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:
Example
Sometimes a given Hamiltonian can be turned into one that looks like the harmonic oscillator Hamiltonian, which is
H
=
a
P
2
+
b
Q
2
.
{\displaystyle H=aP^{2}+bQ^{2}.}
For example, with the Hamiltonian
H
=
1
2
q
2
+
p
2
q
4
2
,
{\displaystyle H={\frac {1}{2q^{2}}}+{\frac {p^{2}q^{4}}{2}},}
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
H
=
Q
2
2
+
P
2
2
,
{\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 third kind,
F
=
F
3
(
p
,
Q
)
.
{\displaystyle F=F_{3}(p,Q).}
To find F explicitly, use the equation for its derivative from the table above,
P
=
−
∂
F
3
∂
Q
,
{\displaystyle P=-{\frac {\partial F_{3}}{\partial Q}},}
and substitute the expression for P from equation (1), expressed in terms of p and Q:
p
Q
2
=
−
∂
F
3
∂
Q
{\displaystyle {\frac {p}{Q^{2}}}=-{\frac {\partial F_{3}}{\partial Q}}}
Integrating this with respect to Q results in an equation for the generating function of the transformation given by equation (1):
To confirm that this is the correct generating function, verify that it matches (1):
q
=
−
∂
F
3
∂
p
=
−
1
Q
{\displaystyle q=-{\frac {\partial F_{3}}{\partial p}}={\frac {-1}{Q}}}
See also
Hamilton–Jacobi equation
Poisson bracket
References
