие от топологии поля.

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