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
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