LECTURE 5. VARIATIONAL INTEGRATORS 397is symplectic if 1 - ,\A is invertible for some real ,. To apply this to our situation,
rewrite the algorithm (5.6) as
Fr(z)-z-TXH (z+~r(z)) =0. (5.8)
Letting S = DFr(z) and A= DXH ( z+~T(z)) we get, by differentiation of equation
(5.8) with respect to z, S - 1 - ~TA(l + S) = O; i.e., (5.7) holds with,\= T/2.
Thus, (5.6) defines a symplectic scheme. +
Example 4. Here is an example of an implicit energy preserving algorithm from
Chorin, Hughes, Marsden and McCracken [1978]. Consider a Hamiltonian system
for q E ]Rn and p E lRn:
. oH
q= op'
. oH
P = - oq.
Define the following implicit scheme
where
_ + fltH(qn+l,Pn+I) - H(qn+l,Pn) ,\
q n+l - q n /\ \ T( Pn+l - Pn ) 'oH
,\ = op (aqn+l + (1 - a)qn, ,Bp n+l + (1 - ,B)pn),
oHμ = oq ('Yqn+l + (1 - 1)q n, DPn+i + (1 - 8)pn),
and where a, ,8, /, 8 are arbitrarily chosen constants in [O, 1].
The proof of conservation of energy is simple: From (5.10), we have(qn+l - qn) T (Pn+l - Pn) = flt(H(qn+l' Pn+I) - H(qn+l, Pn)),
and from (5.11)
(Pn+l - Pn) T (qn+l - qn) = -flt(H(qn+l, Pn) - H(qn, Pn)).
Subtracting (5.15) from (5.14), we obtain
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
H(qn+l, Pn+I) = H(qn, Pn)· (5.16)
This algorithm is checked to be consistent. In general, it is not symplectic. +Example 5. Let us apply the product formula idea to the simple pendulum. The
equations are
:t ( ~ ) = ( ~ ) + ( -s~n 4? ) ·
Each vector field can be integrated explicitly to give maps
and