{{Short description|Function used to generate other functions}} {{About|generating functions in physics|generating functions in mathematics|Generating function}} thumb | right | Generating a sine from a circle. 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 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 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