
Variational equations of motion 153
These are the lengths of the minor and major axis of the elliptical path of the
orbiting body.
10.6 Variational foundation of fluid motion
Until now we have focused on particles of mechanical systems. To provide
a foundation for a later topic we now consider a fluid ”particle” in the form
of an infinitesimally small volume ν. We will assume that this elementary
volume of fluid will not change but, true to the behavior of fluid, could move
by the displacement vector u =(u
x
,u
y
,u
z
). The condition of the unchanged
volume may be expressed as
ν
(
∂u
x
∂x
+
∂u
y
∂y
+
∂u
z
∂z
)dν =
ν
∇·udν =0.
Let us now follow the Hamiltonian avenue again. The ...