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--- theories:classical_mechanics:hamiltonian [2018/04/12 16:10]
bogumilvidovic created
+++ theories:classical_mechanics:hamiltonian [2018/10/11 14:12] (current)
jakobadmin [Abstract]
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 ====== Hamiltonian Mechanics ======
+//see also [[formalisms:hamiltonian_formalism]] and [[equations:hamiltons_equations]]//
 <tabbox Intuitive>
@@ Line 9: / Line 10: @@
 <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>
+Lagrangian mechanics can be formulated geometrically using [[advanced_tools:fiber_bundles|fibre bundles]].
-<note tip>
-The motto in this section is: //the higher the level of abstraction, the better//.
-</note>
+The Hamiltonian function is defined on the cotangent bundle $T^\star(C)$, which is called phase space.
+In contrast, the Lagrangian function is defined on the tangent bundle $T(C)$ of the configuration space $C$.
+The map from $T^\star(C) \leftrightarrow T(C)$ is called [[advanced_tools:legendre_transformation|Legendre transformation]].
+The phase space is endowed with a symplectic structure, called Poisson Bracket. The Poisson Bracket is an operation that eats two scalar fields $\Phi$, $\Psi$ on the manifold and spits out another scalar field $\theta $:
+$$  \theta = \{ \Phi,\Psi \}= \frac{\partial \Phi}{\partial p_a}\frac{\partial \Psi}{\partial q^a}-\frac{\partial \Phi}{\partial q_a}\frac{\partial \Psi}{\partial p^a}.$$
+If we leave the $\Psi$ slot blank, we can use the Poisson bracket to define a differential operator $\{\Phi,\ \}$. This is a vector field and when in acts on $\Psi$, we get $\{\Phi, \Psi \}$. If we use instead of $\Phi$, the Hamiltonian $H$, we get an differential operator $\{H,\ \}$ that 'points along' the trajectories on in phase space $T^\star(C)$ and describes exactly the evolution that we get from Hamilton's equations.
+In this sense, the dynamical evolution of a given system is completely described by the Hamiltonian (= a scalar function).
+----
+  * [[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.
+  * 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/
+----
+----
+<blockquote>
+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 [[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>
+</blockquote>
 <tabbox Why is it interesting?>

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