Editing Hamiltonian mechanics (section)

== Overview ==
=== 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'').

=== From Euler–Lagrange equation to Hamilton's equations ===
In phase space coordinates {{tmath|1= (\boldsymbol{p},\boldsymbol{q}) }}, the (<math>n</math>-dimensional) [[Euler–Lagrange equation]]
<math display="block">\frac{\partial \mathcal L}{\partial \boldsymbol{q}} - \frac{d}{dt}\frac{\partial \mathcal L}{\partial \dot\boldsymbol{q}} = 0</math>
becomes ''Hamilton's equations'' in <math>2n</math> dimensions

{{Equation box 1
|indent =:
|equation = <math>
\frac{\mathrm{d}\boldsymbol{q}}{\mathrm{d}t} = \frac{\partial \mathcal H}{\partial \boldsymbol{p}},\quad
\frac{\mathrm{d}\boldsymbol{p}}{\mathrm{d}t} = -\frac{\partial \mathcal H}{\partial \boldsymbol{q}}.
</math>
|cellpadding= 5
|border
|border colour = #0073CF
|background colour=rgba(0,0,0,0)}}

{{Proof|
The Hamiltonian <math>\mathcal{H}(\boldsymbol{p},\boldsymbol{q})</math> is the [[Legendre transform]] of the Lagrangian <math>\mathcal{L}(\boldsymbol{q},\dot\boldsymbol{q})</math>, thus one has
<math display="block">\mathcal{L}(\boldsymbol{q},\dot\boldsymbol{q})
+ \mathcal{H}(\boldsymbol{p},\boldsymbol{q}) = \boldsymbol{p}\dot\boldsymbol{q}</math>
and thus 
<math display="block">\begin{align}
\frac{\partial \mathcal{H}}{\partial \boldsymbol{p}} &= \dot\boldsymbol{q} \\
\frac{\partial \mathcal{L}}{\partial \boldsymbol{q}} &= -\frac{\partial \mathcal{H}}{\partial \boldsymbol{q}},
\end{align}</math>

Besides, since <math>\boldsymbol{p} = \partial \mathcal{L}/\partial \dot\boldsymbol{q} </math>, the Euler–Lagrange equations yield
<math>
\dot{\boldsymbol{p}} = \frac{\mathrm{d}\boldsymbol{p}}{\mathrm{d}t} = \frac{\partial\mathcal{L}}{\partial\boldsymbol{q}} = -\frac{\partial\mathcal{H}}{\partial\boldsymbol{q}}.
</math>
}}

=== From stationary action principle to Hamilton's equations ===
Let <math> \mathcal P(a,b,\boldsymbol x_a,\boldsymbol x_b)</math> be the set of smooth paths <math>\boldsymbol q: [a,b] \to M</math> for which <math>\boldsymbol q(a) = \boldsymbol x_a</math> and <math>\boldsymbol q(b) = \boldsymbol x_{b}. </math> The [[action (physics)|action functional]] <math> \mathcal S : \mathcal P(a,b,\boldsymbol x_a,\boldsymbol x_b) \to \Reals</math> is defined via
<math display="block"> \mathcal S[\boldsymbol q] = \int_a^b \mathcal L(t,\boldsymbol q(t),\dot{\boldsymbol q}(t))\, dt = \int_a^b \left(\sum^n_{i=1} p_i\dot q^i - \mathcal H(\boldsymbol{p},\boldsymbol{q},t) \right)\, dt,</math>
where {{tmath|1= \boldsymbol{q} = \boldsymbol{q}(t) }}, and <math>\boldsymbol{p} = \partial \mathcal L/\partial \boldsymbol{\dot q}</math> (see above). A path <math>\boldsymbol q \in \mathcal P(a,b,\boldsymbol x_a,\boldsymbol x_b)</math> is a [[Lagrangian mechanics|stationary point]] of <math> \mathcal S</math> (and hence is an equation of motion) if and only if the path <math>(\boldsymbol{p}(t),\boldsymbol{q}(t))</math> in phase space coordinates obeys the Hamilton equations.

=== Basic physical interpretation ===

A simple interpretation of Hamiltonian mechanics comes from its application on a one-dimensional system consisting of one nonrelativistic particle of mass {{mvar|m}}. The value <math>H(p,q)</math> of the Hamiltonian is the total energy of the system, in this case the sum of [[kinetic energy|kinetic]] and [[potential energy]], traditionally denoted {{mvar|T}} and {{mvar|V}}, respectively. Here {{mvar|p}} is the momentum {{mvar|mv}} and {{mvar|q}} is the space coordinate. Then
<math display="block">\mathcal{H} = T + V, \qquad T = \frac{p^2}{2m} , \qquad V = V(q) </math>
{{mvar|T}} is a function of {{mvar|p}} alone, while {{mvar|V}} is a function of {{mvar|q}} alone (i.e., {{mvar|T}} and {{mvar|V}} are [[scleronomic]]).

In this example, the time derivative of {{mvar|q}} is the velocity, and so the first Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum.  The time derivative of the momentum {{mvar|p}} equals the ''Newtonian force'', and so the second Hamilton equation means that the force equals the negative [[gradient]] of potential energy.