Editing Hamiltonian system (section)

== Overview ==

Informally, a Hamiltonian system is a mathematical formalism developed by [[William Rowan Hamilton|Hamilton]] to describe the [[evolution equation|evolution equations]] of a physical system. The advantage of this description is that it gives important insights into the dynamics, even if the [[initial value problem]] cannot be solved analytically. One example is the [[Three-body problem|planetary movement of three bodies]]: while there is no [[closed-form solution]] to the general problem, [[Henri Poincaré|Poincaré]] showed for the first time that it exhibits [[deterministic chaos]].

Formally, a Hamiltonian system is a dynamical system characterised by the scalar function <math>H(\boldsymbol{q},\boldsymbol{p},t)</math>, also known as the Hamiltonian.<ref name=ott>{{cite book|last=Ott|authorlink=Edward Ott|first=Edward|title=Chaos in Dynamical Systems|year=1994|publisher=Cambridge University Press}}</ref> The state of the system, <math>\boldsymbol{r}</math>, is described by the [[generalized coordinates]] <math>\boldsymbol{p}</math> and <math>\boldsymbol{q}</math>, corresponding to generalized momentum and position respectively. Both <math>\boldsymbol{p}</math> and <math>\boldsymbol{q}</math> are real-valued vectors with the same dimension&nbsp;''N''. Thus, the state is completely described by the 2''N''-dimensional vector

:<math>\boldsymbol{r} = (\boldsymbol{q},\boldsymbol{p})</math>

and the evolution equations are given by [[Hamilton's equations]]:

:<math>\begin{align}
& \frac{d\boldsymbol{p}}{dt} = -\frac{\partial H}{\partial \boldsymbol{q}}, \\[5pt]
& \frac{d\boldsymbol{q}}{dt} = +\frac{\partial H}{\partial \boldsymbol{p}}.
\end{align} </math>

The trajectory <math>\boldsymbol{r}(t)</math> is the solution of the [[initial value problem]] defined by Hamilton's equations and the initial condition <math>\boldsymbol{r}(t = 0) = \boldsymbol{r}_0\in\mathbb{R}^{2N}</math>.