{"id":185,"date":"2023-08-11T12:06:53","date_gmt":"2023-08-11T11:06:53","guid":{"rendered":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/?p=185"},"modified":"2023-08-12T11:02:35","modified_gmt":"2023-08-12T10:02:35","slug":"liouvilles-theorem-of-phase-space","status":"publish","type":"post","link":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/2023\/08\/11\/liouvilles-theorem-of-phase-space\/","title":{"rendered":"Liouville&#8217;s Theorem of Phase Space"},"content":{"rendered":"<h3>1. Start from the Abel-Jacobi-Liouville identity<\/h3>\n<p>Recall that if we have a set of differential equation:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"262\" height=\"38\" class=\"alignnone size-full wp-image-188\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/08\/gif.latex_.gif\" alt=\"\" \/><\/p>\n<p>The wronskian evolves as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"301\" height=\"38\" class=\"alignnone size-full wp-image-189\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/08\/2.gif\" alt=\"\" \/><\/p>\n<h3>2. Liouville&#8217;s Theorem<\/h3>\n<p>Now we consider the a time independent Hamiltonian system <strong><em>(q,p)<\/em><\/strong>. It satisfy:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"293\" height=\"39\" class=\"alignnone size-full wp-image-191\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/08\/3.gif\" alt=\"\" \/><\/p>\n<p>Where F is a (n,2) matrix satisfying Hamilton equation for each pi and qi and n is the dimension of the space. This is an initial value problem and thus p and q depends on the initial set <strong><em>(q0,p0)<\/em><\/strong>. Taking derivative for each component about the initial value \u00a0and recombine, and use the fact that derivative w.r.t <strong><em>(q0,p0)<\/em> <\/strong>commutes with time derivative gives:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"337\" height=\"53\" class=\"alignnone size-full wp-image-192\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/08\/4.gif\" alt=\"\" \/><\/p>\n<p>Evaluating the right hand side is a bit dizzy: For every component, for example, for the ith component of <strong><em>q<\/em><\/strong>, this is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"551\" height=\"43\" class=\"alignnone size-full wp-image-195\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/08\/5-1.gif\" alt=\"\" \/><\/p>\n<p>Now is the tricky part: Use the limit that when <em>t<\/em> is small:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"327\" height=\"72\" class=\"alignnone size-full wp-image-196\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/08\/6.gif\" alt=\"\" \/><\/p>\n<p>So the derivatives in (5) tends to zero apart from the derivative of the same component. We thus recognise from that (4) this is in fact, in similar structure to (1).<\/p>\n<p>The trace of <em>A<\/em> is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"372\" height=\"49\" class=\"alignnone size-full wp-image-197\" src=\"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/files\/2023\/08\/7.gif\" alt=\"\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Thus the corresponding <em>W<\/em>, the Jacobian matrix of the mapping on the phase space between initial phase and the infinitesimal phase later, Has zero variation at the moment.<\/p>\n<p>Importantly, one can easily show that <em>W<\/em> is 1 at the beginning. <strong><em>We can apply the same procedure starting from any successive moment and it will always give zero derivative: We thus conclude that W=1 all along and thus, <\/em>the phase space volume conserved<\/strong>.<\/p>\n<p><strong>We observe that the condition that we can obtain this result is that the Hamiltonian is time independent. <\/strong>If not, \u00a0an additional term \u00a0will appears in (5) and it is not guaranteed to have trace of A cancelled in (7).<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>1. Start from the Abel-Jacobi-Liouville identity Recall that if we have a set of differential equation: The wronskian evolves as: 2. Liouville&#8217;s Theorem Now we consider the a time independent Hamiltonian system (q,p). It satisfy: Where F is a (n,2) matrix satisfying Hamilton equation for each pi and qi and n is the dimension of [&hellip;]<\/p>\n","protected":false},"author":1741,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[20,2],"tags":[],"class_list":["post-185","post","type-post","status-publish","format-standard","hentry","category-classical-mechanics","category-ordinary-differential-equations"],"_links":{"self":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/185","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/users\/1741"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/comments?post=185"}],"version-history":[{"count":6,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/185\/revisions"}],"predecessor-version":[{"id":199,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/posts\/185\/revisions\/199"}],"wp:attachment":[{"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/media?parent=185"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/categories?post=185"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs-staging.imperial.ac.uk\/cocteaupedia\/wp-json\/wp\/v2\/tags?post=185"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}