Virtual displacement

In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) formula_1 shows how the mechanical system's trajectory can "hypothetically" (hence the term "virtual") deviate very slightly from the actual trajectory formula_2 of the system without violating the system's constraints. For every time instant formula_3 formula_4 is a vector tangential to the configuration space at the point formula_5 The vectors formula_4 show the directions in which formula_7 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 formula_2 pass through the given point formula_9 at the given time formula_10 i.e. formula_11 then formula_12
Notations.
Let formula_13 be the configuration space of the mechanical system, formula_14 be time instants, formula_15 and
formula_16
The constraints formula_17 formula_18 are here for illustration only. In practice, for each individual system, an individual set of constraints is required.
Definition.
For each path formula_19 and formula_20 a "variation" of formula_2 is a function formula_22 such that, for every formula_23 formula_24 and formula_25 The "virtual displacement" formula_26 formula_27 being the tangent bundle of formula_28 corresponding to the variation formula_29 assigns to every formula_30 the tangent vector
formula_31
In terms of the tangent map,
formula_32
Here formula_33 is the tangent map of formula_34 where formula_35 and formula_36
Examples.
Free particle in R3.
A single particle freely moving in formula_48 has 3 degrees of freedom. The configuration space is formula_49 and formula_50 For every path formula_51 and a variation formula_52 of formula_53 there exists a unique formula_54 such that formula_55 as formula_56
By the definition,
formula_57
which leads to 
formula_58
Free particles on a surface.
formula_59 particles moving freely on a two-dimensional surface formula_60 have formula_61 degree of freedom. The configuration space here is 
formula_62
where formula_63 is the radius vector of the formula_64 particle. It follows that
formula_65
and every path formula_19 may be described using the radius vectors formula_67 of each individual particle, i.e.
formula_68
This implies that, for every formula_69
formula_70
where formula_71 Some authors express this as
formula_72
Rigid body rotating around fixed point.
A rigid body rotating around a fixed point with no additional constraints has 3 degrees of freedom. The configuration space here is formula_73 the special orthogonal group of dimension 3 (otherwise known as 3D rotation group), and formula_50 We use the standard notation formula_75 to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map formula_76 guarantees the existence of formula_77 such that, for every path formula_42 its variation formula_79 and formula_80 there is a unique path formula_81 such that formula_82 and, for every formula_23 formula_84 By the definition,
formula_85
Since, for some function formula_86 formula_87, as formula_88,
formula_89