In
analytical mechanics
In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...
, a branch of
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a virtual displacement (or infinitesimal variation)
shows how the mechanical system's trajectory can ''hypothetically'' (hence the term ''virtual'') deviate very slightly from the actual trajectory
of the system without violating the system's constraints.
For every time instant
is a vector
tangential
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to the
configuration space at the point
The vectors
show the directions in which
can "go" without breaking the constraints.
For example, the virtual displacements of the system consisting of a single particle on a two-dimensional surface fill up the entire tangent plane, assuming there are no additional constraints.
If, however, the constraints require that all the trajectories
pass through the given point
at the given time
i.e.
then
Notations
Let
be the
configuration space of the mechanical system,
be time instants,