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.