equations:hamiltons_equations
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equations:hamiltons_equations [2018/05/05 22:01] jakobadmin [Intuitive] |
equations:hamiltons_equations [2019/02/12 15:21] (current) 129.13.36.189 [Why is it interesting?] |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| + | <blockquote>Hamilton’s Equations show how the $q_i$'s and $p_i$'s undergo a ‘dance to the music of time’, a dance in which, as some $q_i$'s or $p_i$'s increase in value, others decrease in value, but always such as to keep the energy constant (in conservative systems), and always such as to keep the total action minimized, both instant by instant, and over the whole path between ‘surfaces-of-common-action’. This ‘dance’ is governed by one function, $H$, - that is to say, while $H$ is different for different systems (orbiting planets, a statistical ensemble, an electrical circuit, positrons orbiting an atomic antinucleus, a spinning top, juggling pins, a flowing river and so on) yet within any one system there is just one overaching function (there is no need for individual functions, $H_1$, $H_2$,...,$H_n)$. | ||
| + | <cite>The Lazy Universe by Coopersmith</cite> | ||
| + | </blockquote> | ||
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| <blockquote> | <blockquote> | ||
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| {{ :basic_notions:vectorfieldphasespacepenroseemprerornewmind.png?nolink&600 |}} | {{ :basic_notions:vectorfieldphasespacepenroseemprerornewmind.png?nolink&600 |}} | ||
| <cite>page 174ff in "The Emperors new Mind" by R. Penrose</cite></blockquote> | <cite>page 174ff in "The Emperors new Mind" by R. Penrose</cite></blockquote> | ||
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| + | test | ||
| <tabbox Concrete> | <tabbox Concrete> | ||
| Hamiltons equations are | Hamiltons equations are | ||
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| This comes at the cost of doubling the size of the system. | This comes at the cost of doubling the size of the system. | ||
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| + | ---- | ||
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| + | <blockquote>[E]verybody loves Hamilton’s equations: there are just two, and they summarize the entire essence of classical mechanics. | ||
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| + | <cite>[[https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/|John Baez]]</cite></blockquote> | ||
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| + | <blockquote>You get equations like Hamilton's whenever a system *extremizes something subject to constraints*. A moving particle minimizes action; a box of gas maximizes entropy. <cite>[[https://twitter.com/johncarlosbaez/status/1065715514381557761|John Baez]]</cite></blockquote> | ||
| <tabbox Origin> | <tabbox Origin> | ||
| - | "Hamilton’s equations and the [[equations:maxwell_relations|Maxwell relations]]—are mathematically just the same. They both say simply that partial derivatives commute." See https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/ | + | "Hamilton’s equations and the [[formulas:maxwell_relations|Maxwell relations]]—are mathematically just the same. They both say simply that partial derivatives commute." See https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/ |
| and https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/ | and https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/ | ||
equations/hamiltons_equations.1525550471.txt.gz · Last modified: 2018/05/05 20:01 (external edit)
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