https://mizugadro.mydns.jp/t/index.php?title=Classical_mechanics&feed=atom&action=historyClassical mechanics - Revision history2024-03-28T20:05:50ZRevision history for this page on the wikiMediaWiki 1.31.16https://mizugadro.mydns.jp/t/index.php?title=Classical_mechanics&diff=27998&oldid=prevT: Text replacement - "\$([^\$]+)\$" to "\\(\1\\)"2019-07-30T09:26:10Z<p>Text replacement - "\$([^\$]+)\$" to "\\(\1\\)"</p>
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 09:26, 30 July 2019</td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The coordinates of each point are smooth functions of time.</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The coordinates of each point are smooth functions of time.</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For simplicity, assume that coordinate also mean the 3-vector function dependent on time.</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For simplicity, assume that coordinate also mean the 3-vector function dependent on time.</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>This function is interpreted as trajectory. The time is often denoted with letter <del class="diffchange diffchange-inline">$</del>t<del class="diffchange diffchange-inline">$</del>.</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>This function is interpreted as trajectory. The time is often denoted with letter <ins class="diffchange diffchange-inline">\(</ins>t<ins class="diffchange diffchange-inline">\)</ins>.</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Several material points may form a physical body, or physical system. The coordinate of the physical system is assumed to be combination of coordinates of all its material points.</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Several material points may form a physical body, or physical system. The coordinate of the physical system is assumed to be combination of coordinates of all its material points.</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Often, the coordinates are denoted with letter <del class="diffchange diffchange-inline">$</del>\vec x= \{ x[1], x[2], x[2]\}<del class="diffchange diffchange-inline">$</del>.</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Often, the coordinates are denoted with letter <ins class="diffchange diffchange-inline">\(</ins>\vec x= \{ x[1], x[2], x[2]\}<ins class="diffchange diffchange-inline">\)</ins>.</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>If the body counts with several material points, say, <del class="diffchange diffchange-inline">$</del>N<del class="diffchange diffchange-inline">$</del> points,</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>If the body counts with several material points, say, <ins class="diffchange diffchange-inline">\(</ins>N<ins class="diffchange diffchange-inline">\)</ins> points,</div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>then the material points are numbered and denoted as </div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>then the material points are numbered and denoted as </div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>: <del class="diffchange diffchange-inline">$</del>\vec x_n=\vec x_n(t)<del class="diffchange diffchange-inline">$</del>, <del class="diffchange diffchange-inline">$</del>n=1.. N<del class="diffchange diffchange-inline">$</del>.</div></td>
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<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>: <ins class="diffchange diffchange-inline">\(</ins>\vec x_n=\vec x_n(t)<ins class="diffchange diffchange-inline">\)</ins>, <ins class="diffchange diffchange-inline">\(</ins>n=1.. N<ins class="diffchange diffchange-inline">\)</ins>.</div></td>
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<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The component of each vector is not required for formulation of the basic axioms of classical mechanics,</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The component of each vector is not required for formulation of the basic axioms of classical mechanics,</div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>but if necessary, they can be indicated with an additional subscript:</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>but if necessary, they can be indicated with an additional subscript:</div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>: <del class="diffchange diffchange-inline">$</del>\vec x_n=\{ x_n[1], x_n[2], x_n[3] \} = \{ x_{n,1}, x_{n,2}, x_{n,3}\}<del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>: <ins class="diffchange diffchange-inline">\(</ins>\vec x_n=\{ x_n[1], x_n[2], x_n[3] \} = \{ x_{n,1}, x_{n,2}, x_{n,3}\}<ins class="diffchange diffchange-inline">\)</ins></div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, in the last case, it is important to keep the order of subscripts to avoid confusions. </div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, in the last case, it is important to keep the order of subscripts to avoid confusions. </div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Each material point is attributed its universal property, called [[mass]]. The material points may have different masses;</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Each material point is attributed its universal property, called [[mass]]. The material points may have different masses;</div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>in this case, they are numbered and denoted with <del class="diffchange diffchange-inline">$</del>m_n<del class="diffchange diffchange-inline">$</del>, <del class="diffchange diffchange-inline">$</del>n=1.. N<del class="diffchange diffchange-inline">$</del>.</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>in this case, they are numbered and denoted with <ins class="diffchange diffchange-inline">\(</ins>m_n<ins class="diffchange diffchange-inline">\)</ins>, <ins class="diffchange diffchange-inline">\(</ins>n=1.. N<ins class="diffchange diffchange-inline">\)</ins>.</div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In such a way, that material point of mass <del class="diffchange diffchange-inline">$</del>m_n<del class="diffchange diffchange-inline">$</del> has coordinates <del class="diffchange diffchange-inline">$</del>\vec x_n=\vec x_n(t)<del class="diffchange diffchange-inline">$</del>.</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In such a way, that material point of mass <ins class="diffchange diffchange-inline">\(</ins>m_n<ins class="diffchange diffchange-inline">\)</ins> has coordinates <ins class="diffchange diffchange-inline">\(</ins>\vec x_n=\vec x_n(t)<ins class="diffchange diffchange-inline">\)</ins>.</div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Mass of each material point is constant and cannot vary with time. </div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Mass of each material point is constant and cannot vary with time. </div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In some cases, the detailed state of the body is not important for the consideration; then the body is characterised with its mean coordinate called also </div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In some cases, the detailed state of the body is not important for the consideration; then the body is characterised with its mean coordinate called also </div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[center of mass]]; the coordinate of center of mass of a body of <del class="diffchange diffchange-inline">$</del>N<del class="diffchange diffchange-inline">$</del> material points is defined with</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[center of mass]]; the coordinate of center of mass of a body of <ins class="diffchange diffchange-inline">\(</ins>N<ins class="diffchange diffchange-inline">\)</ins> material points is defined with</div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>: <del class="diffchange diffchange-inline">$</del>\displaystyle X(t)= \sum_{n=1}^{N} m_n~ x_n(t)<del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>: <ins class="diffchange diffchange-inline">\(</ins>\displaystyle X(t)= \sum_{n=1}^{N} m_n~ x_n(t)<ins class="diffchange diffchange-inline">\)</ins></div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>At the scientific [[slang]], the coordinate of center of mass of a body can be called simply</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>At the scientific [[slang]], the coordinate of center of mass of a body can be called simply</div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>"coordinate of a body", although it actually consists of 3 components of vector <del class="diffchange diffchange-inline">$</del>\vec X<del class="diffchange diffchange-inline">$</del> and, in principle, may depend on time. </div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>"coordinate of a body", although it actually consists of 3 components of vector <ins class="diffchange diffchange-inline">\(</ins>\vec X<ins class="diffchange diffchange-inline">\)</ins> and, in principle, may depend on time. </div></td>
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<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The time derivative of coordinate is called [[velocity]] and denoted with letter <del class="diffchange diffchange-inline">$</del>\vec v<del class="diffchange diffchange-inline">$</del>;</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The time derivative of coordinate is called [[velocity]] and denoted with letter <ins class="diffchange diffchange-inline">\(</ins>\vec v<ins class="diffchange diffchange-inline">\)</ins>;</div></td>
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<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>: <del class="diffchange diffchange-inline">$</del>\vec v_n(t)=\vec x_n'(t)<del class="diffchange diffchange-inline">$</del>, where prime means derivative.</div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>: <ins class="diffchange diffchange-inline">\(</ins>\vec v_n(t)=\vec x_n'(t)<ins class="diffchange diffchange-inline">\)</ins>, where prime means derivative.</div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>All velocities are differentiable functions, and the time derivative of velocity is called acceleration:</div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>All velocities are differentiable functions, and the time derivative of velocity is called acceleration:</div></td>
</tr>
<tr>
<td class="diff-marker">−</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>: <del class="diffchange diffchange-inline">$</del>\vec w_n(t)=\vec v_n'(t)=\vec x_n''(t)<del class="diffchange diffchange-inline">$</del></div></td>
<td class="diff-marker">+</td>
<td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>: <ins class="diffchange diffchange-inline">\(</ins>\vec w_n(t)=\vec v_n'(t)=\vec x_n''(t)<ins class="diffchange diffchange-inline">\)</ins></div></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td>
</tr>
<tr>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The interaction between material points is characterized with [[force]]s. The properties of masses, coordinates and forces are </div></td>
<td class="diff-marker"> </td>
<td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The interaction between material points is characterized with [[force]]s. The properties of masses, coordinates and forces are </div></td>
</tr>
</table>Thttps://mizugadro.mydns.jp/t/index.php?title=Classical_mechanics&diff=796&oldid=prevMaintenance script at 05:59, 20 June 20132013-06-20T05:59:45Z<p></p>
<p><b>New page</b></p><div>'''Classical mechanics''' ( [[классическая механика]], or "Newtonian mechanics") is [[Science]], branch of [[physics]] and [[mathematics]] based on the concepts of [[coordinate]], [[time]] and [[material points]] ([[point mass]], [[материальная точка]]);<br />
the coordinates are elements of the 3-dimensional [[Euclidean space]], and time is universal real parameter.<br />
The basic elements of the classical mechanics are listed below.<br />
<br />
==Basic concepts==<br />
All the world is supposed to be made of [[material point]]s. These material points are merged in the [[Euclidean space]].<br />
By default, the Catresian orthogonal system of coordinates is assumed (id est, the [[Decart coordinates]]). Each material point coordinates, which form the 3-vector. Components of such a vector are [[real number]]s.<br />
This vector is smooth function an additional coordinate that is interpreted as [[time]].<br />
TIme is also [[real number]].<br />
The coordinates of each point are smooth functions of time.<br />
For simplicity, assume that coordinate also mean the 3-vector function dependent on time.<br />
This function is interpreted as trajectory. The time is often denoted with letter $t$.<br />
<br />
Several material points may form a physical body, or physical system. The coordinate of the physical system is assumed to be combination of coordinates of all its material points.<br />
<br />
Often, the coordinates are denoted with letter $\vec x= \{ x[1], x[2], x[2]\}$.<br />
If the body counts with several material points, say, $N$ points,<br />
then the material points are numbered and denoted as <br />
: $\vec x_n=\vec x_n(t)$, $n=1.. N$.<br />
<br />
The component of each vector is not required for formulation of the basic axioms of classical mechanics,<br />
but if necessary, they can be indicated with an additional subscript:<br />
: $\vec x_n=\{ x_n[1], x_n[2], x_n[3] \} = \{ x_{n,1}, x_{n,2}, x_{n,3}\}$<br />
However, in the last case, it is important to keep the order of subscripts to avoid confusions. <br />
<br />
Each material point is attributed its universal property, called [[mass]]. The material points may have different masses;<br />
in this case, they are numbered and denoted with $m_n$, $n=1.. N$.<br />
In such a way, that material point of mass $m_n$ has coordinates $\vec x_n=\vec x_n(t)$.<br />
Mass of each material point is constant and cannot vary with time. <br />
<br />
In some cases, the detailed state of the body is not important for the consideration; then the body is characterised with its mean coordinate called also <br />
[[center of mass]]; the coordinate of center of mass of a body of $N$ material points is defined with<br />
: $\displaystyle X(t)= \sum_{n=1}^{N} m_n~ x_n(t)$<br />
<br />
At the scientific [[slang]], the coordinate of center of mass of a body can be called simply<br />
"coordinate of a body", although it actually consists of 3 components of vector $\vec X$ and, in principle, may depend on time. <br />
<br />
The time derivative of coordinate is called [[velocity]] and denoted with letter $\vec v$;<br />
: $\vec v_n(t)=\vec x_n'(t)$, where prime means derivative.<br />
<br />
All velocities are differentiable functions, and the time derivative of velocity is called acceleration:<br />
: $\vec w_n(t)=\vec v_n'(t)=\vec x_n''(t)$<br />
<br />
The interaction between material points is characterized with [[force]]s. The properties of masses, coordinates and forces are <br />
assumed to satisfy the [[Laws of Newton]]. Historically, the 4 Laws of Newton are postulated.<br />
<br />
==Laws of Newton==<br />
<br />
<br />
==Keywords==<br />
[[Robert Hooke]],<br />
[[Isaac Newton]],<br />
[[First law of Newton]],<br />
[[Second law of Newton]],<br />
[[Third law of Newton]],<br />
[[Conservation laws]]<br />
<br />
==References==<br />
<references/><br />
<br />
[[Category:Classical mechanics]]<br />
[[Category:Articles in English]]</div>Maintenance script