title: ÐÐ²Ð¸Ð¶ÐµÐ½Ð¸Ðµ ÑÐµÐ»Ð°-ÑÐ¾ÑÐºÐ¸ Ð² ÑÐµÐ½ÑÑÐ°Ð»ÑÐ½Ð¾Ð¼ Ð¿Ð¾ÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½Ð¾Ð¼ Ð¿Ð¾Ð»Ðµ
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: The motion of a body-point in a central potential field
Creators: name: Ð¢ÑÑ, ÐÑÐµÐ²Ð¾Ð»Ð¾Ð´
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2022
Subjects: ÐÐ°ÑÐµÐ¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ
ÐÐ²Ð¸Ð¶ÐµÐ½Ð¸Ðµ
ÐÐµÑ Ð°Ð½Ð¸ÐºÐ°
ÐÐ¾Ð´Ð¾ÑÐ¾Ð´
ÑÐµÐ»Ð¾-ÑÐ¾ÑÐºÐ°
body-point

description: Ð¢ÐµÐ¼Ð° Ð²ÑÐ¿ÑÑÐºÐ½Ð¾Ð¹ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ: Â«ÐÐ²Ð¸Ð¶ÐµÐ½Ð¸Ðµ ÑÐµÐ»Ð°-ÑÐ¾ÑÐºÐ¸ Ð² ÑÐµÐ½ÑÑÐ°Ð»ÑÐ½Ð¾Ð¼ Ð¿Ð¾ÑÐµÐ½ÑÐ¸Ð°Ð»ÑÐ½Ð¾Ð¼ Ð¿Ð¾Ð»ÐµÂ». ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ ÑÑÐ°ÐµÐºÑÐ¾ÑÐ¸Ð¸ ÑÐµÐ»Ð°-ÑÐ¾ÑÐºÐ¸ Ð² ÐºÑÐ»Ð¾Ð½Ð¾Ð²ÑÐºÐ¾Ð¼ Ð¿Ð¾Ð»Ðµ Ð¿ÑÐ¸ÑÑÐ¶ÐµÐ½Ð¸Ñ. ÐÐ°Ð´Ð°ÑÐ¸, ÐºÐ¾ÑÐ¾ÑÑÐµ ÑÐµÑÐ°Ð»Ð¸ÑÑ Ð² ÑÐ¾Ð´Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ: 1. Ð§Ð¸ÑÐ»ÐµÐ½Ð½Ð¾Ðµ Ð¼Ð¾Ð´ÐµÐ»Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð¿ÑÐ¸ ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ Ð½Ð°Ð±Ð¾ÑÐ°Ñ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÐ¾Ð². 2. ÐÑÐµÐ¾Ð±ÑÐ°Ð·Ð¾Ð²Ð°Ð½Ð¸Ðµ Ð¸ÑÑÐ¾Ð´Ð½Ð¾Ð¹ Ð·Ð°Ð´Ð°ÑÐ¸ Ð² ÑÐ¸ÑÑÐµÐ¼Ñ Ð¸Ð· Ð´Ð²ÑÑ ÑÐºÐ°Ð»ÑÑÐ½ÑÑ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹. 3. ÐÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ ÑÐ°ÑÑÐ½ÑÑ ÑÐ»ÑÑÐ°ÐµÐ² Ð¸ Ð½Ð°ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ðµ Ð¾Ð³ÑÐ°Ð½Ð¸ÑÐµÐ½Ð¸Ð¹ Ð½Ð° Ð½Ð°ÑÐ°Ð»ÑÐ½ÑÐµ ÑÑÐ»Ð¾Ð²Ð¸Ñ. 4. ÐÐ½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¾Ðµ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ðµ ÑÐ²Ð¾Ð¹ÑÑÐ² ÑÑÐ°ÐµÐºÑÐ¾ÑÐ¸Ð¸ ÑÐµÐ»Ð°-ÑÐ¾ÑÐºÐ¸. Ð ÑÐ°Ð±Ð¾ÑÐµ Ð±ÑÐ»Ð¸ Ð½Ð°Ð¹Ð´ÐµÐ½Ñ Ð¿ÐµÑÐ²ÑÐµ Ð¸Ð½ÑÐµÐ³ÑÐ°Ð»Ñ Ð´Ð»Ñ ÑÐµÐ»Ð°-ÑÐ¾ÑÐºÐ¸ Ð² ÐºÑÐ»Ð¾Ð½Ð¾Ð²ÑÐºÐ¾Ð¼ Ð¿Ð¾Ð»Ðµ. Ð¡ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸ÐµÐ¼ Ð¿ÐµÑÐ²ÑÑ Ð¸Ð½ÑÐµÐ³ÑÐ°Ð»Ð¾Ð² Ð¸ÑÑÐ¾Ð´Ð½Ð°Ñ ÑÐ¸ÑÑÐµÐ¼Ð° ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹ Ð±ÑÐ»Ð° Ð¿ÑÐµÐ¾Ð±ÑÐ°Ð·Ð¾Ð²Ð°Ð½Ð° Ð² ÑÐ¸ÑÑÐµÐ¼Ñ Ð¸Ð· Ð´Ð²ÑÑ ÑÐºÐ°Ð»ÑÑÐ½ÑÑ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹, ÑÐ´Ð¾Ð±Ð½ÑÑ Ð´Ð»Ñ Ð°Ð½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ. Ð Ð¾Ð±ÑÐµÐ¼ ÑÐ»ÑÑÐ°Ðµ Ð±ÑÐ»Ð¾ Ð¿Ð¾ÐºÐ°Ð·Ð°Ð½Ð¾, ÐºÐ°Ðº Ð¿Ð¾ ÑÐµÑÐµÐ½Ð¸Ñ ÑÐ°ÐºÐ¾Ð¹ ÑÐ¸ÑÑÐµÐ¼Ñ Ð½Ð°Ð¹ÑÐ¸ ÑÐµÑÐµÐ½Ð¸Ðµ Ð¸ÑÑÐ¾Ð´Ð½Ð¾Ð¹ ÑÐ¸ÑÑÐµÐ¼Ñ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ð¹. Ð ÑÐ°ÑÑÐ½Ð¾Ð¼ ÑÐ»ÑÑÐ°Ðµ Ð¿Ð¾ÑÑÐ¾ÑÐ½Ð½Ð¾Ð³Ð¾ ÐºÐ¾Ð»Ð¸ÑÐµÑÑÐ²Ð° Ð´Ð²Ð¸Ð¶ÐµÐ½Ð¸Ñ Ð±ÑÐ»Ð¾ Ð½Ð°Ð¹Ð´ÐµÐ½Ð¾ ÑÑÐ»Ð¾Ð²Ð¸Ðµ Ð¿ÑÐ¾ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ñ ÑÑÐ°ÐµÐºÑÐ¾ÑÐ¸Ð¸ ÑÐµÑÐµÐ· Ð¿Ð¾Ð»ÑÑÐ°. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ Ð±ÑÐ»Ð¸ Ð½Ð°Ð¹Ð´ÐµÐ½Ñ Ð¾Ð³ÑÐ°Ð½Ð¸ÑÐµÐ½Ð¸Ñ Ð½Ð° Ð½Ð°ÑÐ°Ð»ÑÐ½ÑÐµ ÑÑÐ»Ð¾Ð²Ð¸Ñ, Ð¿ÑÐ¸ ÐºÐ¾ÑÐ¾ÑÑÑ ÑÑÐ°ÐµÐºÑÐ¾ÑÐ¸Ñ ÑÐµÐ»Ð°-ÑÐ¾ÑÐºÐ¸ ÑÐ¾ÑÐ¼Ð¸ÑÑÐµÑ Ñ ÑÐµÑÐµÐ½Ð¸ÐµÐ¼ Ð²ÑÐµÐ¼ÐµÐ½Ð¸ ÑÐ¿Ð»Ð¾ÑÐ½Ð¾Ð¹ ÑÐ»Ð¾Ð¹, Ð² Ð»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¼ Ð¿ÑÐ¸Ð±Ð»Ð¸Ð¶ÐµÐ½Ð¸Ð¸ ÑÐ²Ð»ÑÑÑÐ¸Ð¹ÑÑ ÑÑÐµÑÐ¸ÑÐµÑÐºÐ¸Ð¼. Ð¢Ð°ÐºÐ¶Ðµ Ð±ÑÐ»Ð¸ Ð½Ð°Ð¹Ð´ÐµÐ½Ñ Ð¿Ð°ÑÐ°Ð¼ÐµÑÑÑ, Ð¾ÑÐ²ÐµÑÐ°ÑÑÐ¸Ðµ Ð·Ð° ÑÐ¾Ð»ÑÐ¸Ð½Ñ Ð¸ ÑÐ»Ð»Ð¸Ð¿ÑÐ¸ÑÐ½Ð¾ÑÑÑ ÑÐ»Ð¾Ñ. ÐÑÐ»Ð¾ Ð´Ð¾ÐºÐ°Ð·Ð°Ð½Ð¾, ÑÑÐ¾ Ð¿ÑÐ¸ Ð²ÑÐ¿Ð¾Ð»Ð½ÐµÐ½Ð¸Ð¸ Ð¾Ð³ÑÐ°Ð½Ð¸ÑÐµÐ½Ð¸Ð¹ Ð½Ð° Ð½Ð°ÑÐ°Ð»ÑÐ½ÑÐµ ÑÑÐ»Ð¾Ð²Ð¸Ñ, ÑÑÐ°ÐµÐºÑÐ¾ÑÐ¸Ñ ÑÐµÐ»Ð°-ÑÐ¾ÑÐºÐ¸ Ð²ÑÐµÐ³Ð´Ð° Ð»ÐµÐ¶Ð¸Ñ Ð¼ÐµÐ¶Ð´Ñ Ð´Ð²ÑÑ ÑÑÐµÑ ÐºÐ¾Ð½ÐµÑÐ½Ð¾Ð³Ð¾ ÑÐ°Ð´Ð¸ÑÑÐ° Ð¸ Ð¸Ð¼ÐµÐµÑ Ð¼Ð¾Ð½Ð¾ÑÐ¾Ð½Ð½Ð¾ Ð²Ð¾Ð·ÑÐ°ÑÑÐ°ÑÑÐ¸Ð¹ Ð¿Ð¾Ð»ÑÑÐ½ÑÐ¹ ÑÐ³Ð¾Ð».
description: The subject of the graduate qualification work is Â«The motion of a body-point in a central potential fieldÂ». The given work is devoted to the study of the trajectory of a body-point in the Coulomb field of attraction. The research set the following goals: 1. Numerical simulation for different sets of parameters. 2. Transformation of the original problem into a system of two scalar equations. 3. Study of particular cases and finding restrictions on the initial conditions. 4. Analytical study of the properties of the trajectory of a body-point. The first integrals for a body-point in a Coulomb field were found in this work. Using the first integrals, the original system of equations was transformed into a system of two scalar equations, convenient for analytical research. In the general case, it was shown how, by solving such a system, to find a solution to the original system of equations. In the special case of a constant momentum, the condition for the passage of the trajectory through the poles was found. As a result, restrictions were found on the initial conditions under which the trajectory of a body-point forms a continuous layer over time, which is spherical in the linear approximation. The parameters responsible for the thickness and ellipticity of the layer were also found. It was proved that when the restrictions on the initial conditions are met, the trajectory of the body-point always lies between two spheres of finite radius and has a monotonically increasing polar angle.