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Generating function (physics)

{{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 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:{{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}}

class="wikitable" style="margin-left:1.5em;"

! style="background:#ffdead;" | Generating function

! style="background:#ffdead;" | Its derivatives

F = F_1(q, Q, t)

|p = ~~\frac{\partial F_1}{\partial q} \,\! and P = - \frac{\partial F_1}{\partial Q} \,\!

\begin{align} F &= F_2(q, P, t) \\ &= F_1 + QP \end{align}

|p = ~~\frac{\partial F_2}{\partial q} \,\! and Q = ~~\frac{\partial F_2}{\partial P} \,\!

\begin{align} F &= F_3(p, Q, t) \\ &= F_1 - qp \end{align}

|q = - \frac{\partial F_3}{\partial p} \,\! and P = - \frac{\partial F_3}{\partial Q} \,\!

\begin{align} F &= F_4(p, P, t) \\ &= F_1 - qp + QP \end{align}

|q = - \frac{\partial F_4}{\partial p} \,\! and Q = ~~\frac{\partial F_4}{\partial P} \,\!

Example

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

H = aP^2 + bQ^2.

For example, with the Hamiltonian

H = \frac{1}{2q^2} + \frac{p^2 q^4}{2},

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

{{NumBlk||P = pq^2 \text{ and }Q = \frac{-1}{q}. |{{EquationRef|1}}}}

This turns the Hamiltonian into

H = \frac{Q^2}{2} + \frac{P^2}{2},

which is in the form of the harmonic oscillator Hamiltonian.

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

F = F_3(p,Q).

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

P = - \frac{\partial F_3}{\partial Q},

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

\frac{p}{Q^2} = - \frac{\partial F_3}{\partial Q}

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 = F_3(p,Q) = \frac{p}{Q} }}

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

q = - \frac{\partial F_3}{\partial p} = \frac{-1}{Q}

See also

References

{{Reflist}}

Category:Classical mechanics

Category:Hamiltonian mechanics