Computation and Properties of Maps
117
s
y
FIGURE 5.1: Trajectories of particles in a system with horizontal midplane symmetry.
is applied, its coordinates are
x f = m x (x i , a i , y i , b i , l i , δ i ),
a f = m a (x i , a i , y i , b i , l i , δ i ),
y f = m y (x i , a i , y i , b i , l i , δ i ),
b f = m b (x i , a i , y i , b i , l i , δ i ),
l f = m l (x i , a i , y i , b i , l i , δ i ),
δ f = m δ (x i , a i , y i , b i , l i , δ i ).
Under the presence of midplane symmetry, a particle that starts at
(x i , −y i , d i , a i , −b i , t i )
must end at
(x f , −y f , d f , a f , −b f , t f ),
which entails that
x f = m x (x i , a i , −y i , −b i , l i , δ i ),
a f = m a (x i , a i , −y i , −b i , l i , δ i ),
−y f = m y (x i , a i , −y i , −b i , l i , δ i ),
−b f = m b (x i , a i , −y i , −b i , l i , δ i ),
l f = m l (x i , a i , −y i , −b i , l i , δ i ),
δ f = m δ (x i , a i , −y i , −b i , l i , δ i ).
(5.1)
Thus flipping the signs of y i , b i simultaneously flips the signs of y f , b f , but
leaves x f , a f , l f , δ f intact. Flipping the sign of y i , b i simultaneously produces