theories:classical_mechanics:hamiltonian
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theories:classical_mechanics:hamiltonian [2018/05/02 13:18] jakobadmin |
theories:classical_mechanics:hamiltonian [2018/10/11 14:12] (current) jakobadmin [Abstract] |
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| <tabbox Concrete> | <tabbox Concrete> | ||
| - | <note tip> | + | ---- |
| - | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | + | |
| - | </note> | + | **Reading Recommendations** |
| - | + | ||
| + | * The best book on Hamiltonian mechanics is The Lazy Universe by Coopersmith | ||
| <tabbox Abstract> | <tabbox Abstract> | ||
| Lagrangian mechanics can be formulated geometrically using [[advanced_tools:fiber_bundles|fibre bundles]]. | Lagrangian mechanics can be formulated geometrically using [[advanced_tools:fiber_bundles|fibre bundles]]. | ||
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| ---- | ---- | ||
| + | * [[https://core.ac.uk/download/pdf/4887416.pdf|Lectures on Mechanics]] by Marsden | ||
| * See page 471 in Road to Reality by R. Penrose, page 167 in Geometric Methods of Mathematical Physics by B. Schutz and http://philsci-archive.pitt.edu/2362/1/Part1ButterfForBub.pdf. | * See page 471 in Road to Reality by R. Penrose, page 167 in Geometric Methods of Mathematical Physics by B. Schutz and http://philsci-archive.pitt.edu/2362/1/Part1ButterfForBub.pdf. | ||
| * For some more backinfo why there is a symplectic structure in classical mechanics, have a look at https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/ | * For some more backinfo why there is a symplectic structure in classical mechanics, have a look at https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/ | ||
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| a ‘Hamiltonian’ $$H : T^* Q \to \mathbb{R}$$ or a ‘Lagrangian’ $$L : T Q \to \mathbb{R}$$ | a ‘Hamiltonian’ $$H : T^* Q \to \mathbb{R}$$ or a ‘Lagrangian’ $$L : T Q \to \mathbb{R}$$ | ||
| - | Instead, we started with Hamilton’s principal function $$S : Q \to \mathbb{R}$$ where $Q$ is not the usual configuration space describing possible positions for a particle, but the ‘extended’ configuration space, which also includes time. Only this way do Hamilton’s equations, like the [[equations:maxwell_relations|Maxwell relations]], become a trivial consequence of the fact that partial derivatives commute. | + | Instead, we started with Hamilton’s principal function $$S : Q \to \mathbb{R}$$ where $Q$ is not the usual configuration space describing possible positions for a particle, but the ‘extended’ configuration space, which also includes time. Only this way do Hamilton’s equations, like the [[formulas:maxwell_relations|Maxwell relations]], become a trivial consequence of the fact that partial derivatives commute. |
| <cite>https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/</cite> | <cite>https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/</cite> | ||
theories/classical_mechanics/hamiltonian.1525259886.txt.gz · Last modified: 2018/05/02 11:18 (external edit)
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