====== Hamiltonian Mechanics ====== //see also [[formalisms:hamiltonian_formalism]] and [[equations:hamiltons_equations]]// 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. ---- **Reading Recommendations** * The best book on Hamiltonian mechanics is The Lazy Universe by Coopersmith 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). ---- * [[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/ ---- ----
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. https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/