Numerical experiments

Using the modified differential equation for q(t) and p(t) and the definition of x(t), the expression h(f(x(t + h)) — f(x(t — h))) can be written as a series in h2 with coefficients depending on x(t) and x'(t) only. The modified Hamiltonian

H(x, x*) H(q,p) is then constant up to 0(hN) along the numerical solution over bounded time intervals, and by the same argument as in part (d) of the

previous proof, it is constant up to 0(thN) for all t = nh. The first terms of

this modified Hamiltonian are:

H(x, x') = H(x, x1) + (a - ^ + ^ j h 2U',{x)(x')2

+ {a - - t i )T iu ,{* ))2 + 0{h4)-

This proves the statement of Theorem 2.1.