A free motion equation is a differential equation that describes a mechanical system in the absence of external forces, but in the presence only of an inertial force depending on the choice of a reference frame. In non-autonomous mechanics on a configuration space
Q
→
R
, a free motion equation is defined as a second order non-autonomous dynamic equation on
Q
→
R
which is brought into the form
q
¯
t
t
i
=
0
with respect to some reference frame
(
t
,
q
¯
i
)
on
Q
→
R
. Given an arbitrary reference frame
(
t
,
q
i
)
on
Q
→
R
, a free motion equation reads
q
t
t
i
=
d
t
Γ
i
+
∂
j
Γ
i
(
q
t
j
−
Γ
j
)
−
∂
q
i
∂
q
¯
m
∂
q
¯
m
∂
q
j
∂
q
k
(
q
t
j
−
Γ
j
)
(
q
t
k
−
Γ
k
)
,
where
Γ
i
=
∂
t
q
i
(
t
,
q
¯
j
)
is a connection on
Q
→
R
associates with the initial reference frame
(
t
,
q
¯
i
)
. The right-hand side of this equation is treated as an inertial force.
A free motion equation need not exist in general. It can be defined if and only if a configuration bundle
Q
→
R
of a mechanical system is a toroidal cylinder
T
m
×
R
k
.
