
67Integration of Newton’s Equation of Motion
© 2010 Taylor & Francis Group, LLC
xt
tt
m
Ft t
t
() ()=
−
′
d
0
.
We can rewrite this as
xt GttF
t
() (, )()=
d
0
, (3.24)
where we have dened
G(t,t′) = (t − t′)/m. (3.25)
G(t,t′) is called Green’s function for this particular system and is here a function only of the differ-
ence (t – t′). The equation for x(t) is an example of an integral equation, G(t,t′) being a so-called inte-
gral kernel, which, here, converts the applied force function F(t) into the displacement function x(t).
To complete the above example, we assume that F(t) = F
0
, a constant, for t ≥ 0. Then
xt
tt
m
Ft t
F
m
tt t
F
m
tt
() ()
(
=
−
′
′′