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Hamiltonian mechanics is a formulation of mechanics in which the basic datum in a mechanical system is a function $H$, the Hamiltonian of the system, which gives the total energy in the system in terms of the positions and momenta of the objects in the system.
More abstractly, the Hamiltonian is a function on phase space, a manifold whose coordinates are generalised? positions $q^i$ and momenta $p_i$. (Compare this to Lagrangean mechanics?, in which the Lagrangean is a function on state space?, whose coordinates are generalised positions and velocities?.) So to do Hamiltonian mechanics properly, you must ‘mind your $p$s and $q$s’ (blame John Baez for this pun).
To begin with, we often take phase space to be the cotangent bundle of configuration space. (Compare that state space is the tangent bundle of configuration space.) This comes equipped with a natural $2$-form
or simply $\omega = \mathrm{d}p_i \wedge \mathrm{d}q^i$ using the Einstein summation convention. This $2$-form is closed, in fact exact, since it is the differential of the action form $\bar{\mathrm{d}}S = p_i \wedge \mathrm{d}q^i$, and therefore it is a symplectic form.
However, it is also possible to take phase space to be any symplectic manifold, or even any Poisson manifold. In any case, phase space itself gives only the kinematics (in a momentum-based rather than velocity-based sense); you need the Hamiltonian $H$ to get the dynamics.
Hamiltonian mechanics was developed originally for classical mechanics, but it is also the best known formulation of quantum mechanics; many students of physics (and even more so, students of chemistry?) learn it only when they study the latter. This sometimes leads to confusion about the essential differences between classical and quantum physics.
Hamiltonian mechanics is best formalized in terms of symplectic geometry as described for instance in the monograoph
A classical Hamiltonian mechanical system is a pair $((X,\omega), H)$ consisting of a
symplectic manifold $(X,\omega)$
and a Hamiltonian function $H \in C^\infty(X)$.
Here
$X$ is the phase space of the physical system;
a curve $\gamma : \mathbb{R} \to X$ is a trajectory of the physical system in time;
$(X,\omega)$ defines the kinematics of the system;
$H$ is the Hamiltonian that defined the dynamics of the system.
The dynamics is encoded by declaring that those trajectories $\gamma : \mathbb{R} \to X$ are the physically realized trajectories that satisfy the equation
The components of this are Hamilton's equations.
In more detail this equation means that for each $t \in \mathbb{R}$ the 1-form
and the 1-form
coincide.
At first, this formulation of Hamilton’s mechanics is just that, an equivalent reformulation. But as any reformulation in more abstract terms, it serves to
clarify a structure
allow more powerful thinking about that structure
and eventually it bears in it the seed for further developments pointing beyond this structure
Regarding the first point: this formulation of Hamiltonian mechanics makes clear what th meaning of Hamilton’s equations is for systems topologically more interesting than the example $X = \mathbb{R}^{2 n}$ that many introductory physics texts concentrate on-
Regarding the second point: the differential calculus formulation lends itself much more to high-powered arguments than the traditional component-ridden presentation. Of course the latter may still be the preferred method for some concrete computations.
Regarding the second point: after Hamilton’s times people started thinking about what quantization of a classical system should mean. One successful formalization is that of geometric quantization which takes a symplectic manifold with Hamiltonian function on it as input datum.
The impact that this idea of quantization from symplectic geometry eventually had is hard to underestimate. In the hands of Alan Weinstein and his school it led to symplectic groupoids, Courant algebroids and other higher Lie theoretic structures. In the hands of Maxim Kontsevich it led to the theorem on formal deformation quantization and the vast machinery nowadays associated with that.
The symplectic-geometry description of Hamiltonian mechanics is especially well-suited to describe topologically nontrivial phase spaces that are not cotangent bundles.
$n$ vortices on the sphere as finite dimensional limit of 2D Euler equations: the phase space of the system of $n$ vortices is not a cotangent bundle but is $(S^2)^n$ .
traditional Lagrangian mechanics and Hamiltonian mechanics are naturally embedding into local prequantum field theory by the notion of prequantized Lagrangian correspondences