xx INTRODUCTION
A (vertical) vector field on M is a derivation operator (on smooth functions of
t, x) of the form
(0.9) V-H^x)-^.
i
where ai are smooth functions. More generally a (vertical) generalized vector field
(sometimes called Bdcklund vector field) is a derivation operator (on smooth func-
tions of t, x) of the form
(0.10) V:=J2^,x,x',...,x^)~;
i
cf. [114], p. 289.
The total derivative operator is the derivation operator (on smooth functions
of £, x, x',..., x^) defined by
(0.11) A : = |
+
E--°
+1 ) 9
X,
dt ^
l dx9}:
ij
cf. [114], p. 109. It is characterized by the property that
( A £ ) ° ( p r s) = (Lop r s)
for all Lagrangians L and all sections s of IT.
The prolongation of a (vertical) vector field as in Equation 0.9 is defined by
d
(0.12) p r V := ^ ( ^ a , )
dx
or
cf. [114], p. 110; it is the unique (smooth) vector field on
Jr(M)
that commutes
with Dt and coincides with V on functions of t and x.
A vector field as in Equation 0.9 is an infinitesimal symmetry for an R—linear
space C of Lagrangians if
(0.13) (pr V)(C) C £.
If (L^) is a family of Lagrangians and C is the ideal generated by the family (D^L^)
in the ring of all Lagrangians then any infinitesimal symmetry of C in the sense
above is an infinitesimal symmetry of the "system of differential equations Li = 0"
in the sense of [114], p. 161.
The vector field V is a variational infinitesimal symmetry of the Lagrangian L
if
(0.14) (prV)(L) = 0;
cf. [114], p. 253.
The Frechet derivative of a Lagrangian L is the operator on vector fields given
by
(0.15) V = ^a~ - (pr V){L) = (pr V,dL) = ] [ ^ ) ' f^fy
cf. [114], p. 307.
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