Referring to the figure below, the pulleys are assumed to be weightless and frictionless and the ropes massless and inextensible.
Prove that 
if the system is in equilibrium. Find the equation of motion for each mass, i. if 
; and ii. if 
.
Roughwork.
In equilibrium,
.
Thus
.
Mass 
will move down and 
up if , and the reverse if . When makes displacement of 
the vertical, makes 
.
Without gain of speciality, one may obtain the equations of motion by either (a) Newton’s second law, or (b) conservation of mechanical energy, or (c) Euler-Lagrange method.
Take downward positive. When , in one’s mind one can draw two free-body diagrams giving two equations:
Hence,
note that 
. And so
Note the inequalities 
. One can thus find tension in magnitude 
, yet this is left the reader.
The Lagrangian 
of the system is
,
or,
.
where 
and 
vary with time 
as are subjected to acceleration, and 
.
Euler-Lagrange equation reads
.
Computing term by term,
and the result follows.
(to be continued)