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{{Short description|Function used to generate other functions}}
{{About|generating functions in physics|generating functions in mathematics|Generating function}}
In physics, and more specifically in [[Hamiltonian mechanics]], a '''generating function''' is, loosely, a function whose partial derivatives generate the differential equations that determine a system's dynamics. Common examples are the [[partition function (statistical mechanics)|partition function]] of [[statistical mechanics]], the Hamiltonian, and the function which acts as a bridge between two sets of canonical variables when performing a [[canonical transformation]].

==In canonical transformations==
There are four basic generating functions, summarized by the following table:<ref>{{cite book|last1=Goldstein|first1=Herbert|title=Classical Mechanics|last2=Poole|first2=C. P.|last3=Safko|first3=J. L.|publisher=Addison-Wesley|year=2001|isbn=978-0-201-65702-9|edition=3rd|pages=373}}</ref>
{| class="wikitable" style="margin-left:1.5em;"
! style="background:#ffdead;" | Generating function
! style="background:#ffdead;" | Its derivatives
|-
|<math>F = F_1(q, Q, t) </math>
|<math>p = ~~\frac{\partial F_1}{\partial q} \,\!</math> and <math>P = - \frac{\partial F_1}{\partial Q} \,\!</math>
|-
|<math>\begin{align} F &= F_2(q, P, t) \\ &= F_1 + QP \end{align}</math>
|<math>p = ~~\frac{\partial F_2}{\partial q} \,\!</math> and <math>Q = ~~\frac{\partial F_2}{\partial P} \,\!</math>
|-
|<math>\begin{align} F &= F_3(p, Q, t) \\ &= F_1 - qp \end{align}</math>
|<math>q = - \frac{\partial F_3}{\partial p} \,\!</math> and <math> P = - \frac{\partial F_3}{\partial Q} \,\!</math>
|-
|<math>\begin{align} F &= F_4(p, P, t) \\ &= F_1 - qp + QP \end{align}</math>
|<math>q = - \frac{\partial F_4}{\partial p} \,\!</math> and <math> Q = ~~\frac{\partial F_4}{\partial P} \,\!</math>
|}

==Example==
Sometimes a given Hamiltonian can be turned into one that looks like the [[harmonic oscillator]] Hamiltonian, which is

<math display="block">H = aP^2 + bQ^2.</math>

For example, with the Hamiltonian

<math display="block">H = \frac{1}{2q^2} + \frac{p^2 q^4}{2},</math>

where {{mvar|p}} is the generalized momentum and {{mvar|q}} is the [[Generalized coordinates|generalized coordinate]], a good canonical transformation to choose would be

{{NumBlk||<math display="block">P = pq^2 \text{ and }Q = \frac{-1}{q}. </math>|{{EquationRef|1}}}}

This turns the Hamiltonian into

<math display="block">H = \frac{Q^2}{2} + \frac{P^2}{2},</math>

which is in the form of the harmonic oscillator Hamiltonian.

The generating function {{math|''F''}} for this transformation is of the third kind,

<math display="block">F = F_3(p,Q).</math>

To find {{math|''F''}} explicitly, use the equation for its derivative from the table above,

<math display="block">P = - \frac{\partial F_3}{\partial Q},</math>

and substitute the expression for {{mvar|P}} from equation ({{EquationNote|1}}), expressed in terms of {{mvar|p}} and {{mvar|Q}}:

<math display="block">\frac{p}{Q^2} = - \frac{\partial F_3}{\partial Q}</math>

Integrating this with respect to {{mvar|Q}} results in an equation for the generating function of the transformation given by equation ({{EquationNote|1}}):
{{Equation box 1 | indent = : | equation = <math>F_3(p,Q) = \frac{p}{Q}</math> }}

To confirm that this is the correct generating function, verify that it matches ({{EquationNote|1}}):

<math display="block">q = - \frac{\partial F_3}{\partial p} = \frac{-1}{Q}</math>

==See also==
*[[Hamilton–Jacobi equation]]
*[[Poisson bracket]]

==References==
{{Reflist}}

[[Category:Classical mechanics]]
[[Category:Hamiltonian mechanics]]

Generating function (physics) - Revision history

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