{"id":25623,"date":"2022-07-12T00:00:26","date_gmt":"2022-07-12T00:00:26","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25623"},"modified":"2024-05-21T09:42:27","modified_gmt":"2024-05-21T09:42:27","slug":"les-transformations-de-galilee-et-leurs-limitations","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/les-transformations-de-galilee-et-leurs-limitations\/","title":{"rendered":"Les Transformations de Galil\u00e9e et Leurs Limitations"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Les Transformations de Galil\u00e9e et Leurs Limitations<\/h1>\n<p class=\"eq\"><em><strong>R\u00e9sum\u00e9 :<\/strong><br \/>\nLe principe de la relativit\u00e9 sugg\u00e8re que les observations d\u00e9pendent du cadre inertiel, mais de telle mani\u00e8re que les lois physiques restent coh\u00e9rentes. Une approche initiale et intuitive de ce principe provient des Transformations de Galil\u00e9e, qui mod\u00e9lisent comment les observations changent entre les cadres de r\u00e9f\u00e9rence inertiels en m\u00e9canique classique. Dans ce cours, nous \u00e9tudierons ces transformations et leurs propri\u00e9t\u00e9s, et nous verrons \u00e9galement comment elles \u00e9chouent lorsqu&#8217;appliqu\u00e9es au ph\u00e9nom\u00e8ne de propagation des ondes.<\/br><\/em><\/p>\n<p><strong>OBJECTIFS D&#8217;APPRENTISSAGE<\/strong><br \/>\n\u00c0 la fin de ce cours, les \u00e9tudiants seront capables de :\n<\/p>\n<p><\/center><\/p>\n<ol>\n<li><strong>Reconna\u00eetre<\/strong> les concepts fondamentaux des Transformations de Galil\u00e9e, y compris leur formulation de base et leurs principes sous-jacents.<\/li>\n<li><strong>Analyser<\/strong> la g\u00e9om\u00e9trie galil\u00e9enne de l&#8217;espace et du temps et sa s\u00e9paration dans le cadre de la m\u00e9canique classique.<\/li>\n<li><strong>\u00c9valuer<\/strong> les limites des Transformations de Galil\u00e9e lorsqu&#8217;appliqu\u00e9es \u00e0 des ph\u00e9nom\u00e8nes tels que la propagation des ondes et leur pertinence dans l&#8217;avancement vers la th\u00e9orie de la relativit\u00e9 restreinte.<\/li>\n<\/ol>\n<p><center><\/p>\n<p class=\"indx\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\"><strong>Formulation des Transformations de Galil\u00e9e<\/strong><\/a><br \/>\n<a href=\"#2\">La Transformation Inverse<\/a><br \/>\n<a href=\"#3\">Le Temps Absolu et la Somme des Vitesses<\/a><br \/>\n<a href=\"#4\">G\u00e9om\u00e9trie Galil\u00e9enne de l&#8217;Espace et du Temps<\/a><br \/>\n<a href=\"#5\"><strong>La Relativit\u00e9 de Galil\u00e9e et les Lois Physiques<\/strong><\/a><br \/>\n<a href=\"#6\">Appliqu\u00e9e \u00e0 la Dynamique Newtonienne<\/a><br \/>\n<a href=\"#7\">Appliqu\u00e9e \u00e0 la Propagation des Ondes<\/a><br \/>\n<a href=\"#8\">Quel Effet les Transformations de Galil\u00e9e ont-elles sur la Propagation des Ondes ?<\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ku-9nbTSaJg?si=1dmuCtjzBPUy14v_\" title=\"Lecteur vid\u00e9o YouTube\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><br \/>\n<\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Formulation des Transformations de Galil\u00e9e<\/h2>\n<p style=\"text-align:justify;\">La physique newtonienne repose sur le principe de la relativit\u00e9 mod\u00e9lis\u00e9 \u00e0 travers les Transformations de Galil\u00e9e, o\u00f9 le temps est \u00e9tabli comme une coordonn\u00e9e universelle pour tous les observateurs inertiels ; c&#8217;est-\u00e0-dire : <span class=\"katex-eq\" data-katex-display=\"false\">t=t^\\prime<\/span>. Sous cette d\u00e9claration, la transformation lin\u00e9aire reliant les observations de deux cadres de r\u00e9f\u00e9rence inertiels <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> examin\u00e9e dans la classe sur <a href=\"http:\/\/toposuranos.com\/material\/es\/el-principio-de-relatividad-especial\/?fbclid=IwAR2_lpF1hyUlbSzFQ5B5OQpPu1Vhc_20zTGu8D4pxKPsSvzZRvLSzPdwQXU\" rel=\"noopener\" target=\"_blank\">le principe de la relativit\u00e9 restreinte<\/a> prend la forme d&#8217;une transformation lin\u00e9aire :<\/p>\n<p><a name=\"eq1\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nt^\\prime &amp;= At + Bx,\\\\ x^\\prime &amp;= Dt + Ex,\\\\ y^\\prime &amp;= y, \\\\ z^\\prime &amp;=z,\n\n\\end{array}<\/span> <strong>[1]<\/strong><\/p>\n<p style=\"text-align:justify;\">elle prend la forme suivante lorsque les cadres inertiels <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> sont en configuration standard et <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> se d\u00e9place avec la vitesse <span class=\"katex-eq\" data-katex-display=\"false\">v_{ss^\\prime_x}\\hat{x}<\/span> par rapport \u00e0 <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio.png\" alt=\"transformations de coordonn\u00e9es\" width=\"1374\" height=\"741\" class=\"aligncenter size-full wp-image-25502 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio.png\" alt=\"transformations de coordonn\u00e9es\" width=\"1374\" height=\"741\" class=\"aligncenter size-full wp-image-25502 lazyload\" srcset=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio.png 1374w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-300x162.png 300w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-1024x552.png 1024w, https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-768x414.png 768w\" sizes=\"(max-width: 1374px) 100vw, 1374px\" \/><\/noscript><\/center><\/p>\n<p><a name=\"eq2\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rlr}\n\n{}t^\\prime &amp;= t  \\\\ x^\\prime &amp;= x - v_{ss^\\prime_x}t \\\\ y^\\prime &amp;= y \\\\ z^\\prime &amp;= z\n\n\\end{array}\n\n<\/span> <strong>[2]<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>La Transformation Inverse<\/h3>\n<p style=\"text-align:justify;\">D&#8217;une sorte de sym\u00e9trie alg\u00e9brique, nous pouvons \u00e9crire la transformation inverse :<\/p>\n<p><a name=\"eq3\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nt &amp;= t^\\prime \\\\ x &amp;= x^\\prime + v_{ss^\\prime_x}t \\\\ y &amp;= y^\\prime \\\\ z &amp;= z^\\prime \\end{array}\n\n<\/span> <strong>[3]<\/strong><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Le Temps Absolu et la Somme des Vitesses<\/h3>\n<p style=\"text-align:justify;\">D&#8217;apr\u00e8s la premi\u00e8re \u00e9quation des transformations de Galil\u00e9e (l&#8217;une des deux, <a href=\"#eq2\">[2]<\/a> ou <a href=\"#eq3\">[3]<\/a>), il est \u00e9vident que la coordonn\u00e9e temporelle d&#8217;un \u00e9v\u00e9nement ne d\u00e9pend pas du cadre depuis lequel il est observ\u00e9, tandis que la seconde permet d&#8217;obtenir ce qui est couramment compris comme le \u00ab bon sens \u00bb associ\u00e9 \u00e0 la somme des vitesses. Si une particule se d\u00e9place avec une vitesse constante <span class=\"katex-eq\" data-katex-display=\"false\">v_{ss^\\prime_x}<\/span> le long de l&#8217;axe <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span> de <span class=\"katex-eq\" data-katex-display=\"false\">S,<\/span> alors sa vitesse dans <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> est d\u00e9termin\u00e9e par<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle v^\\prime_x = \\frac{dx^\\prime}{dt^\\prime} = \\frac{dx^\\prime}{dt} = \\frac{d}{dt}\\left(x - v_{ss^\\prime_x} t \\right) = v_x - v_{ss^\\prime_x}<\/span>\n<p style=\"text-align:justify;\">D\u00e9rivant dans cette derni\u00e8re expression montre que l&#8217;acc\u00e9l\u00e9ration de toute particule est la m\u00eame dans <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> et dans <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span>, c&#8217;est-\u00e0-dire : <span class=\"katex-eq\" data-katex-display=\"false\">dv^\\prime_x\/dt^\\prime = dv_x\/dt<\/span>.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>La G\u00e9om\u00e9trie Galil\u00e9enne de l&#8217;Espace et du Temps<\/h3>\n<p style=\"text-align:justify;\">Si nous consid\u00e9rons deux \u00e9v\u00e9nements <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> qui ont des coordonn\u00e9es <span class=\"katex-eq\" data-katex-display=\"false\">(t_A,x_A,y_A,z_A)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">(t_B,x_B,y_B,z_B),<\/span> respectivement. Il est facile de voir que les quantit\u00e9s <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta t = t_B - t_A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta r^2 = \\Delta x^2 + \\Delta y^2 + \\Delta z^2<\/span> sont s\u00e9par\u00e9ment invariantes sous les transformations de Galil\u00e9e, nous amenant \u00e0 consid\u00e9rer l&#8217;espace et le temps comme des entit\u00e9s s\u00e9par\u00e9es. D&#8217;autre part, <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta r^2<\/span> sugg\u00e8re que c&#8217;est une propri\u00e9t\u00e9 g\u00e9om\u00e9trique de l&#8217;espace lui-m\u00eame. Nous reconnaissons <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta r^2<\/span> comme le carr\u00e9 de la distance entre les \u00e9v\u00e9nements dans l&#8217;espace euclidien. Ceci d\u00e9finit la g\u00e9om\u00e9trie de l&#8217;espace et du temps dans le contexte de la m\u00e9canique newtonienne.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>La Relativit\u00e9 de Galil\u00e9e et les Lois Physiques<\/h2>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Appliqu\u00e9e \u00e0 la Dynamique Newtonienne<\/h3>\n<p style=\"text-align:justify;\">Dans la section pr\u00e9c\u00e9dente, nous avons vu que, dans le contexte de la physique newtonienne, deux cadres inertiels diff\u00e9rents observeront toujours les m\u00eames acc\u00e9l\u00e9rations. Ceci, coupl\u00e9 \u00e0 la seconde loi de Newton, implique que tous les cadres inertiels observeront toujours la m\u00eame dynamique. C&#8217;est-\u00e0-dire :<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F_x = m\\frac{dv_x}{dt}= m\\frac{dv^\\prime_x}{dt^\\prime} = F^\\prime_x.<\/span> <\/p>\n<p style=\"text-align:justify;\">Cette derni\u00e8re expression nous indique que <strong>la physique ne change pas lors de l&#8217;ex\u00e9cution des transformations de Galil\u00e9e,<\/strong> ce qui revient \u00e0 dire que : la physique est la m\u00eame pour tous les observateurs inertiels.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Appliqu\u00e9e \u00e0 la Propagation des Ondes<\/h3>\n<p style=\"text-align:justify;\">Alors que la persistance de la physique face aux changements d&#8217;observateurs inertiels est attendue, d&#8217;abord parce que c&#8217;est ce que nous observons en nous d\u00e9pla\u00e7ant, et ensuite parce que c&#8217;est ce qui a \u00e9t\u00e9 obtenu \u00e0 travers des calculs pr\u00e9c\u00e9dents, ce n&#8217;est pas toujours le cas. Le cas le plus notable d&#8217;un ph\u00e9nom\u00e8ne qui ne se conserve pas sous les transformations galil\u00e9ennes est la propagation des ondes ; g\u00e9n\u00e9ralement, l&#8217;\u00e9quation qui mod\u00e9lise la propagation d&#8217;une onde <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> dans l&#8217;espace et le temps est sous la forme<\/p>\n<p><a name=\"eq4\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\nabla^2 \\psi = \\frac{1}{v_0^2}\\frac{\\partial^2 \\psi}{\\partial t^2}<\/span> <strong>[4]<\/strong><\/p>\n<p style=\"text-align:justify;\">o\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">v_0<\/span> est la vitesse de propagation de l&#8217;onde.<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h4>Quel effet les transformations de Galil\u00e9e ont-elles sur la propagation des ondes ?<\/h4>\n<p style=\"text-align:justify;\">Pour cela, il y a une r\u00e9ponse courte et une r\u00e9ponse longue. La r\u00e9ponse courte est que \u00abm\u00eame en observant le m\u00eame ph\u00e9nom\u00e8ne, diff\u00e9rents observateurs inertiels verront une &#8216;physique diff\u00e9rente'\u00bb. La r\u00e9ponse longue implique d&#8217;examiner comment l&#8217;\u00e9quation de propagation des ondes change lorsque la transformation de Galil\u00e9e est appliqu\u00e9e ; pour ce faire, nous prenons d&#8217;abord l&#8217;\u00e9quation <a href=\"#eq3\">[4]<\/a> et l&#8217;\u00e9tendons sur chacune de ses coordonn\u00e9es en obtenant :<\/p>\n<p><a name=\"eq5\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial ^2 \\psi}{\\partial x^2} + \\frac{\\partial ^2 \\psi}{\\partial y^2} + \\frac{\\partial ^2 \\psi}{\\partial z^2} = \\frac{1}{v_0^2} \\frac{\\partial ^2 \\psi}{\\partial t^2}.<\/span> <strong>[5]<\/strong><\/p>\n<p>Avec cette \u00e9quation en main, nous devons maintenant utiliser les \u00e9quations de <a href=\"#eq3\">[3]<\/a> pour r\u00e9exprimer les d\u00e9riv\u00e9es dans l&#8217;autre cadre inertiel.<\/p>\n<h5>Transformation des premi\u00e8res d\u00e9riv\u00e9es<\/h5>\n<p style=\"text-align:justify;\">En suivant les expressions de <a href=\"#eq3\">[3]<\/a> et en d\u00e9rivant chaque variable par rapport aux variables prim\u00e9es, nous obtenons :<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial x^\\prime}{\\partial x} = \\frac{\\partial y^\\prime}{\\partial y} = \\frac{\\partial z^\\prime}{\\partial z} = \\frac{\\partial t^\\prime}{\\partial t} = 1<\/span>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial x^\\prime}{\\partial t} = - v_{x_0}<\/span>\n<p style=\"text-align:justify;\">Alors que toutes les autres sont annul\u00e9es :<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial t^\\prime}{\\partial x} = \\frac{\\partial t^\\prime}{\\partial y} = \\frac{\\partial t^\\prime}{\\partial z} = \\frac{\\partial x^\\prime}{\\partial y} = \\frac{\\partial x^\\prime}{\\partial z} = \\frac{\\partial y^\\prime}{\\partial x} = \\frac{\\partial y^\\prime}{\\partial z} = \\frac{\\partial y^\\prime}{\\partial t} = \\frac{\\partial z^\\prime}{\\partial x} = \\frac{\\partial z^\\prime}{\\partial y} = \\frac{\\partial z^\\prime}{\\partial t} = 0\n\n<\/span>\n<p style=\"text-align:justify;\">Avec cela en main, nous pouvons maintenant calculer les d\u00e9riv\u00e9es de <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> en utilisant la r\u00e8gle de la cha\u00eene :<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{\\partial \\psi}{\\partial x} = \\frac{\\partial \\psi}{\\partial x^\\prime} \\underbrace{\\frac{\\partial x^\\prime}{\\partial x}}_{=1} + \\frac{\\partial \\psi}{\\partial y^\\prime} \\underbrace{\\frac{\\partial y^\\prime}{\\partial x}}_{=0} +\n\n\\frac{\\partial \\psi}{\\partial z^\\prime} \\underbrace{\\frac{\\partial z^\\prime}{\\partial x}}_{=0} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\underbrace{\\frac{\\partial t^\\prime}{\\partial x}}_{=0} =  \\frac{\\partial \\psi}{\\partial x^\\prime}.<\/span>\n<p>Et de mani\u00e8re analogue pour les deux autres variables spatiales :<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{\\partial \\psi}{\\partial y} =  \\frac{\\partial \\psi}{\\partial y^\\prime}.<\/span>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{\\partial \\psi}{\\partial z} =  \\frac{\\partial \\psi}{\\partial z^\\prime}.<\/span>\n<p style=\"text-align:justify;\">Cependant, la d\u00e9riv\u00e9e temporelle montrera certaines diff\u00e9rences :<\/p>\n<p id=\"eq\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\frac{\\partial \\psi}{\\partial t} &amp;= \\displaystyle\\frac{\\partial \\psi}{\\partial x^\\prime}\\underbrace{\\frac{\\partial x^\\prime}{\\partial t}}_{=-v_{x_0}} + \\frac{\\partial \\psi}{\\partial y^\\prime}\\underbrace{\\frac{\\partial y^\\prime}{\\partial t}}_{=0} + \\frac{\\partial \\psi}{\\partial z^\\prime}\\underbrace{\\frac{\\partial z^\\prime}{\\partial t}}_{=0} + \\frac{\\partial \\psi}{\\partial t^\\prime}\\underbrace{\\frac{\\partial t^\\prime}{\\partial t}}_{=1}\\\\ &amp;=\\displaystyle -v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime},\n\n\\end{array}\n\n<\/span>\n<h5>Transformation des secondes d\u00e9riv\u00e9es<\/h5>\n<p style=\"text-align:justify;\">Pour la partie spatiale, nous pouvons continuer sans difficult\u00e9s majeures, les r\u00e9sultats sont :<\/p>\n<p><a name=\"eq6\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial^2 \\psi}{\\partial x^2} = \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2}.<\/span> <strong>[6]<\/strong><\/p>\n<p><a name=\"eq7\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial^2 \\psi}{\\partial y^2} = \\frac{\\partial^2 \\psi}{\\partial {y^\\prime}^2}<\/span> <strong>[7]<\/strong><\/p>\n<p><a name=\"eq8\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial^2 \\psi}{\\partial z^2} = \\frac{\\partial^2 \\psi}{\\partial {z^\\prime}^2}<\/span> <strong>[8]<\/strong><\/p>\n<p style=\"text-align:justify;\">Mais la partie temporelle, comme nous pouvions d\u00e9j\u00e0 l&#8217;anticiper \u00e0 partir des premi\u00e8res d\u00e9riv\u00e9es, montre de grandes diff\u00e9rences :<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle\\frac{\\partial^2 \\psi}{\\partial t^2} &amp;=\\displaystyle \\frac{\\partial}{\\partial t}\\left( -v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\right)\\\\ &amp; \\displaystyle = -v_{x_0} \\frac{\\partial }{\\partial t} \\left(\\frac{\\partial \\psi}{\\partial x^\\prime} \\right) + \\frac{\\partial }{\\partial t} \\left(\\frac{\\partial \\psi}{\\partial t^\\prime} \\right)\\\\ &amp;\\displaystyle = -v_{x_0} \\frac{\\partial }{\\partial x^\\prime} \\left(\\frac{\\partial \\psi}{\\partial t} \\right) + \\frac{\\partial }{\\partial t^\\prime} \\left(\\frac{\\partial \\psi}{\\partial t} \\right)\\\\ &amp;\\displaystyle = -v_{x_0} \\frac{\\partial }{\\partial x^\\prime} \\left(-v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\right) + \\frac{\\partial }{\\partial t^\\prime} \\left(-v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\right) \\end{array}<\/span>\n<p><a name=\"eq9\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle\\frac{\\partial^2 \\psi}{\\partial t^2}  = v_{x_0}^2 \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2} - 2v_{x_0}\\frac{\\partial^2 \\psi}{\\partial x^\\prime \\partial t^\\prime} + \\frac{\\partial^2 \\psi}{\\partial {t^\\prime}^2}.<\/span> <strong>[9]<\/strong><\/p>\n<h5>Application des Transformations de Galil\u00e9e \u00e0 la Propagation des Ondes<\/h5>\n<p style=\"text-align:justify;\">Ainsi, il est possible de r\u00e9aliser la transformation de Galil\u00e9e sur l&#8217;\u00e9quation de propagation des ondes en rempla\u00e7ant les \u00e9quations [<a href=\"#eq6\">6<\/a>,<a href=\"#eq7\">7<\/a>,<a href=\"#eq8\">8<\/a>] et [<a href=\"#eq9\">9<\/a>] par [<a href=\"#eq5\">5<\/a>], ce qui donne :<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2} + \\frac{\\partial^2 \\psi}{\\partial {y^\\prime}^2} + \\frac{\\partial^2 \\psi}{\\partial {z^\\prime}^2} = \\frac{1}{v_0^2} \\left(\\color{red}{ v_{x_0}^2 \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2} - 2v_{x_0}\\frac{\\partial^2 \\psi}{\\partial x^\\prime \\partial t^\\prime}} + \\frac{\\partial^2 \\psi}{\\partial {t^\\prime}^2} \\right).\n\n<\/span> <strong>[10]<\/strong><\/p>\n<p style=\"text-align:justify;\">On observe que la forme de la propagation des ondes ne se maintient pas sous les transformations de Galil\u00e9e en raison de l&#8217;apparition des termes suppl\u00e9mentaires marqu\u00e9s en rouge. Bien que cela n&#8217;ait pas de cons\u00e9quences majeures pour l&#8217;instant, dans les cours futurs, nous verrons que c&#8217;est pr\u00e9cis\u00e9ment le point qui \u00abrompt\u00bb, pour ainsi dire, avec la physique classique, ouvrant la voie \u00e0 la relativit\u00e9 sp\u00e9ciale.<\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\">\n<h2>Conclusions<\/h2>\n<p style=\"text-align:justify;\">\n        Les Transformations de Galil\u00e9e, fondamentales en m\u00e9canique classique, \u00e9tablissent un cadre pour comprendre comment les observations changent entre diff\u00e9rents cadres de r\u00e9f\u00e9rence inertiels. Gr\u00e2ce \u00e0 cette \u00e9tude, nous avons reconnu le concept de temps absolu et l&#8217;addition des vitesses comme piliers de la g\u00e9om\u00e9trie galil\u00e9enne de l&#8217;espace et du temps. Cependant, nous avons d\u00e9couvert des limitations significatives de ces transformations, en particulier dans leur application \u00e0 la propagation des ondes. Cette analyse souligne la n\u00e9cessit\u00e9 d&#8217;une approche plus complexe pour d\u00e9crire l&#8217;univers physique, nous conduisant vers la relativit\u00e9 sp\u00e9ciale et au-del\u00e0 de l&#8217;intuition classique. En r\u00e9sum\u00e9, bien que les Transformations de Galil\u00e9e fournissent une base solide en physique classique, leur inad\u00e9quation dans certains ph\u00e9nom\u00e8nes met en \u00e9vidence l&#8217;\u00e9volution constante de notre compr\u00e9hension de l&#8217;univers.\n    <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Les Transformations de Galil\u00e9e et Leurs Limitations R\u00e9sum\u00e9 : Le principe de la relativit\u00e9 sugg\u00e8re que les observations d\u00e9pendent du cadre inertiel, mais de telle mani\u00e8re que les lois physiques restent coh\u00e9rentes. Une approche initiale et intuitive de ce principe provient des Transformations de Galil\u00e9e, qui mod\u00e9lisent comment les observations changent entre les cadres de [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25587,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":47,"footnotes":""},"categories":[647,703],"tags":[],"citadela-post-location":[],"class_list":["post-25623","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physique","category-relativite"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Les Transformations de Galil\u00e9e et Leurs Limitations - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez les Transformations de Galil\u00e9e en M\u00e9canique Classique et Leurs Limitations Intrigantes dans Cette 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