I'm a classical music lover.
Let me define Adagio, in music, means slowly. In Ballet, Adage, means slow, unfolding movements. A type of slow flowing, balletic movement being performed. Usually done with a beautiful music.
In a classical ballet class, Adage often come after pirouettes, this portion of the lesson concentrates on slow movement to improve the dancer's ability to control the leg and increase extension. Flexibility is always a plus, dancers to bring the leg into high positions with control and ease.
Leg extensions are at full classical height of 90 degrees, can be taken higher, though only when appropriate and without distorting the purity of their line. In adage, it is absolutely important for dancers to focus on the process by which the limbs reach their full extension rather than on how high they rise their legs. Always remember QUALITY not quantity. The quality of sustaining and lengthening the limbs outward, radiating from the centre of the whole body. Classical leg extensions achieve their beauty shaped by the dancer;s breathing, as an expression of spiritual expansiveness, never by force or strain.
For Advance students, Adage section are more complex co-ordination, using all alignments and directions of travel, a fuller range of body poses, and different ways in which the limbs can move in space from one position to another. Sometimes, it require great co-ordination, balance, control and concentration. It require strength to perform more intricate movements with the appearance of ease.
That is what Ballerina do, to make difficult movement, look so easy.