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theories:classical_mechanics:hamiltonian [2018/05/04 09:43] jakobadmin ↷ Page moved from theories:hamiltonian_mechanics to theories:classical_mechanics:hamiltonian_mechanics |
theories:classical_mechanics:hamiltonian [2018/05/13 09:17] jakobadmin ↷ Links adapted because of a move operation |
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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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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> |