In analytical mechanics, a branch of applied mathematics and physics, 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
is a vector tangential to the configuration space at the point
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
δ γ ( τ ) = 0.
be the configuration space of the mechanical system,
be time instants,
consists of smooth functions on
In practice, for each individual system, an individual set of constraints is required.
being the tangent bundle of
the tangent vector
In terms of the tangent map,
is the tangent map of
A single particle freely moving in
has 3 degrees of freedom.
The configuration space is
δ γ ( t ) = σ ( t ) ∈
particles moving freely on a two-dimensional surface
degree of freedom.
The configuration space here is
Some authors express this as
A rigid body rotating around a fixed point with no additional constraints has 3 degrees of freedom.
The configuration space here is
the special orthogonal group of dimension 3 (otherwise known as 3D rotation group), and
to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices.
guarantees the existence of
( t , ϵ ) = γ ( t ) exp (