Question on derivation of a property of Poisson brackets

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27, 2024 at 4:45 PM">Nov 27, 2024</time> </li> <li> <dl class="tagList tagList--thread-1067161 "> <dt> <i class="fa--xf fal fa-tags" aria-hidden="true" title="Tags"></i> <span class="u-srOnly">Tags</span> </dt> <dd> <span class="js-tagList"> <a href="/tags/classical-mechanics/" data-xf-init="preview-tooltip" data-preview-url="/tags/classical-mechanics/preview" class="tagItem tagItem--tag_classical-mechanics" dir="auto"> Classical mechanics </a> <a href="/tags/hamiltonian-formalism/" data-xf-init="preview-tooltip" data-preview-url="/tags/hamiltonian-formalism/preview" class="tagItem tagItem--tag_hamiltonian-formalism" dir="auto"> Hamiltonian formalism </a> <a href="/tags/poisson-brackets/" data-xf-init="preview-tooltip" data-preview-url="/tags/poisson-brackets/preview" class="tagItem tagItem--tag_poisson-brackets" dir="auto"> Poisson brackets </a> </span> </dd> </dl> </li> </ul> </div> </div> <script src="https://www.physicsforums.com/check-for-tex.js" defer></script> <div class="block 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fal fa-thumbs-up fa-fw" aria-hidden="true" data-xf-init="tooltip" title="Reaction score"></i></dt> <dd>0</dd> </dl> </div> <span class="message-userArrow"></span> </section> </div> <div class="message-cell message-cell--main"> <div class="message-main js-quickEditTarget"> <div class="message-content js-messageContent"> <div class="message-userContent lbContainer js-lbContainer " data-lb-id="post-7134718" data-lb-caption-desc="simonstar76 · Nov 27, 2024 at 4:45 PM"> <div class="message-fields message-fields--before"> <dl class="pairs pairs--columns pairs--fixedSmall pairs--customField" data-field="summary"> <dt>TL;DR Summary</dt> <dd>The textbook Classical Mechanics by Deriglazov presents a proof of the fact that Poisson brackets are at most function of time in a canonical transformation. While the proof seems convincing for a 2D phase space, it appears to the OP at least incomplete for greater dimensions</dd> </dl> </div> <article class="message-body js-selectToQuote"> <div > <div class="bbWrapper">The book <a href="https://books.google.it/books?id=QAQ6DQAAQBAJ" target="_blank" class="link link--external" rel="nofollow ugc noopener">Classical Mechanics</a> by Alexei Deriglazov defines as canonical a transformation Z=Z(z,t) that preserves the Hamiltonian form of the equation of motion for any H. After taking the divergence of the vector equation relating the components of the time derivative of Z in the two coordinate systems, he writes (p. 116, I modify slightly the notation) the following equation (it is intended that repeated indices sum):<br /> <br /> $$<br /> \frac{\partial{}}{\partial{Z_k}}\left(\left\{Z_k,Z_l\right\}_z|_{z(Z,t)}\right)\frac{\partial{H(z(Z,t))}}{\partial{Z_l}}+\frac{\partial{}}{\partial{Z_k}}\left(\frac{\partial{Z(z,t)}}{\partial{t}}\Big{|}_{z(Z,t)}\right)=0.<br /> $$<br /> <br /> He asserts that, since H is arbitrary, the partial derivatives of each Poisson bracket<br /> <br /> $$<br /> \left\{Z_k,Z_l\right\}_z\Big{|}_{z(Z,t)}<br /> <br /> $$<br /> <br /> with respect to the Z coordinates must be 0 and so we can at most write (the substitution z=z(Z,t) can be omitted):<br /> <br /> $$<br /> \left\{Z_k,Z_l\right\}_z=c_{kl}\left(t\right)<br /> $$<br /> <br /> The derivation seems confirmed by an explicit calculation that the author does in the following chapter (p. 138 and 142), where he considers the phase space of dimension 2d=2. For the case e.g. of a time-independent transformation, he takes the derivatives of the two equations for the time derivatives of Q and P and adds them, according to the discussion above, obtaining (without explicitly showing the variable dependency):<br /> <br /> $$<br /> \frac{\partial{}}{\partial{Q}}\left(\left\{Q,P\right\}\right)\frac{\partial{H}}{\partial{P}}-\frac{\partial{}}{\partial{P}}\left(\left\{Q,P\right\}\right)\frac{\partial{H}}{\partial{Q}}=0<br /> $$<br /> <br /> In this case, the arbitrariness of H forces us to conclude that both partial derivatives of {Q,P} with respect to Q and P are 0, thus verifying the assertion <i>for this case</i> (there's no time dependence here so we can conclude {Q,P}=c=constant).<br /> <br /> Turning back to the general assertion for 2d>2, how can such an approach lead to the desired conclusion (i.e. that all partial derivatives of the {Z<br /> <sub>i</sub>,Z<sub>j</sub>} with respect to Z<sub>k</sub> are 0)<br /> if we have only one equation in the 2d arbitrary partial derivatives of H which potentially multiply 2d<br /> <sup>2</sup>(d-1)/2 partial derivatives of the Poisson brackets ? The d terms would collect many different partial Poisson bracket derivatives, leaving us with no possibility to conclude that each of the four partial derivative, for every Poisson bracket, is zero.</div> </div> <div class="js-selectToQuoteEnd"> </div> <div class="googleads_fixed"> <script async src="https://pagead2.googlesyndication.com/pagead/js/adsbygoogle.js?client=ca-pub-6580726045122001" crossorigin="anonymous"></script>  <ins class="adsbygoogle" style="display:block" data-ad-client="ca-pub-6580726045122001" data-ad-slot="3049969336" data-ad-format="auto" data-full-width-responsive="true"></ins> <script> (adsbygoogle = window.adsbygoogle || []).push({}); </script> </div> </article> </div> </div> <footer class="message-footer"> <div class="reactionsBar js-reactionsList "> </div> <div class="js-historyTarget message-historyTarget toggleTarget" data-href="trigger-href"></div> </footer> <script> if (typeof parseNewPost == 'function') { $(document).ajaxStop(parseNewPost(7134718)); } 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Are you sure you want to submit this form?", attach: "Attach files", rich_text_box: "Rich text box", close: "Close", link_copied_to_clipboard: "Link copied to clipboard.", text_copied_to_clipboard: "Text copied to clipboard.", loading: "Loading…", you_have_exceeded_maximum_number_of_selectable_items: "You have exceeded the maximum number of selectable items.", processing: "Processing", 'processing...': "Processing…", showing_x_of_y_items: "Showing {count} of {total} items", showing_all_items: "Showing all items", no_items_to_display: "No items to display", number_button_up: "Increase", number_button_down: "Decrease", push_enable_notification_title: "Push notifications enabled successfully at Physics Forums: Science Discussion, Homework Help, Articles", push_enable_notification_body: "Thank you for enabling push notifications!", pull_down_to_refresh: "Pull down to refresh", release_to_refresh: "Release to refresh", refreshing: "Refreshing…" }); </script> <form style="display:none" hidden="hidden"> <input type="text" name="_xfClientLoadTime" value="" id="_xfClientLoadTime" title="_xfClientLoadTime" tabindex="-1" /> </form>  <script type="application/ld+json"> { "@context": "https://schema.org", "@type": "WebPage", "url": "https://www.physicsforums.com/threads/question-on-derivation-of-a-property-of-poisson-brackets.1067161/", "mainEntity": { "@type": "DiscussionForumPosting", "@id": "https://www.physicsforums.com/threads/question-on-derivation-of-a-property-of-poisson-brackets.1067161/", "headline": "Question on derivation of a property of Poisson brackets", "datePublished": "2024-11-27T22:45:09+00:00", "keywords": "Classical mechanics, Hamiltonian formalism, Poisson brackets", "url": "https://www.physicsforums.com/threads/question-on-derivation-of-a-property-of-poisson-brackets.1067161/", "articleSection": "Classical Physics", "author": { "@type": "Person", "@id": "https://www.physicsforums.com/members/simonstar76.755321/", "name": "simonstar76", "url": "https://www.physicsforums.com/members/simonstar76.755321/" }, "interactionStatistic": [ { "@type": "InteractionCounter", "interactionType": "https://schema.org/ViewAction", "userInteractionCount": 585 }, { "@type": "InteractionCounter", "interactionType": "https://schema.org/CommentAction", "userInteractionCount": 0 }, { "@type": "InteractionCounter", "interactionType": "https://schema.org/LikeAction", "userInteractionCount": 0 } ], "articleBody": "The book Classical Mechanics by Alexei Deriglazov defines as canonical a transformation Z=Z(z,t) that preserves the Hamiltonian form of the equation of motion for any H. After taking the divergence of the vector equation relating the components of the time derivative of Z in the two coordinate systems, he writes (p. 116, I modify slightly the notation) the following equation (it is intended that repeated indices sum):\n\n$$\n\\frac{\\partial{}}{\\partial{Z_k}}\\left(\\left\\{Z_k,Z_l\\right\\}_z|_{z(Z,t)}\\right)\\frac{\\partial{H(z(Z,t))}}{\\partial{Z_l}}+\\frac{\\partial{}}{\\partial{Z_k}}\\left(\\frac{\\partial{Z(z,t)}}{\\partial{t}}\\Big{|}_{z(Z,t)}\\right)=0.\n$$\n\nHe asserts that, since H is arbitrary, the partial derivatives of each Poisson bracket\n\n$$\n\\left\\{Z_k,Z_l\\right\\}_z\\Big{|}_{z(Z,t)}\n\n$$\n\nwith respect to the Z coordinates must be 0 and so we can at most write (the substitution z=z(Z,t) can be omitted):\n\n$$\n\\left\\{Z_k,Z_l\\right\\}_z=c_{kl}\\left(t\\right)\n$$\n\nThe derivation seems confirmed by an explicit calculation that the author does in the following chapter (p. 138 and 142), where he considers the phase space of dimension 2d=2. For the case e.g. of a time-independent transformation, he takes the derivatives of the two equations for the time derivatives of Q and P and adds them, according to the discussion above, obtaining (without explicitly showing the variable dependency):\n\n$$\n\\frac{\\partial{}}{\\partial{Q}}\\left(\\left\\{Q,P\\right\\}\\right)\\frac{\\partial{H}}{\\partial{P}}-\\frac{\\partial{}}{\\partial{P}}\\left(\\left\\{Q,P\\right\\}\\right)\\frac{\\partial{H}}{\\partial{Q}}=0\n$$\n\nIn this case, the arbitrariness of H forces us to conclude that both partial derivatives of {Q,P} with respect to Q and P are 0, thus verifying the assertion for this case (there's no time dependence here so we can conclude {Q,P}=c=constant).\n\nTurning back to the general assertion for 2d>2, how can such an approach lead to the desired conclusion (i.e. that all partial derivatives of the {Z\ni,Zj} with respect to Zk are 0)\nif we have only one equation in the 2d arbitrary partial derivatives of H which potentially multiply 2d\n2(d-1)/2 partial derivatives of the Poisson brackets ? The d terms would collect many different partial Poisson bracket derivatives, leaving us with no possibility to conclude that each of the four partial derivative, for every Poisson bracket, is zero." }, "publisher": { "@type": "Organization", "name": "Physics Forums: Science Discussion, Homework Help, Articles", "alternateName": "Physics Forums", "description": "Join Physics Forums, where students, scientists, and enthusiasts come together to explore and discuss the current understanding and practice of various scientific fields.", "url": "https://www.physicsforums.com", "logo": "https://www.physicsforums.com/styles/physicsforums/images/physics-forums-logo-og.png" } } </script> <script type="text/javascript" src="https://www.physicsforums.com/copyright.js" defer></script> <script> function parseNewPost(postid) { var newpost = document.getElementById("js-post-" + postid), npa = [newpost]; MathJax.typeset(npa); } function parsePreview() { var preview = document.getElementsByClassName("xfPreview"); MathJax.typeset(preview); } function parseNewConversationPost(postid) { var newpost = document.getElementsByClassName("convMessage-" + postid); MathJax.typeset(newpost); } function parseNewProfilePost(postid) { var newpost = document.getElementById("js-profilePost-" + postid), npa = [newpost]; MathJax.typeset(npa); } function parseTooltip() { MathJax.typeset(); } </script> </body> </html>

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Question on derivation of a property of Poisson brackets