Number Theory: An Introduction to Mathematics

4 Applications 481

Suppose further that there exists aninvariant region X⊆Rd.Thatis,Xis the
closure of a bounded connected open set andx ∈ Ximpliesφt(x)∈X. Then the
mapTt:X → Xgiven byTtx =φt(x)is defined for everyt ∈ Rand satisfies
Tt+sx=Tt(Tsx).
Suppose finally that divf = 0foreveryx ∈ Rd,wherex =(x 1 ,...,xd),
f=(f 1 ,...,fd)and

divf:=

∑d

k= 1

∂fk/∂xk.

Then, by a theorem due to Liouville, the mapTtsends an arbitrary region into a region
of the same volume. (For the statement and proof of Liouville’s theorem see, for exam-
ple, V.I. Arnold,Mathematical methods of classical mechanics, Springer-Verlag, New
York, 1978.) It follows that ifBis the family of Borel subsets ofXandμLebesgue
measure, normalized so thatμ(X)=1, thenTtis a measure-preserving transformation
of the probability space (X,B,μ).
An important special case is the Hamiltonian system of ordinary differential equa-
tions

dpi/dt=−∂H/∂qi, dqi/dt=∂H/∂pi (i= 1 ,...,n),

whereH(p 1 ,...,pn,q 1 ,...,qn)is a twice continuously differentiable real-valued
function. The divergence does indeed vanish identically in this case, since

−

∑n

i= 1

∂^2 H/∂pi∂qi+

∑n

i= 1

∂^2 H/∂qi∂pi= 0.

Furthermore, for anyh∈R, the energy surfaceX:H(p,q)=his invariant, since

dH[p(t),q(t)]/dt=

∑n

i= 1

∂H/∂pi(−∂H/∂qi)+

∑n

i= 1

∂H/∂qi∂H/∂pi= 0.

It is not difficult to show that ifσis the volume element onXinduced by the Euclidean
metric‖‖onR^2 n,andif

∇H=(∂H/∂p 1 ,...,∂H/∂pn,∂H/∂q 1 ,...,∂H/∂qn)

is the gradient ofH, then the mapsTtpreserve the measureμonXdefined by

μ(B)=

∫

B

dσ/‖∇H‖.

IfXis compact, this measure can be normalized and we obtain a family of measure-
preserving transformationsTt(t∈R)of the corresponding probability space.

(vi) Many problems arising in mechanics may be reduced by a change of variables to
the geometric problem ofgeodesic flow.IfMis a smooth Riemannian manifold then
the set of all pairs (x,v), wherex∈Mandvis a unit vector in the tangent space to