Let be any functions of two variables . Then the expression
(1)
|
is called a Lagrange bracket (Lagrange 1808; Whittaker 1944, p. 298).
The Lagrange brackets are anticommutative,
(2)
|
(Plummer 1960, p. 136).
If are any functions of variables , then
(3)
|
where the summation on the right-hand side is taken over all pairs of variables in the set .
But if the transformation from to is a contact transformation, then
(4)
|
giving
(5)
| |||
(6)
| |||
(7)
| |||
(8)
|
Furthermore, these may be regarded as partial differential equations which must be satisfied by , considered as function of in order that the transformation from one set of variables to the other may be a contact transformation.
Let be independent functions of the variables . Then the Poisson bracket is connected with the Lagrange bracket by
(9)
|
where is the Kronecker delta. But this is precisely the condition that the determinants formed from them are reciprocal (Whittaker 1944, p. 300; Plummer 1960, p. 137).