Editing Hamiltonian mechanics (section)

=== Phase space coordinates (''p'', ''q'') and Hamiltonian ''H'' ===
Let <math>(M, \mathcal L)</math> be a [[Lagrangian mechanics|mechanical system]] with  [[configuration space (physics)|configuration space]] <math>M</math> and smooth [[Lagrangian_mechanics#Lagrangian|Lagrangian]] <math> \mathcal L.</math> Select a standard coordinate system <math>(\boldsymbol{q},\boldsymbol{\dot q})</math> on <math>M.</math> The quantities <math>\textstyle p_i(\boldsymbol{q},\boldsymbol{\dot q},t) ~\stackrel{\text{def}}{=}~ {\partial \mathcal L}/{\partial \dot q^i}</math> are called ''momenta''. (Also ''generalized momenta'', ''conjugate momenta'', and ''canonical momenta''). For a time instant <math>t,</math> the [[Legendre transformation#Legendre transformation on manifolds|Legendre transformation]] of <math>\mathcal{L}</math> is defined as the map <math>(\boldsymbol{q}, \boldsymbol{\dot q}) \to \left(\boldsymbol{p},\boldsymbol{q}\right) </math> which is assumed to have a smooth inverse <math>(\boldsymbol{p},\boldsymbol{q}) \to (\boldsymbol{q},\boldsymbol{\dot q}).</math> For a system with <math>n</math> degrees of freedom, the Lagrangian mechanics defines the ''energy function''
<math display="block">E_{\mathcal L}(\boldsymbol{q},\boldsymbol{\dot q},t)\, \stackrel{\text{def}}{=}\, \sum^n_{i=1} \dot q^i \frac{\partial \mathcal L}{\partial \dot q^i} - \mathcal L.</math>

The Legendre transform of <math>\mathcal{L}</math> turns <math>E_{\mathcal L}</math> into a function <math> \mathcal H(\boldsymbol{p},\boldsymbol{q},t)</math> known as the {{em|Hamiltonian}}. The Hamiltonian satisfies
<math display="block">
 \mathcal H\left(\frac{\partial \mathcal L}{\partial \boldsymbol{\dot q}},\boldsymbol{q},t\right) = E_{\mathcal L}(\boldsymbol{q},\boldsymbol{\dot q},t)
</math>
which implies that
<math display="block">
 \mathcal H(\boldsymbol{p},\boldsymbol{q},t) = \sum^n_{i=1} p_i\dot q^i - \mathcal L(\boldsymbol{q},\boldsymbol{\dot q},t),
</math>
where the velocities <math>\boldsymbol{\dot q} = (\dot q^1,\ldots, \dot q^n)</math> are found from the (<math>n</math>-dimensional) equation <math>\textstyle \boldsymbol{p} = {\partial \mathcal L}/{\partial \boldsymbol{\dot q}}</math> which, by assumption, is uniquely solvable for {{tmath|1= \boldsymbol{\dot q} }}. The (<math>2n</math>-dimensional) pair <math>(\boldsymbol{p},\boldsymbol{q})</math> is called ''phase space coordinates''. (Also ''canonical coordinates'').