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 [uimg]http://mathworld.wolfram.com/images/equations/LagrangeBracket/inline7.gif[/img]. 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).
be any functions of two variables 
. Then the expression 
are any functions of 
variables 
, then
in the set [uimg]http://mathworld.wolfram.com/images/equations/LagrangeBracket/inline7.gif[/img].
to 
is a contact transformation , then 
, considered as function of 
in order that the transformation from one set of variables to the other may be a contact transformation .
be 
independent functions of the variables 
. Then the Poisson bracket 
is connected with the Lagrange bracket 
by 
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).