How To Find Generating Function Of Canonical Transformation

Function whose partial derivatives generate the differential equations that determine the dynamics of a system

This article is most generating functions in physics. For generating functions in mathematics, run into Generating function.

In physics, and more specifically in Hamiltonian mechanics, a generating office is, loosely, a function whose partial derivatives generate the differential equations that determine a system'due south dynamics. Common examples are the partition office of statistical mechanics, the Hamiltonian, and the part which acts every bit a bridge betwixt two sets of canonical variables when performing a approved transformation.

In canonical transformations [edit]

There are iv basic generating functions, summarized past the post-obit tabular array:^[1]

Generating function	Its derivatives
${\displaystyle F=F_{1}(q,Q,t)\,\!}$	${\displaystyle p=~~{\frac {\partial F_{1}}{\partial q}}\,\!}$ and ${\displaystyle P=-{\frac {\partial F_{1}}{\fractional Q}}\,\!}$
${\displaystyle F=F_{2}(q,P,t)=F_{1}+QP\,\!}$	${\displaystyle p=~~{\frac {\partial F_{2}}{\partial q}}\,\!}$ and ${\displaystyle Q=~~{\frac {\partial F_{ii}}{\partial P}}\,\!}$
${\displaystyle F=F_{3}(p,Q,t)=F_{1}-qp\,\!}$	${\displaystyle q=-{\frac {\partial F_{3}}{\fractional p}}\,\!}$ and ${\displaystyle P=-{\frac {\partial F_{three}}{\partial Q}}\,\!}$
${\displaystyle F=F_{four}(p,P,t)=F_{1}-qp+QP\,\!}$	${\displaystyle q=-{\frac {\partial F_{4}}{\partial p}}\,\!}$ and ${\displaystyle Q=~~{\frac {\fractional F_{iv}}{\partial P}}\,\!}$

Instance [edit]

Sometimes a given Hamiltonian tin be turned into ane that looks similar the harmonic oscillator Hamiltonian, which is

${\displaystyle H=aP^{2}+bQ^{ii}.}$

For example, with the Hamiltonian

${\displaystyle H={\frac {ane}{2q^{ii}}}+{\frac {p^{two}q^{4}}{2}},}$

where p is the generalized momentum and q is the generalized coordinate, a practiced canonical transformation to choose would be

${\displaystyle P=pq^{two}{\text{ and }}Q={\frac {-one}{q}}.\,}$

(i)

This turns the Hamiltonian into

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

which is in the form of the harmonic oscillator Hamiltonian.

The generating function F for this transformation is of the 3rd kind,

${\displaystyle F=F_{3}(p,Q).}$

To find F explicitly, use the equation for its derivative from the table to a higher place,

${\displaystyle P=-{\frac {\partial F_{three}}{\partial Q}},}$

and substitute the expression for P from equation (1), expressed in terms of p and Q:

${\displaystyle {\frac {p}{Q^{two}}}=-{\frac {\partial F_{iii}}{\fractional Q}}}$

Integrating this with respect to Q results in an equation for the generating function of the transformation given past equation (1):

${\displaystyle F_{3}(p,Q)={\frac {p}{Q}}}$

To confirm that this is the correct generating role, verify that it matches (1):

${\displaystyle q=-{\frac {\partial F_{3}}{\partial p}}={\frac {-ane}{Q}}}$

See also [edit]

• Hamilton–Jacobi equation
• Poisson bracket

References [edit]

1. ^ Goldstein, Herbert; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (tertiary ed.). Addison-Wesley. p. 373. ISBN978-0-201-65702-9.

Further reading [edit]

• Goldstein, Herbert; Poole, C. P.; Safko, J. 50. (2001). Classical Mechanics (3rd ed.). Addison-Wesley. ISBN978-0-201-65702-9.

Source: https://en.wikipedia.org/wiki/Generating_function_(physics)

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