formalisms:hamiltonian_formalism
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formalisms:hamiltonian_formalism [2018/04/13 11:34] bogumilvidovic [Why is it interesting?] |
formalisms:hamiltonian_formalism [2023/04/02 03:34] (current) 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/blog/classical-mechanics|Fun with Symmetry]]. | ||
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | <WRAP group> | + | |
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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, | ||
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| --> How is the Hamiltonian Formalism related to the Newtonian Formalism?# | --> How is the Hamiltonian Formalism related to the Newtonian Formalism?# | ||
| - | <blockquote>"Recall that we derived Hamilton’s equations for a particle moving in a force field $F = -dV/dx$ by writing down the equations of motion in the form $$ m \dot{x} = p , \quad \dot{p} = - \frac{\partial V}{\partial x} .$$ The observant reader will have noticed that these two equations are just one way to express Newton’s second law. More generally for a system of N point-like particles moving in three-dimensional physical space, Newton’s second law would be $$ m \dot{x_j} = p_j , \quad \dot{p}_j = - \frac{\partial V}{\partial x_j} .$$" <cite>The symplectic egg in classical and quantum mechanics by Maurice A. de Gosson</cite></blockquote> | + | <blockquote>Recall that we derived Hamilton’s equations for a particle moving in a force field $F = -dV/dx$ by writing down the equations of motion in the form $$ m \dot{x} = p , \quad \dot{p} = - \frac{\partial V}{\partial x} .$$ The observant reader will have noticed that these two equations are just one way to express Newton’s second law. More generally for a system of N point-like particles moving in three-dimensional physical space, Newton’s second law would be $$ m \dot{x_j} = p_j , \quad \dot{p}_j = - \frac{\partial V}{\partial x_j} .$$ <cite>The symplectic egg in classical and quantum mechanics by Maurice A. de Gosson</cite></blockquote> |
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formalisms/hamiltonian_formalism.1523612095.txt.gz · Last modified: 2018/04/13 09:34 (external edit)
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