Constraint force C and virtual displacement δr for a particle of mass m confined to a curve. The resultant non-constraint force is N. The components of virtual displacement are related by a constraint equation.
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
of the system without violating the system's constraints.[1][2][3]: 263 For every time instant
is a vector tangential 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
Let
be the configuration space of the mechanical system,
be time instants,
consists of smooth functions on
, and
The constraints
are here for illustration only. In practice, for each individual system, an individual set of constraints is required.
For each path
and
a variation of
is a function
such that, for every
and
The virtual displacement
being the tangent bundle of
corresponding to the variation
assigns[1] to every
the tangent vector
In terms of the tangent map,
Here
is the tangent map of
where
and
- Coordinate representation. If
are the coordinates in an arbitrary chart on
and
then ![{\displaystyle \delta \gamma (t)=\sum _{i=1}^{n}{\frac {d[q_{i}(\Gamma (t,\epsilon ))]}{d\epsilon }}{\Biggl |}_{\epsilon =0}\cdot {\frac {d}{dq_{i}}}{\Biggl |}_{\gamma (t)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03040a9a0724da8c0b16cc0b6559e5bec3cd5059)
- If, for some time instant
and every
then, for every
![{\displaystyle \delta \gamma (\tau )=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84c4ef65f2fc64f6f070d0495611328c22afb41e)
- If
then ![{\displaystyle \delta {\frac {d\gamma }{dt}}={\frac {d}{dt}}\delta \gamma .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5593a6a90a7272e4d2b85b1d5289d359188bf980)
Free particle in R3
[edit] A single particle freely moving in
has 3 degrees of freedom. The configuration space is
and
For every path
and a variation
of
there exists a unique
such that
as
By the definition,
which leads to
Free particles on a surface
[edit]
particles moving freely on a two-dimensional surface
have
degree of freedom. The configuration space here is
where
is the radius vector of the
particle. It follows that
and every path
may be described using the radius vectors
of each individual particle, i.e.
This implies that, for every
where
Some authors express this as
Rigid body rotating around fixed point
[edit] 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
We use the standard notation
to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map
guarantees the existence of
such that, for every path
its variation
and
there is a unique path
such that
and, for every
By the definition,
Since, for some function
, as
,