Editing Hamiltonian mechanics (section)

== Deriving Hamilton's equations ==
Hamilton's equations can be derived by a calculation with the [[Lagrangian mechanics|Lagrangian]] {{tmath|1= \mathcal L }}, generalized positions {{mvar|q<sup>i</sup>}}, and generalized velocities {{math|{{overset|lh=0.3|⋅|''q''}}<sup>''i''</sup>}}, where {{tmath|1= i = 1,\ldots,n }}.<ref>This derivation is along the lines as given in {{Harvnb|Arnol'd|1989|pp=65–66}}</ref> Here we work [[On shell and off shell|off-shell]], meaning {{tmath|1= q^i }}, {{tmath|1= \dot{q}^i }}, {{tmath|1= t }} are independent coordinates in phase space, not constrained to follow any equations of motion (in particular, <math>\dot{q}^i</math> is not a derivative of {{tmath|1= q^i }}). The [[total differential]] of the Lagrangian is:
<math display="block">\mathrm{d} \mathcal{L} 
= \sum_i \left ( \frac{\partial \mathcal{L}}{\partial q^i} \mathrm{d} q^i + \frac{\partial \mathcal{L}}{\partial \dot{q}^i} \, \mathrm{d} \dot{q}^i \right )
+ \frac{\partial \mathcal{L}}{\partial t} \, \mathrm{d}t \ .</math>
The generalized momentum coordinates were defined as {{tmath|1= p_i = \partial \mathcal{L}/\partial \dot{q}^i }}, so we may rewrite the equation as:
<math display="block"> \begin{align}
\mathrm{d} \mathcal{L} 
=& \sum_i \left( \frac{\partial \mathcal{L}}{\partial q^i} \, \mathrm{d} q^i 
+ p_i \mathrm{d} \dot{q}^i \right) + \frac{\partial \mathcal{L}}{\partial t}\mathrm{d}t
\\
=& 
\sum_i \left( \frac{\partial \mathcal{L}}{\partial q^i} \, \mathrm{d}q^i + \mathrm{d}( p_i \dot{q}^i) - \dot{q}^i \, \mathrm{d} p_i \right) 
+ \frac{\partial \mathcal{L}}{\partial t} \, \mathrm{d}t\,.
\end{align}
</math>

After rearranging, one obtains:
<math display="block">\mathrm{d}\! \left ( \sum_i p_i \dot{q}^i - \mathcal{L} \right ) 
= \sum_i \left( - \frac{\partial \mathcal{L}}{\partial q^i} \, \mathrm{d} q^i + \dot{q}^i \mathrm{d}p_i \right) 
- \frac{\partial \mathcal{L}}{\partial t} \, \mathrm{d}t\ .</math>

The term in parentheses on the left-hand side is just the Hamiltonian <math display="inline">\mathcal H = \sum p_i \dot{q}^i - \mathcal L</math> defined previously, therefore:
<math display="block">\mathrm{d} \mathcal{H} 
= \sum_i \left( - \frac{\partial \mathcal{L}}{\partial q^i} \, \mathrm{d} q^i + \dot{q}^i \, \mathrm{d} p_i \right) 
- \frac{\partial \mathcal{L}}{\partial t} \, \mathrm{d}t\ .</math>

One may also calculate the total differential of the Hamiltonian <math>\mathcal H</math> with respect to coordinates {{tmath|1= q^i }}, {{tmath|1= p_i }}, {{tmath|1= t }} instead of {{tmath|1= q^i }}, {{tmath|1= \dot{q}^i }}, {{tmath|1= t }}, yielding:
<math display="block">\mathrm{d} \mathcal{H}
=\sum_i \left( \frac{\partial \mathcal{H}}{\partial q^i} \mathrm{d} q^i +
\frac{\partial \mathcal{H}}{\partial p_i} \mathrm{d} p_i \right) 
+ \frac{\partial \mathcal{H}}{\partial t} \, \mathrm{d}t\ .</math>

One may now equate these two expressions for {{tmath|1= d\mathcal H }}, one in terms of {{tmath|1= \mathcal L }}, the other in terms of {{tmath|1= \mathcal H }}:
<math display="block">\sum_i \left( - \frac{\partial \mathcal{L}}{\partial q^i} \mathrm{d} q^i + \dot{q}^i \mathrm{d} p_i \right) 
- \frac{\partial \mathcal{L}}{\partial t} \, \mathrm{d}t
\ =\ 
\sum_i \left( \frac{\partial \mathcal{H}}{\partial q^i} \mathrm{d} q^i +
\frac{\partial \mathcal{H}}{\partial p_i} \mathrm{d} p_i \right) 
+ \frac{\partial \mathcal{H}}{\partial t} \, \mathrm{d}t\ .</math>

Since these calculations are off-shell, one can equate the respective coefficients of {{tmath|1= \mathrm{d}q^i }}, {{tmath|1= \mathrm{d}p_i}}, {{tmath|1= \mathrm{d}t }} on the two sides:
<math display="block">\frac{\partial \mathcal{H}}{\partial q^i} = - \frac{\partial \mathcal{L}}{\partial q^i} \quad, 
\quad \frac{\partial \mathcal{H}}{\partial p_i} = \dot{q}^i \quad, 
\quad \frac{\partial \mathcal{H}}{\partial t } = - {\partial \mathcal{L} \over \partial t}\ . </math>

On-shell, one substitutes parametric functions <math>q^i=q^i(t)</math> which define a trajectory in phase space with velocities {{tmath|1= \dot q^i = \tfrac{d}{dt}q^i(t) }}, obeying [[Euler–Lagrange equation|Lagrange's equations]]:
<math display="block">\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial \mathcal{L}}{\partial \dot{q}^i} - \frac{\partial \mathcal{L}}{\partial q^i} = 0\ . </math>

Rearranging and writing in terms of the on-shell <math>p_i = p_i(t) </math> gives:
<math display="block">\frac{\partial \mathcal{L}}{\partial q^i} = \dot{p}_i\ . </math>

Thus Lagrange's equations are equivalent to Hamilton's equations:
<math display="block">\frac{\partial \mathcal{H}}{\partial q^i} =- \dot{p}_i \quad , 
\quad \frac{\partial \mathcal{H}}{\partial p_i} = \dot{q}^i \quad , 
\quad \frac{\partial \mathcal{H}}{\partial t} = - \frac{\partial \mathcal{L}}{\partial t}\, .</math>

In the case of time-independent <math>\mathcal H</math> and {{tmath|1= \mathcal L }}, i.e. {{tmath|1= \partial\mathcal H/\partial t = -\partial\mathcal L/\partial t = 0 }}, Hamilton's equations consist of {{math|2''n''}} first-order [[differential equation]]s, while Lagrange's equations consist of {{mvar|n}} second-order equations. Hamilton's equations usually do not reduce the difficulty of finding explicit solutions, but important theoretical results can be derived from them, because coordinates and momenta are independent variables with nearly symmetric roles.

Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, so that some coordinate <math>q_i </math> does not occur in the Hamiltonian (i.e. a ''cyclic coordinate''), the corresponding momentum coordinate <math>p_i </math> is conserved along each trajectory, and that coordinate can be reduced to a constant in the other equations of the set. This effectively reduces the problem from {{mvar|n}} coordinates to {{math|(''n'' − 1)}} coordinates: this is the basis of [[symplectic reduction]] in geometry. In the Lagrangian framework, the conservation of momentum also follows immediately, however all the generalized velocities <math>\dot q_i </math> still occur in the Lagrangian, and a system of equations in {{mvar|n}} coordinates still has to be solved.<ref name=Goldstein>{{harvnb|Goldstein|Poole|Safko|2002|pp=347–349}}</ref>

The Lagrangian and Hamiltonian approaches provide the groundwork for deeper results in classical mechanics, and suggest analogous formulations in [[quantum mechanics]]: the [[path integral formulation]] and the [[Schrödinger equation]].