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Concept
Virtual displacement
Summary
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. 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 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
If, for some time instant and every then, for every
If then
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
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
A rigid body rotating around a fixed point with no additional constraints has 3 degrees of freedom.
Official source
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