title: ÐÐ²ÑÑ ÑÐ°ÐºÑÐ¾ÑÐ½Ð°Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ñ Ð² Ð·Ð°Ð´Ð°ÑÐµ Ð¾ Ð±ÑÐ°Ñ Ð¸ÑÑÐ¾Ñ ÑÐ¾Ð½Ðµ
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: Two-factor optimization criterion in the brachistochrone problem
Creators: name: Ð¡ÑÐ²Ð¾ÑÐ¾Ð², Ð¡ÐµÑÐ³ÐµÐ¹
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: 2020
Subjects: ÐÑÐ¸Ð²ÑÐµ (Ð¼Ð°Ñ. )
Ð·Ð°Ð´Ð°ÑÐ° Ð¾ Ð±ÑÐ°Ñ Ð¸ÑÑÐ¾Ñ ÑÐ¾Ð½Ðµ
Ð±ÑÐ°Ñ Ð¸ÑÑÐ¾Ñ ÑÐ¾Ð½Ð°
ÑÐ¸ÐºÐ»Ð¾Ð¸Ð´Ð°
Ð´Ð²ÑÑ ÑÐ°ÐºÑÐ¾ÑÐ½ÑÐ¹ ÐºÑÐ¸ÑÐµÑÐ¸Ð¹ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸
ÑÐ¸ÐºÐ»Ð¾Ð¸Ð´Ð°Ð»ÑÐ½ÑÐ¹ Ð¿ÑÐ¾ÑÐ¸Ð»Ñ
Ð´Ð²ÑÑ ÑÐ°ÐºÑÐ¾ÑÐ½Ð°Ñ Ð±ÑÐ°Ñ Ð¸ÑÑÐ¾Ñ ÑÐ¾Ð½Ð°
ÑÐ°ÑÐ¸Ð¾Ð½Ð°Ð»ÑÐ½ÑÐµ Ð¿ÑÐ¾ÑÐ¸Ð»Ð¸
problem on brachistochron
cycloid
brachistochron
two-factory optimization criterion
cycloidal profile
two-factor brachistochronium
rational profiles

description: Ð¢ÐµÐ¼Ð° Ð²ÑÐ¿ÑÑÐºÐ½Ð¾Ð¹ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÑ: Â«ÐÐ²ÑÑÑÐ°ÐºÑÐ¾ÑÐ½ÑÐ¹ ÐºÑÐ¸ÑÐµÑÐ¸Ð¹ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð² Ð·Ð°Ð´Ð°ÑÐµ Ð¾ Ð±ÑÐ°ÑÐ¸ÑÑÐ¾ÑÑÐ¾Ð½ÐµÂ». ÐÐ°Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¿Ð¾ÑÐ²ÑÑÐµÐ½Ð° Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð¸Ñ Ð´Ð²ÑÑÑÐ°ÐºÑÐ¾ÑÐ½Ð¾Ð³Ð¾ ÐºÑÐ¸ÑÐµÑÐ¸Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð² Ð·Ð°Ð´Ð°ÑÐµ Ð¾ Ð±ÑÐ°ÑÐ¸ÑÑÐ¾ÑÑÐ¾Ð½Ðµ Ð¸ Ð¾ÑÐµÐ½ÐºÐµ ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾ÑÑÐ¸ Ð¿ÑÐ¾ÑÐ¸Ð»Ñ Ð±ÑÐ°ÑÐ¸ÑÑÐ¾ÑÑÐ¾Ð½Ñ. 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ÑÐ²Ð¾Ð´ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ Ð´Ð»Ñ ÐºÐ»Ð°ÑÑÐ¸ÑÐµÑÐºÐ¾Ð¹ Ð±ÑÐ°ÑÐ¸ÑÑÐ¾ÑÑÐ¾Ð½Ñ Ð¸ Ð¾Ð¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð¸Ðµ ÑÐ¾ÑÐ¼ÑÐ»Ñ Ð´Ð»Ñ Ð½Ð°ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ñ Ð²ÑÐµÐ¼ÐµÐ½Ð¸ ÑÐ¾ÑÐºÐ°Ð»ÑÐ·ÑÐ²Ð°Ð½Ð¸Ñ Ð¾Ñ ÑÐ¾ÑÐºÐ¸ A Ð´Ð¾ ÑÐ¾ÑÐºÐ¸ B, ÐºÐ¾ÑÐ¾ÑÑÐµ Ð½Ð°ÑÐ¾Ð´ÑÑÑÑ Ð½Ð° Ð¾Ð´Ð½Ð¾Ð¹ Ð³Ð¾ÑÐ¸Ð·Ð¾Ð½ÑÐ°Ð»Ð¸. 2. Ð Ð°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ð¸Ðµ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¾ Ð±ÑÐ°ÑÐ¸ÑÑÐ¾ÑÑÐ¾Ð½Ðµ Ð·Ð°Ð´Ð°Ð½Ð½Ð¾Ð¹ Ð´Ð»Ð¸Ð½Ñ. 3. ÐÐ¾Ð½ÑÑÑÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ðµ Ð´Ð²ÑÑÑÐ°ÐºÑÐ¾ÑÐ½Ð¾Ð³Ð¾ ÐºÑÐ¸ÑÐµÑÐ¸Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸ Ð¸ Ð½Ð°ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ðµ Ð¾Ð¿ÑÐ¸Ð¼Ð°Ð»ÑÐ½Ð¾Ð¹ ÑÑÐ°ÐµÐºÑÐ¾ÑÐ¸Ð¸ Ð¿Ð¾ Ð´Ð°Ð½Ð½Ð¾Ð¼Ñ ÐºÑÐ¸ÑÐµÑÐ¸Ñ. 4. ÐÑÐµÐ½ÐºÐ° ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾ÑÑÐ¸ ÑÐ¸ÐºÐ»Ð¾Ð¸Ð´Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ Ð¿ÑÐ¾ÑÐ¸Ð»Ñ. 5. ÐÑÐµÐ½ÐºÐ° ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾ÑÑÐ¸ Ð´Ð²ÑÑÑÐ°ÐºÑÐ¾ÑÐ½Ð¾Ð¹ Ð±ÑÐ°ÑÐ¸ÑÑÐ¾ÑÑÐ¾Ð½Ñ. ÐÑÐ»Ð¸ Ð¿ÑÐ¾Ð²ÐµÐ´ÐµÐ½Ñ Ð°Ð½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¸Ðµ Ð¸ ÑÐ¸ÑÐ»ÐµÐ½Ð½ÑÐµ ÑÐ°ÑÑÐµÑÑ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¾ Ð±ÑÐ°ÑÐ¸ÑÑÐ¾ÑÑÐ¾Ð½Ðµ Ð² ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ Ð¿Ð¾ÑÑÐ°Ð½Ð¾Ð²ÐºÐ°Ñ. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ Ð±ÑÐ»Ð¾ Ð¿Ð¾Ð»ÑÑÐµÐ½Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ Ð´Ð»Ñ ÐºÐ»Ð°ÑÑÐ¸ÑÐµÑÐºÐ¾Ð¹ Ð±ÑÐ°ÑÐ¸ÑÑÐ¾ÑÑÐ¾Ð½Ñ. ÐÐ¿ÑÐµÐ´ÐµÐ»ÐµÐ½Ð° ÑÐ¾ÑÐ¼ÑÐ»Ð° Ð´Ð»Ñ Ð½Ð°ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ñ Ð²ÑÐµÐ¼ÐµÐ½Ð¸ Ð´Ð²Ð¸Ð¶ÐµÐ½Ð¸Ñ ÑÐ¾ÑÐºÐ¸ Ð¿Ð¾ ÑÐ¸ÐºÐ»Ð¾Ð¸Ð´Ðµ. Ð¡ÐºÐ¾Ð½ÑÑÑÑÐ¸ÑÐ¾Ð²Ð°Ð½ Ð´Ð²ÑÑÑÐ°ÐºÑÐ¾ÑÐ½ÑÐ¹ ÐºÑÐ¸ÑÐµÑÐ¸Ð¹ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸. ÐÑÐ²ÐµÐ´ÐµÐ½Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ Ð±ÑÐ°ÑÐ¸ÑÑÐ¾ÑÑÐ¾Ð½Ñ Ð·Ð°Ð´Ð°Ð½Ð½Ð¾Ð¹ Ð´Ð»Ð¸Ð½Ñ. ÐÐ°Ð¹Ð´ÐµÐ½ Ð¾Ð¿ÑÐ¸Ð¼ÑÐ¼ Ð¿Ð¾ Ð´Ð²ÑÑÑÐ°ÐºÑÐ¾ÑÐ½Ð¾Ð¼Ñ ÐºÑÐ¸ÑÐµÑÐ¸Ñ Ð¾Ð¿ÑÐ¸Ð¼Ð¸Ð·Ð°ÑÐ¸Ð¸. ÐÑÐ¿Ð¾Ð»Ð½ÐµÐ½Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ðµ Ð´Ð²ÑÑÑÐ°ÐºÑÐ¾ÑÐ½Ð¾Ð¹ Ð±ÑÐ°ÑÐ¸ÑÑÐ¾ÑÑÐ¾Ð½Ñ Ð¸ ÑÐ¸ÐºÐ»Ð¾Ð¸Ð´Ñ. ÐÑÐ¾Ð²ÐµÐ´ÐµÐ½Ð° Ð¾ÑÐµÐ½ÐºÐ° ÑÑÑÐµÐºÑÐ¸Ð²Ð½Ð¾ÑÑÐ¸ ÑÐ¸ÐºÐ»Ð¾Ð¸Ð´Ð°Ð»ÑÐ½Ð¾Ð³Ð¾ Ð¿ÑÐ¾ÑÐ¸Ð»Ñ Ð¸ Ð´Ð²ÑÑÑÐ°ÐºÑÐ¾ÑÐ½Ð¾Ð¹ Ð±ÑÐ°ÑÐ¸ÑÑÐ¾ÑÑÐ¾Ð½Ñ. ÐÐ¾Ð»ÑÑÐµÐ½Ð½ÑÐµ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð¼Ð¾Ð³ÑÑ Ð¿ÑÐ¸Ð¼ÐµÐ½ÑÑÑÑÑ Ð¿ÑÐ¸ Ð¿ÑÐ¾ÐµÐºÑÐ¸ÑÐ¾Ð²Ð°Ð½Ð¸Ð¸ Ð¿Ð¾Ð´Ð·ÐµÐ¼Ð½ÑÑ ÑÐ¾Ð½Ð½ÐµÐ»ÐµÐ¹.
description: Theme of the final qualification work: âTwo-factor optimization criterion in the brachistochrone problemâ. This work is devoted to the study of a two-factor optimization criterion in the brachistochrone problem and the evaluation of the effectiveness of the brachistochrone profile. Tasks that were solved during the study: 1. Derivation of the equation for the classical brachistochrone and determination of the formula for finding the sliding time from point A to point B, which are on the same horizontal. 2. Consideration of the problem of a brachistochrone of a given length. 3. Designing a two-factor optimization criterion and finding the optimal trajectory by this criterion. 4. Evaluation of the effectiveness of the cycloidal profile. 5. Evaluation of the effectiveness of two-factor brachistochron. Analytical and numerical calculations of the brachistochrone problem were carried out in various formulations. As a result, an equation for the classical brachistochron was obtained. A formula is determined for finding the time of a point moving along a cycloid. A two-factor optimization criterion is designed. The equation of a brachistochron of a given length is derived. The optimum was found by the two-factor optimization criterion. A comparison of two-factor brachistochron and cycloid. The effectiveness of the cycloidal profile and two-factor brachistochron was assessed. The results can be applied in the design of underground tunnels.