Abstract
Normally in mathematics and physics only point particle systems, which are either finite or countable, are studied. We introduce new formal mathematical object called regular continuum system of point particles (with continuum number of particles). Initially each particle is characterized by the pair: (initial coordinate, initial velocity) in \(R^{2d}\). Moreover, all initial coordinates are different and fill up some domain in \(R^{d}\). Each particle moves via normal newtonian dynamics under influence of sone external force, but there is no interaction between particles. If the external force is bounded then trajectories of any two particles in the phase space do not intersect. More exactly, at any time moment any two particles have either different coordinates or different velocities. The system is called regular if there are no particle collisions in the coordinate space. The regularity condition is necessary for the velocity of the particle, situated at a given time at a given space point, were uniquely defined. Then the classical Euler equation for the field of velocities has rigorous meaning. Though the continuum of particles is in fact a continuum medium, the crucial notion of regularity was not studied in mathematical literature. It appeared that the seeming simplicity of the object (absence of interaction) is delusive. Even for simple external forces we could not find simple necessary and sufficient regularity conditions. However, we found a rich list of examples, one dimensional and many dimensional, where we get regularity conditions on different time intervals. In conclusion we formulate many perspective problems for regular systems with interaction.
Highlights
The system is called regular if there are no particle collisions in the coordinate space
The regularity condition is necessary for the velocity of the particle, situated at a given time at a given space point, were uniquely defined
Though the continuum of particles is a continuum medium, the crucial notion of regularity was not studied in mathematical literature
Summary
Дадим сначала точное определение объекта, который мы будем здесь изучать. Регулярная континуальная система MT точечных частиц отождествляется с набором подмножеств Λt ∈ Rd в моменты времени t ∈ [0, T ), 0 < T ≤ ∞. При этом Λ0 предполагается замыканием открытой области в Rd с кусочно гладкой границей ∂Λ0. Каждая точка этой области рассматривается как “материальная частица” бесконечно малой массы. Каждая точка (частица) x ∈ Λ0 описывает свою траекторию в Rd: y(t, x) = Ut(x), где y(0, x) = x - начальное положение этой частицы. Мы назовем MT системой без взаимодействия, если y(t, x) определяются решениями уравнений d2y(t, x) dy(0, x) dt2 = Fx(y(t, x)), y(0, x) = x, dt = v(x). Что v(x) и m(x) достаточно гладко зависят от x ∈ Λ0, а F (y) гладкая или кусочно гладкая по y. Всегда предполагается, что каждое уравнение (1) имеет единственное решение на всем рассматриваемом интервале [0, T ). Понимание сплошной среды как состоящей из континуума частиц бесконечно малой массы хорошо известно математикам, см. Понимание сплошной среды как состоящей из континуума частиц бесконечно малой массы хорошо известно математикам, см. например [7], стр
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
