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20.1767 5.37686 19.4863 5.37686C18.796 5.37686 18.2363 4.81722 18.2363 4.12686ZM19.4863 1.37686C17.9675 1.37686 16.7363 2.60808 16.7363 4.12686C16.7363 4.43206 16.786 4.72564 16.8778 4.99996H7C4.79086 4.99996 3 6.79082 3 8.99996V19C3 21.2091 4.79086 23 7 23H17C19.2091 23 21 21.2091 21 19V8.99996C21 8.16642 20.745 7.39244 20.3089 6.75173C21.4258 6.40207 22.2363 5.35911 22.2363 4.12686C22.2363 2.60808 21.0051 1.37686 19.4863 1.37686ZM7 6.49996H16.6319L14.6964 8.43547C14.4035 8.72837 14.4035 9.20324 14.6964 9.49613C14.9893 9.78903 15.4641 9.78903 15.757 9.49613L18.3547 6.89846C19.0438 7.34362 19.5 8.11852 19.5 8.99996V19C19.5 20.3807 18.3807 21.5 17 21.5H7C5.61929 21.5 4.5 20.3807 4.5 19V8.99996C4.5 7.61924 5.61929 6.49996 7 6.49996ZM9.25 12C9.25 11.5857 9.58579 11.25 10 11.25H12H14C14.4142 11.25 14.75 11.5857 14.75 12C14.75 12.4142 14.4142 12.75 14 12.75H12H10C9.58579 12.75 9.25 12.4142 9.25 12ZM9.25 17C9.25 16.5857 9.58579 16.25 10 16.25H12H14C14.4142 16.25 14.75 16.5857 14.75 17C14.75 17.4142 14.4142 17.75 14 17.75H12H10C9.58579 17.75 9.25 17.4142 9.25 17Z" fill="currentColor"/> </svg> Аннотация</a>Аннотация<a href="/c/dinamika-c3b94d/annotation" class="bre-article-menu__list-item tw-mx-auto tw-flex lg:tw-hidden"><svg viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"> <path fill-rule="evenodd" clip-rule="evenodd" d="M18.2363 4.12686C18.2363 3.43651 18.796 2.87686 19.4863 2.87686C20.1767 2.87686 20.7363 3.43651 20.7363 4.12686C20.7363 4.81722 20.1767 5.37686 19.4863 5.37686C18.796 5.37686 18.2363 4.81722 18.2363 4.12686ZM19.4863 1.37686C17.9675 1.37686 16.7363 2.60808 16.7363 4.12686C16.7363 4.43206 16.786 4.72564 16.8778 4.99996H7C4.79086 4.99996 3 6.79082 3 8.99996V19C3 21.2091 4.79086 23 7 23H17C19.2091 23 21 21.2091 21 19V8.99996C21 8.16642 20.745 7.39244 20.3089 6.75173C21.4258 6.40207 22.2363 5.35911 22.2363 4.12686C22.2363 2.60808 21.0051 1.37686 19.4863 1.37686ZM7 6.49996H16.6319L14.6964 8.43547C14.4035 8.72837 14.4035 9.20324 14.6964 9.49613C14.9893 9.78903 15.4641 9.78903 15.757 9.49613L18.3547 6.89846C19.0438 7.34362 19.5 8.11852 19.5 8.99996V19C19.5 20.3807 18.3807 21.5 17 21.5H7C5.61929 21.5 4.5 20.3807 4.5 19V8.99996C4.5 7.61924 5.61929 6.49996 7 6.49996ZM9.25 12C9.25 11.5857 9.58579 11.25 10 11.25H12H14C14.4142 11.25 14.75 11.5857 14.75 12C14.75 12.4142 14.4142 12.75 14 12.75H12H10C9.58579 12.75 9.25 12.4142 9.25 12ZM9.25 17C9.25 16.5857 9.58579 16.25 10 16.25H12H14C14.4142 16.25 14.75 16.5857 14.75 17C14.75 17.4142 14.4142 17.75 14 17.75H12H10C9.58579 17.75 9.25 17.4142 9.25 17Z" fill="currentColor"/> </svg> Аннотация</a></div><div class="tw-grow tw-basis-0 max-md:tw-max-w-[80px]"><a href="/c/dinamika-c3b94d/references" class="bre-article-menu__list-item"><svg viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"> <path fill-rule="evenodd" clip-rule="evenodd" d="M5 3.25C3.48122 3.25 2.25 4.48122 2.25 6V16.1667C2.25 17.6854 3.48122 18.9167 5 18.9167H9.3C9.80519 18.9167 10.2974 19.1269 10.6662 19.5139C11.0362 19.9022 11.25 20.4361 11.25 21C11.25 21.4142 11.5858 21.75 12 21.75C12.4142 21.75 12.75 21.4142 12.75 21C12.75 20.4227 12.9564 19.8833 13.3026 19.4973C13.6464 19.114 14.0941 18.9167 14.5412 18.9167H19C20.5188 18.9167 21.75 17.6855 21.75 16.1667V6C21.75 4.48122 20.5188 3.25 19 3.25H15.3882C14.2627 3.25 13.2022 3.74922 12.4341 4.60572C12.266 4.79308 12.1147 4.99431 11.9809 5.20674C11.8358 4.98777 11.6713 4.78092 11.4885 4.58908C10.6758 3.73626 9.56568 3.25 8.4 3.25H5ZM12.75 17.993C13.2735 17.6237 13.8929 17.4167 14.5412 17.4167H19C19.6904 17.4167 20.25 16.857 20.25 16.1667V6C20.25 5.30964 19.6904 4.75 19 4.75H15.3882C14.7165 4.75 14.0534 5.04681 13.5507 5.60725C13.0457 6.17037 12.75 6.95001 12.75 7.77778V17.993ZM11.25 18.0438V7.77778C11.25 6.96341 10.9414 6.18924 10.4026 5.62389C9.86506 5.05976 9.14388 4.75 8.4 4.75H5C4.30964 4.75 3.75 5.30964 3.75 6V16.1667C3.75 16.857 4.30964 17.4167 5 17.4167H9.3C10.0044 17.4167 10.6825 17.64 11.25 18.0438Z" fill="currentColor"/> </svg> Библиография</a>Библиография<a href="/c/dinamika-c3b94d/references" class="bre-article-menu__list-item tw-mx-auto tw-flex lg:tw-hidden"><svg viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"> <path fill-rule="evenodd" clip-rule="evenodd" d="M5 3.25C3.48122 3.25 2.25 4.48122 2.25 6V16.1667C2.25 17.6854 3.48122 18.9167 5 18.9167H9.3C9.80519 18.9167 10.2974 19.1269 10.6662 19.5139C11.0362 19.9022 11.25 20.4361 11.25 21C11.25 21.4142 11.5858 21.75 12 21.75C12.4142 21.75 12.75 21.4142 12.75 21C12.75 20.4227 12.9564 19.8833 13.3026 19.4973C13.6464 19.114 14.0941 18.9167 14.5412 18.9167H19C20.5188 18.9167 21.75 17.6855 21.75 16.1667V6C21.75 4.48122 20.5188 3.25 19 3.25H15.3882C14.2627 3.25 13.2022 3.74922 12.4341 4.60572C12.266 4.79308 12.1147 4.99431 11.9809 5.20674C11.8358 4.98777 11.6713 4.78092 11.4885 4.58908C10.6758 3.73626 9.56568 3.25 8.4 3.25H5ZM12.75 17.993C13.2735 17.6237 13.8929 17.4167 14.5412 17.4167H19C19.6904 17.4167 20.25 16.857 20.25 16.1667V6C20.25 5.30964 19.6904 4.75 19 4.75H15.3882C14.7165 4.75 14.0534 5.04681 13.5507 5.60725C13.0457 6.17037 12.75 6.95001 12.75 7.77778V17.993ZM11.25 18.0438V7.77778C11.25 6.96341 10.9414 6.18924 10.4026 5.62389C9.86506 5.05976 9.14388 4.75 8.4 4.75H5C4.30964 4.75 3.75 5.30964 3.75 6V16.1667C3.75 16.857 4.30964 17.4167 5 17.4167H9.3C10.0044 17.4167 10.6825 17.64 11.25 18.0438Z" fill="currentColor"/> </svg> Библиография</a></div><div class="tw-grow tw-basis-0 max-md:tw-max-w-[80px]"><a href="/c/dinamika-c3b94d/versions" class="bre-article-menu__list-item"><svg viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"> <path fill-rule="evenodd" clip-rule="evenodd" d="M10.9565 3.85864H7.51619C7.8045 3.2057 8.4577 2.75 9.21734 2.75H13.5652H14.7687C15.4365 2.75 16.0697 3.0466 16.4972 3.55959L19.2502 6.86313C19.5871 7.26748 19.7717 7.77718 19.7717 8.30354V10.6957V16.7826C19.7717 17.5422 19.316 18.1954 18.663 18.4838V13.3043V10.9122C18.663 10.0349 18.3555 9.18542 17.7939 8.51149L15.0409 5.20795C14.3284 4.35298 13.273 3.85864 12.1601 3.85864H10.9565ZM14.913 22.7499C16.6051 22.7499 18.0354 21.6293 18.5022 20.0898C20.0762 19.8113 21.2717 18.4365 21.2717 16.7826V10.6957V8.30354C21.2717 7.42628 20.9641 6.57678 20.4025 5.90285L17.6496 2.59931C16.9371 1.74434 15.8817 1.25 14.7687 1.25H13.5652H9.21734C7.56341 1.25 6.18869 2.44548 5.91016 4.01947C4.37062 4.48633 3.25 5.91662 3.25 7.60864V18.9999C3.25 21.071 4.92893 22.7499 7 22.7499H14.913ZM7 5.35864C5.75736 5.35864 4.75 6.366 4.75 7.60864V18.9999C4.75 20.2426 5.75736 21.2499 7 21.2499H14.913C16.1557 21.2499 17.163 20.2426 17.163 18.9999V13.3043V10.9122C17.163 10.7991 17.1545 10.6867 17.1378 10.5761H15.3043C13.9296 10.5761 12.8152 9.46164 12.8152 8.08694V5.45611C12.6051 5.39215 12.3845 5.35864 12.1601 5.35864H10.9565H7ZM14.3152 6.68014V8.08694C14.3152 8.63322 14.758 9.07607 15.3043 9.07607H16.3118L14.3152 6.68014ZM6.72827 13.3043C6.72827 12.8901 7.06406 12.5543 7.47827 12.5543H14.4348C14.849 12.5543 15.1848 12.8901 15.1848 13.3043C15.1848 13.7185 14.849 14.0543 14.4348 14.0543H7.47827C7.06406 14.0543 6.72827 13.7185 6.72827 13.3043ZM7.47827 16.9022C7.06406 16.9022 6.72827 17.238 6.72827 17.6522C6.72827 18.0664 7.06406 18.4022 7.47827 18.4022H10.9565C11.3707 18.4022 11.7065 18.0664 11.7065 17.6522C11.7065 17.238 11.3707 16.9022 10.9565 16.9022H7.47827Z" fill="currentColor"/> </svg> Версии</a>Версии<a href="/c/dinamika-c3b94d/versions" class="bre-article-menu__list-item tw-mx-auto tw-flex lg:tw-hidden"><svg viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"> <path fill-rule="evenodd" clip-rule="evenodd" d="M10.9565 3.85864H7.51619C7.8045 3.2057 8.4577 2.75 9.21734 2.75H13.5652H14.7687C15.4365 2.75 16.0697 3.0466 16.4972 3.55959L19.2502 6.86313C19.5871 7.26748 19.7717 7.77718 19.7717 8.30354V10.6957V16.7826C19.7717 17.5422 19.316 18.1954 18.663 18.4838V13.3043V10.9122C18.663 10.0349 18.3555 9.18542 17.7939 8.51149L15.0409 5.20795C14.3284 4.35298 13.273 3.85864 12.1601 3.85864H10.9565ZM14.913 22.7499C16.6051 22.7499 18.0354 21.6293 18.5022 20.0898C20.0762 19.8113 21.2717 18.4365 21.2717 16.7826V10.6957V8.30354C21.2717 7.42628 20.9641 6.57678 20.4025 5.90285L17.6496 2.59931C16.9371 1.74434 15.8817 1.25 14.7687 1.25H13.5652H9.21734C7.56341 1.25 6.18869 2.44548 5.91016 4.01947C4.37062 4.48633 3.25 5.91662 3.25 7.60864V18.9999C3.25 21.071 4.92893 22.7499 7 22.7499H14.913ZM7 5.35864C5.75736 5.35864 4.75 6.366 4.75 7.60864V18.9999C4.75 20.2426 5.75736 21.2499 7 21.2499H14.913C16.1557 21.2499 17.163 20.2426 17.163 18.9999V13.3043V10.9122C17.163 10.7991 17.1545 10.6867 17.1378 10.5761H15.3043C13.9296 10.5761 12.8152 9.46164 12.8152 8.08694V5.45611C12.6051 5.39215 12.3845 5.35864 12.1601 5.35864H10.9565H7ZM14.3152 6.68014V8.08694C14.3152 8.63322 14.758 9.07607 15.3043 9.07607H16.3118L14.3152 6.68014ZM6.72827 13.3043C6.72827 12.8901 7.06406 12.5543 7.47827 12.5543H14.4348C14.849 12.5543 15.1848 12.8901 15.1848 13.3043C15.1848 13.7185 14.849 14.0543 14.4348 14.0543H7.47827C7.06406 14.0543 6.72827 13.7185 6.72827 13.3043ZM7.47827 16.9022C7.06406 16.9022 6.72827 17.238 6.72827 17.6522C6.72827 18.0664 7.06406 18.4022 7.47827 18.4022H10.9565C11.3707 18.4022 11.7065 18.0664 11.7065 17.6522C11.7065 17.238 11.3707 16.9022 10.9565 16.9022H7.47827Z" fill="currentColor"/> </svg> Версии</a></div></div></div></nav><div><meta itemprop="image primaryImageOfPage" content="https://i.bigenc.ru/resizer/resize?sign=7JK17_fgqWlQEFIk2cRkfA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=120"><article itemscope itemprop="mainEntity" itemtype="https://schema.org/Article"><div itemprop="publisher" itemscope itemtype="https://schema.org/Organization"><meta itemprop="name" content="Автономная некоммерческая организация «Национальный научно-образовательный центр «Большая российская энциклопедия»"><meta itemprop="address" content="Покровский бульвар, д. 8, стр. 1А, Москва, 109028"><meta itemprop="telephone" content="+7 (495) 781-15-95"><meta itemprop="logo" 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stroke-linecap="round"/></svg> <div class="bre-article-loc-short">Динамика материальной точки</div></div></nav><div class="article-sidebar -hide-on-desktop-s"><div class="article-sidebar-button -show-on-tablet -hide-on-desktop-s">Информация<svg viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"><path d="M6 9l6 6 6-6" stroke="currentColor" stroke-width="1.5" stroke-linecap="round"/></svg> <div class="article-sidebar-text -show-on-tablet -hide-on-desktop-s">Динамика (в физике)</div></div><div class="article-sidebar-wrapper -hide-on-tablet"><header class="bre-article-header -hide-on-tablet"><div class="bre-label__wrap"><a href="/t/terms" class="bre-label _link">Термины</a>Термины</div><h1 class="bre-article-header-title">Динамика (в физике)</h1></header><section class="-hide-on-tablet tw-h-14 md:tw-h-20"></section><meta itemprop="name" content="Физика"><meta itemprop="caption" content="Физика. Научно-образовательный портал «Большая российская энциклопедия»"><img src="https://i.bigenc.ru/resizer/resize?sign=7JK17_fgqWlQEFIk2cRkfA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=120" onerror="this.setAttribute('data-error', 1)" alt="Физика" data-nuxt-img sizes="320px" srcset="https://i.bigenc.ru/resizer/resize?sign=7JK17_fgqWlQEFIk2cRkfA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=120 120w,https://i.bigenc.ru/resizer/resize?sign=Jf8Ovt6NK1CJRMEXmLmu9w&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=320 320w,https://i.bigenc.ru/resizer/resize?sign=9FmjZNIS1_JG-eBy3nkCow&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=480 480w,https://i.bigenc.ru/resizer/resize?sign=W0YAxakNej-ihBYTmKOUhA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=640 640w,https://i.bigenc.ru/resizer/resize?sign=bU5vxPnJKBxMhvgLEjl-uA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=768 768w,https://i.bigenc.ru/resizer/resize?sign=CO7eqX0CglCAJmsuCYDJxQ&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=1024 1024w,https://i.bigenc.ru/resizer/resize?sign=SNrDJXfJeDUaOjs9TGABPA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=1280 1280w,https://i.bigenc.ru/resizer/resize?sign=A65s2m2zZF6hgTDDppMoDA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=1920 1920w" title="Физика" class="" itemprop="contentUrl"><div class="article-sidebar-meta"><dl class="tw-mt-0"><dt>Области знаний:</dt><dd>Классическая механика Ньютона</dd></dl></div></div></div><div class="bre-article-page__container"><div class="bre-article-page__content bre-article-content"><header class="bre-article-header -show-on-tablet"><div class="bre-label__wrap"><a href="/t/terms" class="bre-label _link">Термины</a>Термины</div><h1 class="bre-article-header-title">Динамика (в физике)</h1></header><section class="tw-flex"><div class="-show-on-tablet tw-h-14 md:tw-h-20"><div><div><div itemprop="interactionStatistic" itemscope itemtype="https://schema.org/InteractionCounter"><meta itemprop="interactionType" content="https://schema.org/ViewAction"><meta itemprop="userInteractionCount" content=""></div><div itemprop="interactionStatistic" itemscope itemtype="https://schema.org/InteractionCounter"><meta itemprop="interactionType" content="https://schema.org/ShareAction"><meta itemprop="userInteractionCount" content=""></div><div itemprop="interactionStatistic" itemscope itemtype="https://schema.org/InteractionCounter"><meta itemprop="interactionType" content="https://schema.org/LikeAction"><meta itemprop="userInteractionCount" content=""></div></div></div></div></section><div class="js-preview-link-root"><div itemprop="articleBody" class="bre-article-body"><section><section>Дина́мика (от греч. δύναμις – возможность, сила), раздел <a href="/c/mekhanika-abd757" class="bre-preview-link" itemprop="url" data-external="false">механики</a>, посвящённый изучению изменения характеристик движения материальных тел под действием приложенных к ним <a href="/c/sila-77a176" class="bre-preview-link" itemprop="url" data-external="false">сил</a>. Основы динамики свободной <a href="/c/material-naia-tochka-a6ca99" class="bre-preview-link" itemprop="url" data-external="false">материальной точки</a> заложены в начале 17 в. <a href="/c/galilei-galileo-0b9095" class="bre-preview-link" itemprop="url" data-external="false">Г. Галилеем</a>, который рассмотрел падение тел под действием <a href="/c/sila-tiazhesti-c50b1b" class="bre-preview-link" itemprop="url" data-external="false">силы тяжести</a> и сформулировал <a href="/c/zakon-inertsii-6b6b01" class="bre-preview-link" itemprop="url" data-external="false">закон инерции</a>. В 1687 г. <a href="/c/n-iuton-isaak-0f3dbe" class="bre-preview-link" itemprop="url" data-external="false">И. Ньютон</a> дал чёткую формулировку трёх основных <a href="/c/zakony-mekhaniki-n-iutona-84aa92" class="bre-preview-link" itemprop="url" data-external="false">законов динамики</a>. В 18 в. существенный вклад в постановку и решение общих задач динамики внесли <a href="/c/eiler-leonard-0e881f" class="bre-preview-link" itemprop="url" data-external="false">Л. Эйлер</a>, <a href="/c/dalamber-zhan-leron-22730e" class="bre-preview-link" itemprop="url" data-external="false">Ж. Л. Д’ Аламбер</a> и <a href="/c/lagranzh-zhozef-lui-f66cf7" class="bre-preview-link" itemprop="url" data-external="false">Ж.-Л. Лагранж</a>.Динамика – важная составляющая математического естествознания, сформировавшая правила и приёмы построения механико-математических моделей <a href="/c/mekhanicheskoe-dvizhenie-6727b4" class="bre-preview-link" itemprop="url" data-external="false">механического движения</a>. Для описания движения объектов реального мира применяют различные модели, в которых объекты принимают за материальную точку, <a href="/c/tviordoe-telo-e5875e" class="bre-preview-link" itemprop="url" data-external="false">твёрдое тело</a> и т. п.<h2 id="h2_dinamika_material'noi_tochki">Динамика материальной точки</h2>Динамика, основанная на положениях Галилея и Ньютона, называется классической, или ньютоновской. Она описывает движение объектов, размерами которых можно пренебречь (материальных точек), со скоростями, много меньшими <a href="/c/skorost-sveta-a676e2" class="bre-preview-link" itemprop="url" data-external="false">скорости света</a> (движение микрочастиц рассматривается в <a href="/c/kvantovaia-mekhanika-6d0792" class="bre-preview-link" itemprop="url" data-external="false">квантовой механике</a>, движение со скоростями, близкими к скорости света, – в <a href="/c/reliativistskaia-mekhanika-50571e" class="bre-preview-link" itemprop="url" data-external="false">релятивистской механике</a>). В классической динамике аксиоматически вводятся понятия неподвижного пространства и абсолютного времени, одинакового во всех точках пространства и не зависящего от выбора конкретной системы координат.Классическая динамика базируется на трёх основных законах – <a href="/c/zakony-mekhaniki-n-iutona-84aa92" class="bre-preview-link" itemprop="url" data-external="false">законах механики Ньютона</a>. Первый из них, называемый также законом инерции, вводит понятие <a href="/c/inertsial-naia-sistema-otschiota-bd89b3" class="bre-preview-link" itemprop="url" data-external="false">инерциальной системы отсчёта</a>, в которой материальная точка покоится или движется прямолинейно и равномерно, если на неё не действуют другие тела или влияние этих тел скомпенсировано. Меру механического действия одного тела на другое называют <a href="/c/sila-77a176" class="bre-preview-link" itemprop="url" data-external="false">силой</a>. Второй закон устанавливает, что действие силы <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">{F}</annotation></semantics></math>F на материальную точку массой <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">{m}</annotation></semantics></math>m вызывает ускорение <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">{w}</annotation></semantics></math>w точки, определяемое равенством<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi>w</mi><mo>=</mo><mi>F</mi><mi mathvariant="normal">/</mi><mi>m</mi></mrow><mi mathvariant="normal">.</mi><mspace width="1em"/><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{w=F/m}.\quad (1)</annotation></semantics></math>w=F/m.(1)Третий закон динамики устанавливает, что при взаимодействии двух материальных точек возникает пара сил, равных по величине и противоположных по направлению (см. <a href="/c/zakon-deistviia-i-protivodeistviia-3f828f" class="bre-preview-link" itemprop="url" data-external="false">закон действия и противодействия</a>). Если к телу приложено несколько сил, то в соответствии с принципом независимости действия сил каждая из них сообщает телу такое же <a href="/c/uskorenie-37472d" class="bre-preview-link" itemprop="url" data-external="false">ускорение</a>, какое она сообщила бы, если бы действовала одна. Поэтому в качестве <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">{F}</annotation></semantics></math>F в уравнении (1)рассматривается <a href="/c/ravnodeistvuiushchaia-b38de4" class="bre-preview-link" itemprop="url" data-external="false">равнодействующая</a> сил, действующих на тело.Динамика решает 2 класса задач: прямые и обратные. Прямая задача динамики состоит в определении движения точки, происходящего под действием заданных сил. Сила <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">{F}</annotation></semantics></math>F считается заданной, если известна её зависимость от времени <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">{t}</annotation></semantics></math>t и векторов <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">r</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{r}</annotation></semantics></math>r и <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">v</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{v}</annotation></semantics></math>v, определяющих положение и скорость материальной точки:<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mi>F</mi><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi mathvariant="bold-italic">r</mi><mo separator="true">,</mo><mi mathvariant="bold-italic">v</mi><mo separator="true">,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="normal">.</mi><mspace width="1em"/><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{F=F(\boldsymbol{r},\boldsymbol{v},t)}.\quad (2)</annotation></semantics></math>F=F(r,v,t).(2)В этом случае равенство (1) превращается в <a href="/c/differentsial-noe-uravnenie-7e57fc" class="bre-preview-link" itemprop="url" data-external="false">дифференциальное уравнение</a> движения точки. Его решение описывает зависимость вектора <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">r</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{r}</annotation></semantics></math>r от времени и начальных условий:<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">r</mi><mo>=</mo><mi mathvariant="bold-italic">r</mi><mo stretchy="false">(</mo><mi>t</mi><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">r</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi mathvariant="bold-italic">v</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi><mspace width="1em"/><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\boldsymbol{r}=\boldsymbol{r}(t,\boldsymbol{r}_{0},\boldsymbol{v}_{0}).\quad (3)</annotation></semantics></math>r=r(t,r0,v0).(3)Примером подобной задачи может служить задача по определению <a href="/c/traektoriia-079f31" class="bre-preview-link" itemprop="url" data-external="false">траектории</a> движения снаряда по его начальной скорости (силы тяжести и сопротивление воздуха считаются известными).Обратная задача динамики состоит в определении силы, обеспечивающей заданное движение: по семейству законов движения, описываемых выражением (3), требуется восстановить зависимость силы (2) от перечисленных аргументов. Классическим примером решения этой задачи является открытие И. Ньютоном <a href="/c/zakon-vsemirnogo-tiagoteniia-ac9aa2" class="bre-preview-link" itemprop="url" data-external="false">закона всемирного тяготения</a>. Рассматривая <a href="/c/zakony-keplera-be033d" class="bre-preview-link" itemprop="url" data-external="false">законы движения планет Кеплера</a>, Ньютон пришёл к выводу, что эти движения происходят под действием силы, обратно пропорциональной квадрату расстояния от <a href="/c/solntse-6df72e" class="bre-preview-link" itemprop="url" data-external="false">Солнца</a> до планеты и не зависящей ни от времени, ни от скоростей движения планет.В ряде задач динамики удобно использовать различные динамические меры движения точки: <a href="/c/impul-s-c287b2" class="bre-preview-link" itemprop="url" data-external="false">импульс</a> (количество движения) <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">K</mi><mo>=</mo><mi>m</mi><mi mathvariant="bold-italic">v</mi></mrow><annotation encoding="application/x-tex">{\boldsymbol{K}=m\boldsymbol{v}}</annotation></semantics></math>K=mv, <a href="/c/moment-impul-sa-f669a3" class="bre-preview-link" itemprop="url" data-external="false">момент импульса</a> (<a href="/c/kineticheskii-moment-1e6be6" class="bre-preview-link" itemprop="url" data-external="false">кинетический момент</a>) относительно начала координат <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">G</mi><mo>=</mo><mi mathvariant="bold-italic">r</mi><mo>×</mo><mi>m</mi><mi mathvariant="bold-italic">v</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{G}=\boldsymbol{r}×m\boldsymbol{v}</annotation></semantics></math>G=r×mv, <a href="/c/kineticheskaia-energiia-06ee5a" class="bre-preview-link" itemprop="url" data-external="false">кинетическую энергию</a> <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>T</mtext><mo>=</mo><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">\text{T}=mv^{2}/2</annotation></semantics></math>T=mv2/2. При помощи этих мер уравнение (1) можно записать в виде закона изменения импульса <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi mathvariant="bold-italic">K</mi><mi mathvariant="normal">/</mi><mi>d</mi><mi>t</mi><mo>=</mo><mi mathvariant="bold-italic">F</mi></mrow><annotation encoding="application/x-tex">d\boldsymbol{K}/dt=\boldsymbol{F}</annotation></semantics></math>dK/dt=F, или закона изменения момента импульса <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi mathvariant="bold-italic">G</mi><mi mathvariant="normal">/</mi><mi>d</mi><mi>t</mi><mo>=</mo><mi mathvariant="bold-italic">r</mi><mo>×</mo><mi mathvariant="bold-italic">F</mi><mo>=</mo><mi mathvariant="bold-italic">M</mi></mrow><annotation encoding="application/x-tex">d\boldsymbol{G}/dt=\boldsymbol{r}×\boldsymbol{F}=\boldsymbol{M}</annotation></semantics></math>dG/dt=r×F=M, где <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">M</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{M}</annotation></semantics></math>M – момент силы относительно начала координат, или закона изменения энергии <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>T</mi><mi mathvariant="normal">/</mi><mi>d</mi><mi>t</mi><mo>=</mo><mi>F</mi><mi>v</mi><mo>=</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">{dT/dt=Fv=N}</annotation></semantics></math>dT/dt=Fv=N, где <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">{N}</annotation></semantics></math>N– мощность силы <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">F</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{F}</annotation></semantics></math>F.<h2 id="h2_dinamika_sistemы_svoбodnыh_tochek">Динамика системы свободных точек</h2>Движение системы свободных материальных точек с массами <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m_{i}</annotation></semantics></math>mi можно описать совокупностью уравнений вида (1), вводя суммарные меры движения: импульс системы точек <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">K</mi><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub><mo>=</mo><mi>m</mi><msub><mi mathvariant="bold-italic">v</mi><mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{K}=\sum\limits_{i}m_{i}\boldsymbol{v}_{i}=m\boldsymbol{v}_{c}</annotation></semantics></math>K=i∑mivi=mvc, где <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">{m}</annotation></semantics></math>m – общая масса системы, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">v</mi><mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{v}_{c}</annotation></semantics></math>vc – скорость центра масс системы; главный кинетический момент системы <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">G</mi><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi mathvariant="bold-italic">r</mi><mi>i</mi></msub><mo>×</mo><msub><mi>m</mi><mi>i</mi></msub><msub><mi mathvariant="bold-italic">v</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{G}=\sum\limits_{i}\boldsymbol{r}_{i}×m_{i}\boldsymbol{v}_{i}</annotation></semantics></math>G=i∑ri×mivi; кинетическую энергию <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>m</mi><mi>i</mi></msub><msubsup><mi mathvariant="bold-italic">v</mi><mi>i</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">T=\frac{1}{2}\sum\limits_{i}m_{i}\boldsymbol{v}^2_{i}</annotation></semantics></math>T=21i∑mivi2. Соотношения между суммарными мерами движения и силами, приложенными к точкам, называются общими теоремами динамики. К этим теоремам относятся следующие:1. Теорема об изменении импульса системы: изменение импульса системы за любой промежуток времени равняется геометрической сумме импульсов, действующих на систему внешних сил. Следствиями этой теоремы являются <a href="/c/zakon-sokhraneniia-impul-sa-d06b11" class="bre-preview-link" itemprop="url" data-external="false">закон сохранения импульса</a> системы и теорема о движении центра масс системы.2. Теорема об изменении главного кинетического момента системы: <a href="/c/proizvodnaia-68fd90" class="bre-preview-link" itemprop="url" data-external="false">производная</a> по времени от главного кинетического момента системы относительно любого неподвижного центра (или оси) равна сумме моментов действующих внешних сил относительно того же центра (или оси). Следствием данной теоремы является закон сохранения главного кинетического момента системы при равенстве нулю суммы моментов внешних сил.3. Теорема об изменении кинетической энергии системы: изменение кинетической энергии системы при любом её перемещении равняется сумме работ всех приложенных сил на том же перемещении. Для случая, когда все приложенные силы потенциальны, из теоремы вытекает <a href="/c/zakon-sokhraneniia-energii-c12c59" class="bre-preview-link" itemprop="url" data-external="false">закон сохранения механической энергии</a>.<h2 id="h2_dinamika_sistem_so_svyazyami">Динамика систем со связями</h2>В моделях, описывающих различные движения, происходящие в природе и технике, объекты рассматриваются как системы материальных точек и твёрдых тел, соединённых <a href="/c/mekhanicheskie-sviazi-30fcc2" class="bre-preview-link" itemprop="url" data-external="false">механическими связями</a>. В этих случаях в задачу динамики входит определение не только закона движения системы связанных точек и тел, но и сил реакции связей. Последние добавляются в уравнение (1), записываемое для каждой точки системы. Для систем с т. н. идеальными связями (для которых сумма элементарных работ всех реакций при любом возможном перемещении системы равна нулю) Ж. Д’Аламбер и Ж. Лагранж разработали общие методы составления уравнений движения, не содержащих реакций связей (см. <a href="/c/printsip-dalambera-8e46a8" class="bre-preview-link" itemprop="url" data-external="false">принцип Д’Аламбера</a> и <a href="/c/printsip-dalambera-lagranzha-e0c686" class="bre-preview-link" itemprop="url" data-external="false">принцип Д’Аламбера– Лагранжа</a>). Эти методы приводят к несколько иной формулировке общих теорем динамики (добавляются условия, налагаемые на связи), а <a href="/c/zakony-sokhraneniia-5e4f61" class="bre-preview-link" itemprop="url" data-external="false">законы сохранения</a> динамических мер приобретают математически строгую форму интегралов уравнений движения.Кинетическая энергия – скалярная величина, обладающая определённой универсальностью. Ж. Лагранж ввёл понятие <a href="/c/obobshchionnye-koordinaty-a8a098" class="bre-preview-link" itemprop="url" data-external="false">обобщённых координат</a> и записал кинетическую энергию в виде функции от обобщённых скоростей и обобщённых координат. Используя эту функцию, Лагранж вывел новую форму уравнений движения механических <a href="/c/golonomnye-sistemy-aa1f75" class="bre-preview-link" itemprop="url" data-external="false">голономных систем</a> (см. <a href="/c/uravneniia-lagranzha-d8b586" class="bre-preview-link" itemprop="url" data-external="false">уравнения Лагранжа</a>, <a href="/c/uravneniia-gamil-tona-8a716e" class="bre-preview-link" itemprop="url" data-external="false">уравнения Гамильтона</a>, <a href="/c/variatsionnye-printsipy-mekhaniki-7eccf8" class="bre-preview-link" itemprop="url" data-external="false">вариационные принципы механики</a>). Изучением свойств этих уравнений и их решений занимается <a href="/c/analiticheskaia-mekhanika-6d57cc" class="bre-preview-link" itemprop="url" data-external="false">аналитическая механика</a>, методы которой нашли широкое применение в различных областях физики.<h2 id="h2_dinamika_otnositel'nogo_dvizheniya">Динамика относительного движения</h2>Многие задачи механики сводятся к изучению движения одного объекта относительно другого, с которым нельзя связать инерциальную систему координат (например, движение тела относительно вращающейся Земли). В этом случае уравнение относительного движения материальной точки можно свести к виду (1), если к числу сил <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">{F}</annotation></semantics></math>F добавить силы инерции: переносную <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>e</mi></msub><mo>=</mo><mo>−</mo><mi>m</mi><msub><mi mathvariant="bold-italic">w</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">{F}_{e}=−m\boldsymbol{w}_{e}</annotation></semantics></math>Fe=−mwe и <a href="/c/sila-koriolisa-fa00d2" class="bre-preview-link" itemprop="url" data-external="false">кориолисову</a> <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>c</mi></msub><mo>=</mo><mo>−</mo><mi>m</mi><msub><mi mathvariant="bold-italic">w</mi><mi>c</mi></msub></mrow><annotation encoding="application/x-tex">{F}_{c} =−m\boldsymbol{w}_{c}</annotation></semantics></math>Fc=−mwc, где <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">w</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{w}_{e}</annotation></semantics></math>we, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">w</mi><mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\boldsymbol{w}_{c}</annotation></semantics></math>wc – соответственно переносное и кориолисово ускорения. Примерами задач динамики относительного движения могут служить задачи экспериментального доказательства вращения Земли (падение тела на вращающейся Земле с отклонением к востоку, <a href="/c/maiatnik-fuko-5028d1" class="bre-preview-link" itemprop="url" data-external="false">маятник Фуко</a>), задачи описания движения <a href="/c/liotchik-kosmonavt-bba5eb" class="bre-preview-link" itemprop="url" data-external="false">космонавта</a> относительно космической станции и др. На эффектах относительного движения основан предложенный <a href="/c/uatt-dzheims-e8197f" class="bre-preview-link" itemprop="url" data-external="false">Дж. Уаттом</a> <a href="/c/tsentrobezhnaia-sila-ffdd3a" class="bre-preview-link" itemprop="url" data-external="false">центробежный</a> регулятор угловой скорости вращения, используемый в технике.<h2 id="h2_dinamika_tvyordogo_tela">Динамика твёрдого тела</h2>В этом разделе динамики рассматриваются движения, в которых тело нельзя считать материальной точкой. Простейшая задача такого типа – задача о вращении абсолютно твёрдого тела вокруг неподвижной оси <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">{L}</annotation></semantics></math>L. В этом случае тело имеет одну <a href="/c/stepeni-svobody-958e77" class="bre-preview-link" itemprop="url" data-external="false">степень свободы</a>, его положение определяется одной обобщённой координатой – углом поворота <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex"> \varphi </annotation></semantics></math>φ. Производная <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex"> \varphi </annotation></semantics></math>φ по времени называется <a href="/c/uglovaia-skorost-9586b2" class="bre-preview-link" itemprop="url" data-external="false">угловой скоростью</a> <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex"> \omega </annotation></semantics></math>ω. В рассматриваемой задаче роль уравнения (1) играет уравнение вращения твёрдого тела: <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi mathvariant="bold-italic">I</mi><mi>ε</mi></msub><mo>=</mo><mi mathvariant="bold-italic">M</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{I}_{ε}=\boldsymbol{M}</annotation></semantics></math>Iε=M, где <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex"> \varepsilon </annotation></semantics></math>ε – <a href="/c/uglovoe-uskorenie-93d07c" class="bre-preview-link" itemprop="url" data-external="false">угловое ускорение</a> тела, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">I</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{I}</annotation></semantics></math>I – <a href="/c/moment-inertsii-tela-9ca691" class="bre-preview-link" itemprop="url" data-external="false">момент инерции</a> тела, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">M</mi></mrow><annotation encoding="application/x-tex">\boldsymbol{M}</annotation></semantics></math>M – <a href="/c/vrashchaiushchii-moment-cf93d0" class="bre-preview-link" itemprop="url" data-external="false">вращающий момент</a> (момент сил, приложенных к телу) относительно оси <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">{L}</annotation></semantics></math>L. Если <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="bold-italic">M</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\boldsymbol{M} = 0</annotation></semantics></math>M=0, то тело совершает вращение с постоянной угловой скоростью (угловое ускорение равно нулю).Эта задача применяется при моделировании вращающихся элементов машин (<a href="/c/rotor-743f14" class="bre-preview-link" itemprop="url" data-external="false">роторов</a>, <a href="/c/makhovik-79657d" class="bre-preview-link" itemprop="url" data-external="false">маховиков</a> и т. п.). В технических приложениях динамики твёрдого тела важно учитывать также силы реакции опор, на которых закреплена ось <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">{L}</annotation></semantics></math>L. Величина этих сил растёт пропорционально квадрату угловой скорости. Для машин с высокооборотными маховиками реакции настолько велики, что способны вызвать <a href="/c/deformatsiia-92bf2c" class="bre-preview-link" itemprop="url" data-external="false">деформацию</a> опор или оси и <a href="/c/vibratsiia-961b67" class="bre-preview-link" itemprop="url" data-external="false">вибрацию</a> машины. Для уменьшения вибраций (например, в автомобильном колесе) производится изменение распределения масс маховика – его балансировка, что достигается приближением центра масс к оси вращения (статическая балансировка) или приближением т. н. главной оси инерции тела к оси вращения (динамическая балансировка).Более сложная типовая задача этого раздела динамики – вращение твёрдого тела вокруг неподвижной точки <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">{O}</annotation></semantics></math>O. Для решения таких задач Л. Эйлер ввёл систему <a href="/c/dekartova-sistema-koordinat-4de670" class="bre-preview-link" itemprop="url" data-external="false">декартовых координат</a> <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mrow><mi>x</mi><mi>y</mi><mi>z</mi></mrow></mrow><annotation encoding="application/x-tex">{O}{xyz}</annotation></semantics></math>Oxyz, связанную с вращающимся телом. В данной задаче тело имеет 3 степени свободы, а его положение в выбранной системе координат часто определяют 3 углами (<a href="/c/ugly-eilera-2d95dd" class="bre-preview-link" itemprop="url" data-external="false">углами Эйлера</a>): углом <a href="/c/nutatsiia-3badf8" class="bre-preview-link" itemprop="url" data-external="false">нутации</a>, углом <a href="/c/pretsessiia-361a1c" class="bre-preview-link" itemprop="url" data-external="false">прецессии</a> и углом собственного вращения. Производные по времени от углов Эйлера связаны с проекциями вектора мгновенной угловой скорости вращения тела кинематическими уравнениями Эйлера. Направив оси <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">{x}</annotation></semantics></math>x, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">{y}</annotation></semantics></math>y, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">{z}</annotation></semantics></math>z по <a href="/c/glavnye-osi-inertsii-0e45be" class="bre-preview-link" itemprop="url" data-external="false">главным осям инерции</a> тела, Эйлер придал системе динамических уравнений вращения тела компактный и симметричный вид:<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>x</mi></msub><mi>d</mi><msub><mi>ω</mi><mi>x</mi></msub><mi mathvariant="normal">/</mi><mi>d</mi><mi>t</mi><mo>+</mo><mo stretchy="false">(</mo><msub><mi>I</mi><mi>z</mi></msub><mo>−</mo><msub><mi>I</mi><mi>y</mi></msub><mo stretchy="false">)</mo><msub><mi>ω</mi><mi>y</mi></msub><msub><mi>ω</mi><mi>z</mi></msub><mo>=</mo><msub><mi>M</mi><mi>x</mi></msub><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">{I}_{x}dω_{x}/dt+(I_{z}−I_{y})ω_{y}ω_{z}=M_{x},</annotation></semantics></math>Ixdωx/dt+(Iz−Iy)ωyωz=Mx,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>y</mi></msub><mi>d</mi><msub><mi>ω</mi><mi>y</mi></msub><mi mathvariant="normal">/</mi><mi>d</mi><mi>t</mi><mo>+</mo><mo stretchy="false">(</mo><msub><mi>I</mi><mi>x</mi></msub><mo>−</mo><msub><mi>I</mi><mi>z</mi></msub><mo stretchy="false">)</mo><msub><mi>ω</mi><mi>z</mi></msub><msub><mi>ω</mi><mi>x</mi></msub><mo>=</mo><msub><mi>M</mi><mi>y</mi></msub><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">I_{y}dω_{y}/dt+(I_{x}−I_{z})ω_{z}ω_{x}=M_{y},</annotation></semantics></math>Iydωy/dt+(Ix−Iz)ωzωx=My,<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>z</mi></msub><mi>d</mi><msub><mi>ω</mi><mi>z</mi></msub><mi mathvariant="normal">/</mi><mi>d</mi><mi>t</mi><mo>+</mo><mo stretchy="false">(</mo><msub><mi>I</mi><mi>y</mi></msub><mo>−</mo><msub><mi>I</mi><mi>x</mi></msub><mo stretchy="false">)</mo><msub><mi>ω</mi><mi>x</mi></msub><msub><mi>ω</mi><mi>y</mi></msub><mo>=</mo><msub><mi>M</mi><mi>z</mi></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">I_{z}dω_{z}/dt+(I_{y}−I_{x})ω_{x}ω_{y}=M_{z}.</annotation></semantics></math>Izdωz/dt+(Iy−Ix)ωxωy=Mz.Здесь <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">I_{x}</annotation></semantics></math>Ix, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">I_{y}</annotation></semantics></math>Iy, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>I</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">I_{z}</annotation></semantics></math>Iz – главные моменты инерции тела относительно осей <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>O</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">O_{x}</annotation></semantics></math>Ox, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>O</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">O_{y}</annotation></semantics></math>Oy, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>O</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">O_{z}</annotation></semantics></math>Oz; <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>M</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">M_{x}</annotation></semantics></math>Mx, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>M</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">M_{y}</annotation></semantics></math>My, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>M</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">M_{z}</annotation></semantics></math>Mz – моменты сил, приложенных к телу, относительно тех же осей; <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">ω_{x}</annotation></semantics></math>ωx, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>y</mi></msub></mrow><annotation encoding="application/x-tex">ω_{y}</annotation></semantics></math>ωy, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>ω</mi><mi>z</mi></msub></mrow><annotation encoding="application/x-tex">ω_{z}</annotation></semantics></math>ωz – проекции вектора мгновенной угловой скорости. Так как вращающие моменты могут зависеть от времени, углов Эйлера и угловых скоростей, решения этих уравнений известны лишь при частных предположениях о действующих силах и расположении масс в теле.Задача о движении свободного твёрдого тела, имеющего 6 степеней свободы, обсуждается в связи с проблемами моделирования поступательно-вращательного движения небесных тел, <a href="/c/raketa-f5791c" class="bre-preview-link" itemprop="url" data-external="false">ракет</a>, снарядов и других объектов. Для решения таких задач часто выбирается система координат, связанная с центром масс тела и движущаяся поступательно. Относительно такой системы координат рассматривается вращательное движение тела с применением методов динамики твёрдого тела.Помимо установления общих методов изучения движения под действием сил в динамике рассматриваются также специальные задачи: динамика гироскопических систем (см. <a href="/c/giroskop-648dcc" class="bre-preview-link" itemprop="url" data-external="false">гироскоп</a>), теория <a href="/c/kolebaniia-ef7a5d" class="bre-preview-link" itemprop="url" data-external="false">колебаний</a> механических систем, теория <a href="/c/ustoichivost-dvizheniia-1687f7" class="bre-preview-link" itemprop="url" data-external="false">устойчивости движения</a>, <a href="/c/mekhanika-tel-peremennoi-massy-fd1254" class="bre-preview-link" itemprop="url" data-external="false">механика тел переменной массы</a>, <a href="/c/teoriia-udara-194d8a" class="bre-preview-link" itemprop="url" data-external="false">теория удара</a> и др. В результате применения моделей динамики к изучению движения конкретных объектов возник ряд самостоятельных дисциплин: <a href="/c/nebesnaia-mekhanika-5830f4" class="bre-preview-link" itemprop="url" data-external="false">небесная механика</a>, <a href="/c/dinamika-mekhanizmov-i-mashin-5b0308" class="bre-preview-link" itemprop="url" data-external="false">динамика механизмов и машин</a>, <a href="/c/dinamika-poliota-ed2069" class="bre-preview-link" itemprop="url" data-external="false">динамика полёта</a> <a href="/c/letatel-nyi-apparat-ed614a" class="bre-preview-link" itemprop="url" data-external="false">летательных аппаратов</a>, динамика транспортных средств и др. С помощью законов динамики изучается также движение сплошной среды – упруго и пластически деформируемых тел, а также жидкостей и газов (см. <a href="/c/uprugost-7e7de2" class="bre-preview-link" itemprop="url" data-external="false">упругость</a>, <a href="/c/plastichnost-784fa2" class="bre-preview-link" itemprop="url" data-external="false">пластичность</a>, <a href="/c/gidrodinamika-31c37b" class="bre-preview-link" itemprop="url" data-external="false">гидродинамика</a>, <a href="/c/dinamika-razrezhennykh-gazov-7f45b2" class="bre-preview-link" itemprop="url" data-external="false">динамика разреженных газов</a>, <a href="/c/aerodinamika-c63b82" class="bre-preview-link" itemprop="url" data-external="false">аэродинамика</a>, <a href="/c/gazovaia-dinamika-f54512" class="bre-preview-link" itemprop="url" data-external="false">газовая динамика</a>).<a href="/a/va-samsonov-ea4e75" class="">Самсонов Виталий Александрович</a></section></section></div><meta itemprop="description" content="Дина́мика, раздел механики, посвящённый изучению изменения характеристик движения материальных тел под действием приложенных к ним сил. Динамика –...">Опубликовано 6 июня 2023 г. в 16:24 (GMT+3). Последнее обновление 27 марта 2024 г. в 16:19 (GMT+3).<button type="button" class="b-button tw-gap-2 b-button--link -text-button-text tw-rounded-lg tw-cursor-pointer" data-v-cfbedafc>Связаться с редакцией</button></div></div><div class="bre-tags-wrap"><a href="/l/razdely-mekhaniki-0bcf7d" class="bre-article-tag bre-article-tag__link _default _no-border" data-v-063d9480>#Разделы механики</a></div></div><aside class="bre-article-page__sidebar -show-on-desktop-s" style=""><nav class="bre-article-loc lg:tw-sticky"><div class="bre-article-loc-button">Содержание<svg viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"><path d="M6 9l6 6 6-6" stroke="currentColor" stroke-width="1.5" stroke-linecap="round"/></svg> <div class="bre-article-loc-short">Динамика материальной точки</div></div></nav><div class="bre-article-page__sidebar-wrapper"><div class="article-sidebar"><div class="article-sidebar-button -show-on-tablet -hide-on-desktop-s">Информация<svg viewBox="0 0 24 24" fill="none" xmlns="http://www.w3.org/2000/svg"><path d="M6 9l6 6 6-6" stroke="currentColor" stroke-width="1.5" stroke-linecap="round"/></svg> <div class="article-sidebar-text -show-on-tablet -hide-on-desktop-s"></div></div><div class="article-sidebar-wrapper -hide-on-tablet"><meta itemprop="name" content="Физика"><meta itemprop="caption" content="Физика. Научно-образовательный портал «Большая российская энциклопедия»"><img src="https://i.bigenc.ru/resizer/resize?sign=7JK17_fgqWlQEFIk2cRkfA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=120" onerror="this.setAttribute('data-error', 1)" alt="Физика" data-nuxt-img sizes="320px" srcset="https://i.bigenc.ru/resizer/resize?sign=7JK17_fgqWlQEFIk2cRkfA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=120 120w,https://i.bigenc.ru/resizer/resize?sign=Jf8Ovt6NK1CJRMEXmLmu9w&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=320 320w,https://i.bigenc.ru/resizer/resize?sign=9FmjZNIS1_JG-eBy3nkCow&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=480 480w,https://i.bigenc.ru/resizer/resize?sign=W0YAxakNej-ihBYTmKOUhA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=640 640w,https://i.bigenc.ru/resizer/resize?sign=bU5vxPnJKBxMhvgLEjl-uA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=768 768w,https://i.bigenc.ru/resizer/resize?sign=CO7eqX0CglCAJmsuCYDJxQ&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=1024 1024w,https://i.bigenc.ru/resizer/resize?sign=SNrDJXfJeDUaOjs9TGABPA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=1280 1280w,https://i.bigenc.ru/resizer/resize?sign=A65s2m2zZF6hgTDDppMoDA&filename=vault/2a7425e2f5716d70117aeb7f30155e04.webp&width=1920 1920w" title="Физика" class="" itemprop="contentUrl"><div class="article-sidebar-meta"><dl class="tw-mt-0"><dt>Области знаний:</dt><dd>Классическая механика Ньютона</dd></dl></div></div></div></div></aside></div></article></div></div><div></div></main><footer class="bre-footer" itemscope itemprop="hasPart" itemtype="https://schema.org/WPFooter"><meta itemprop="copyrightNotice" content="&copy;&nbsp;АНО БРЭ, 2022&nbsp;&mdash;&nbsp;2024. 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Свидетельство о регистрации СМИ ЭЛ № ФС77-84198, выдано Федеральной службой по надзору в сфере связи, информационных технологий и массовых коммуникаций (Роскомнадзор) 15 ноября 2022 года. ISSN: 2949-2076</li><li class="_footer-text bre-inline-menu__item">Учредитель: Автономная некоммерческая организация «Национальный научно-образовательный центр «Большая российская энциклопедия» Главный редактор: Кравец С. Л. 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