Schur orthogonality relations: Difference between revisions
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'''Generating functions''' which arise in [[Hamiltonian mechanics]] are quite different from [[generating functions]] in mathematics. In physics, a generating function acts as a bridge between two sets of canonical variables when performing a [[canonical transformation]]. | |||
==Details== | |||
There are four basic generating functions, summarized by the following table: | |||
{| border="1" cellpadding="5" cellspacing="0" | |||
! 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>F= F_2(q, P, t) - QP \,\!</math> | |||
|<math>p = ~~\frac{\partial F_2}{\partial q} \,\!</math> and <math>Q = ~~\frac{\partial F_2}{\partial P} \,\!</math> | |||
|- | |||
|<math>F= F_3(p, Q, t) + qp \,\!</math> | |||
|<math>q = - \frac{\partial F_3}{\partial p} \,\!</math> and <math> P = - \frac{\partial F_3}{\partial Q} \,\!</math> | |||
|- | |||
|<math>F= F_4(p, P, t) + qp - QP \,\!</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>H = aP^2 + bQ^2.</math> | |||
For example, with the Hamiltonian | |||
:<math>H = \frac{1}{2q^2} + \frac{p^2 q^4}{2},</math> | |||
where ''p'' is the generalized momentum and ''q'' is the generalized coordinate, a good canonical transformation to choose would be | |||
{{NumBlk|:|<math>P = pq^2 \text{ and }Q = \frac{-1}{q}. \,</math>|{{EquationRef|1}}}} | |||
This turns the Hamiltonian into | |||
:<math>H = \frac{Q^2}{2} + \frac{P^2}{2},</math> | |||
which is in the form of the harmonic oscillator Hamiltonian. | |||
The generating function ''F'' for this transformation is of the third kind, | |||
:<math>F = F_3(p,Q).</math> | |||
To find ''F'' explicitly, use the equation for its derivative from the table above, | |||
:<math>P = - \frac{\partial F_3}{\partial Q},</math> | |||
and substitute the expression for ''P'' from equation ({{EquationNote|1}}), expressed in terms of ''p'' and ''Q'': | |||
: <math>\frac{p}{Q^2} = - \frac{\partial F_3}{\partial Q}</math> | |||
Integrating this with respect to ''Q'' results in an equation for the generating function of the transformation given by equation ({{EquationNote|1}}): | |||
::{|cellpadding="2" style="border:2px solid #ccccff" | |||
|<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>q = - \frac{\partial F_3}{\partial p} = \frac{-1}{Q}</math> | |||
==See also== | |||
*[[Hamilton-Jacobi equation]] | |||
*[[Poisson bracket]] | |||
==References== | |||
*{{cite book | author=Goldstein, Herbert | title=Classical Mechanics | publisher=Addison Wesley | year=2002 | isbn=978-0-201-65702-9}} | |||
[[Category:Classical mechanics]] | |||
[[Category:Hamiltonian mechanics]] | |||
{{classicalmechanics-stub}} | |||
Revision as of 10:27, 8 November 2013
Generating functions which arise in Hamiltonian mechanics are quite different from generating functions in mathematics. In physics, a generating function acts as a bridge between two sets of canonical variables when performing a canonical transformation.
Details
There are four basic generating functions, summarized by the following table:
| Generating Function | Its Derivatives |
|---|---|
| and | |
| and | |
| and | |
| and |
Example
Sometimes a given Hamiltonian can be turned into one that looks like the harmonic oscillator Hamiltonian, which is
For example, with the Hamiltonian
where p is the generalized momentum and q is the generalized coordinate, a good canonical transformation to choose would be
This turns the Hamiltonian into
which is in the form of the harmonic oscillator Hamiltonian.
The generating function F for this transformation is of the third kind,
To find F explicitly, use the equation for its derivative from the table above,
and substitute the expression for P from equation (Template:EquationNote), expressed in terms of p and Q:
Integrating this with respect to Q results in an equation for the generating function of the transformation given by equation (Template:EquationNote):
To confirm that this is the correct generating function, verify that it matches (Template:EquationNote):
See also
References
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