formalisms:hamiltonian_formalism
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formalisms:hamiltonian_formalism [2018/12/19 11:00] jakobadmin ↷ Links adapted because of a move operation |
formalisms:hamiltonian_formalism [2020/12/26 23:47] edi [Concrete] |
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| ====== Hamiltonian Formalism ====== | ====== Hamiltonian Formalism ====== | ||
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
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| $$H \equiv \sum_j {\dot q}_j p_j - L$$ | $$H \equiv \sum_j {\dot q}_j p_j - L$$ | ||
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| + | ---- | ||
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| + | **Graphical Summary of Newtonian, Lagrangian, and Hamiltonian Formalism** | ||
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| + | [{{ :frameworks:newton_lagrange_hamilton.jpg?nolink }}] | ||
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| + | For some concrete examples worked out in all three frameworks see [[https://esackinger.wordpress.com/|Fun with Symmetry]]. | ||
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | <WRAP group> | + | |
| - | <WRAP half column> | + | |
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| - | </WRAP> | ||
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| - | <WRAP half column> | ||
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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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| - | </WRAP> | ||
| - | </WRAP> | ||
| <blockquote>In fact, | <blockquote>In fact, | ||
| with suitable choices of H, Hamilton's equations still hold true for any system | with suitable choices of H, Hamilton's equations still hold true for any system | ||
| of classical equations whatever, not just for Newton's equations. In particular, this will be the case for the Maxwell(-Lorentz) theory that we shall be | of classical equations whatever, not just for Newton's equations. In particular, this will be the case for the Maxwell(-Lorentz) theory that we shall be | ||
| - | considering shortly. Hamilton's equations also hold true for [[theories:special_relativity|special relativity]]. | + | considering shortly. Hamilton's equations also hold true for [[models:special_relativity|special relativity]]. |
| Even[[models:general_relativity| general relativity]] can, if due care is exercised, be subsumed into the | Even[[models:general_relativity| general relativity]] can, if due care is exercised, be subsumed into the | ||
| Hamiltonian framework. Moreover, as we shall see later with [[equations:schroedinger_equation|Schrodinger's | Hamiltonian framework. Moreover, as we shall see later with [[equations:schroedinger_equation|Schrodinger's | ||
formalisms/hamiltonian_formalism.txt · Last modified: 2023/04/02 03:34 by edi
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