An extremely attractive but very difficult problem of fluid dynamics is to
find a regular way of constructing functionals which grow monotonically
with time. Existence of such a functional would mean that an initially
`small' solution gradually becomes `large', which is closely related to
such fundamental problems as instability of fluid flows,
magnetohydrodynamic dynamo, etc. One such functional known for a long
time, it can be called `virial'. First it appeared in classical mechanics
in the `virial equation'. It was first introduced in the 'virial theorem'
and then extensively used in proving of instability in finite-dimensional
mechanical systems. Then the `virial equality' had been used in problems
of fluid equilibria instability in astrophysical applications and for
fluid possessing a free surface. At the same time, the rational reason,
why monotonically increasing `virial' functions or functionals should
exist, is unknown (and if they do exist what does define their
mathematical structure). In the present paper, we discuss one possible
general way of introducing the `virial functional' in fluid dynamics from
a universal viewpoint. Our approach is based on the equalities underlying
the Hamilton principle of least action. We can introduce the notion of
`virial' (and show that it is useful) for a fluid system as soon as we
know the expression for its Lagrangian. Technically, we consider families
of vector fields depending on a scalar parameter. The variations of
velocities (and other unknown fields) correspond to derivatives with
respect to that parameter. The `virial' appears in the equality for the
second derivative of Lagrangian with respect to that parameter (or in
other words in the equation for second variation of the action
functional). It shows the close link to the solution of the well-known
`Jacobi equation for geodesic deviations'.
We use the `virial' to obtain results concerning nonlinear and linear
{\it a priori} estimates of solutions. Particular cases of an inviscid
incompressible fluid (with and without free surface), stratified fluid,
and ideal magnetohydrodynamics have been considered. Our approach recovers
and unifies all known instability results obtained using the `virial'. In
particular, it shows that the `horseshoe' instability considered earlier
by Arnold can be discovered and studied using the direct Lyapunov method
with the ` virial' as the Lyapunov functional.
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