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Generating functions arise in Hamiltonian mechanics are quite different from generating functions in mathematics. They act as a bridge between two sets of variables.
Details
There are four basic generating functions, summarized by the following table.
Generating Function
It's Derivatives
F
1
=
F
1
(
q
,
Q
,
t
)
{\displaystyle F_{1}=F_{1}(q,Q,t)\,}
p
=
∂
F
1
∂
q
{\displaystyle p={\frac {\partial F_{1}}{\partial q}}\,}
P
=
−
∂
F
1
∂
Q
{\displaystyle P=-{\frac {\partial F_{1}}{\partial Q}}\,}
F
2
=
F
2
(
q
,
P
,
t
)
−
Q
P
{\displaystyle F_{2}=F_{2}(q,P,t)-QP\,}
p
=
∂
F
2
∂
q
{\displaystyle p={\frac {\partial F_{2}}{\partial q}}\,}
Q
=
∂
F
2
∂
P
{\displaystyle Q={\frac {\partial F_{2}}{\partial P}}\,}
F
3
=
F
3
(
p
,
Q
,
t
)
+
q
P
{\displaystyle F_{3}=F_{3}(p,Q,t)+qP\,}
q
=
−
∂
F
3
∂
p
{\displaystyle q=-{\frac {\partial F_{3}}{\partial p}}\,}
P
=
−
∂
F
3
∂
Q
{\displaystyle P=-{\frac {\partial F_{3}}{\partial Q}}\,}
F
4
=
F
4
(
p
,
P
,
t
)
+
q
P
−
Q
P
{\displaystyle F_{4}=F_{4}(p,P,t)+qP-QP\,}
q
=
−
∂
F
4
∂
p
{\displaystyle q=-{\frac {\partial F_{4}}{\partial p}}\,}
Q
=
∂
F
4
∂
P
{\displaystyle Q={\frac {\partial F_{4}}{\partial P}}\,}
See also
References
Goldstein, Herbert (2002). Classical Mechanics . Addison Wesley. ISBN 0-201-65702-3. 
