Libration takes place when a body on periodic motion does not have enough energy to complete a full rotation. When it reaches the end points the sign of
changes. Librational motion is illustrated in the phase diagram below, the librations taking place between
and
The time to travel betweenand
is given by
This is equal to the time
taken to travel betweenandfor a mechanical system in which energy is conserved. The total period is then
Example: For the linear oscillator,
and the turning points are
Substituting
changes the upper and lower limits to
and
respectively . The integrand becomes
and using
gives
