✎ Hamiltonian Mechanics [The Physics Travel Guide]

You've loaded an old revision of the document! If you save it, you will create a new version with this data.

====== Hamiltonian Mechanics ======

<note tip>
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
</note>
  
<tabbox Concrete>

<note tip>
In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
</note>
 
<tabbox Abstract> 
Lagrangian mechanics can be formulated geometrically using [[advanced_tools:fiber_bundles|fibre bundles]].

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).

----

* 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/
<tabbox Why is it interesting?>

</tabbox>

Please solve the following equation to prove you're human. 201 +8 = Please keep this field empty:

Edit summary

Note: By editing this page you agree to license your content under the following license: CC Attribution-Share Alike 4.0 International

User Tools

Site Tools

Page Tools