0.3. COMPARISON WITH OTHE R THEORIE S xxi

The Enter-Lagrange form attached to a Lagrangian L is the 1—form

cf. [114], p. 246.

Noether's Theorem in this context is the (trivial) statement that for any La-

grangian L(t,x,x', ...,x^) and any vector field V as in Equation 0.9 there exists

G = G(£, x, x\..., x ^ " 1 ) ) such that

(0.17) (V, EL(L)) - (pr V)(L) = DtG.

If V is a variational infinitesimal symmetry of L (i.e. Equation 0.14 holds) then G

is called the conservation law attached to V; it is unique up to an additive constant

and it has the property that if the function t i— x(t) satisfies the Euler-Lagrange

equations

(0.18) ^-

) E

L(L) (*,,(*), |

W

, . . , ^ ( 0 ) = 0

for i = 1,..., n then the function

«»o('.*(').f(0.....^(«))

is constant.

Finally the Hamiltonian vector field attached to a function H — H(£, x, #') and

the vector fields d/dxi is the vector field

V

(—

— _

dH d

\

( ] ^\dx77~dx~~ dxidx'J'

All the concepts above have arithmetic analogues some of which will play a key

role in the proofs of our main theorems. The arithmetic analogues of jet spaces

of manifolds, of total derivatives, and of Lagrangians were already encountered in

our discussion of "arithmetic jet spaces". The arithmetic analogue of prolongation

of vector fields (and certain related operators) will play a key role in controlling

S—invariants in our theory. The arithmetic analogues of Frechet derivatives will

be the tangent maps in £—geometry. In some of our basic 5—geometric examples

we will prove the existence of "arithmetic (variational) infinitesimal symmetries",

we will discover remarkable systems of commuting "arithmetic Hamiltonian vector

fields", and we will compute "arithmetic Euler-Lagrange equations".

In order to clarify the analogy between the usual differential equations and

arithmetic differential equations it will be useful to discuss an intermediate step

which is provided by the Ritt-Kolchin differential algebra [117], [84].

0.3.2. Differential algebra. The presentation here follows [13]. In differen-

tial algebra one usually starts with a field K equipped with a derivation 5 : K — K\

recall that, by definition, S is an additive map satisfying the usual Leibniz rule. We

shall assume, for simplicity, that K has characteristic zero and is algebraically

closed. In applications K usually contains the field C(t) of rational functions in

one variable and S extends the derivation d/dt. (For the "most general" case of this

theory, in which K is replaced by any ring and S is replaced by a "Hasse-Schmidt

(0.16)

EL(L)

:= J2