Mathematical Physics Vol 1

7.3 Linear and quasilinear first order PDE

353

Further, given that z = z ( x , y ) , its differential

∂ z ∂ x

∂ z ∂ y

d z =

d x +

d y = p d x + q d y

(7.68)

represents the Pfaffian equation

ϕ ( x , y , z , c 1 ) d x + ψ ( x , y , z , c 1 ) d y − d z = 0 . (7.69) In order for this equation to be integrable, i.e. reduced to a complete differential, it is necessary to satisfy the condition of integrability (7.46), which in this case is reduced to ( P = p , Q = q , R = − 1) ∂ p ∂ y + q ∂ p ∂ z − ∂ q ∂ x + p ∂ q ∂ z = 0 . (7.70) By integrating the equation (7.69), another arbitrary constant is obtained, which finally yields a complete solution in the form v = v ( x , y , z , c 1 , c 2 ) . Let us now find the relations and constraints from which we could determine the function g . To that end, we shall first define some concepts.

Definition Two functions f and g are said to be in involution if [ f , g ]= 0.

The notation [ f , g ] introduced here is determined by the expression

∂ f ∂ q ∂ g ∂ q

∂ f ∂ y ∂ g ∂ y

∂ f ∂ z ∂ g ∂ z

∂ f ∂ p ∂ g ∂ p

∂ f ∂ x ∂ g ∂ x

∂ f ∂ z ∂ g ∂ z

+ q

+ p

[ f , g ]=

(7.71)

+

.

+ q

+ p

The expression (7.71) is known in literature as Mayer 7 bracket . In particular, if functions f and g do not depend explicitly of z , i.e.

∂ f ∂ z

∂ g ∂ z

= 0, the Mayer

=

bracket comes down to

∂ f ∂ q ∂ g ∂ q

∂ f ∂ y ∂ g ∂ y

∂ f ∂ p ∂ g ∂ p

∂ f ∂ x ∂ g ∂ x

( f , g )=

(7.72)

+

.

The expression ( f , g ) , defined by the relation (7.72) is known in literature as Poisson bracket . 8

Theorem24 Functions p and q , determined by equations (7.66) form a total differential (7.69) iff the functions f and g are in involution.

7 Mayer 8 Denis Poisson (1781-1840), French mathematician. He dealt with rational mechanics, probability calculus and mathematical physics. He laid the foundations of magnetism.