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.

One degree of freedom.
Two degrees of freedom.
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.

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

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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.

Definition

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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  

Properties

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  • 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  

Examples

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Free particle in R3

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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

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  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

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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  ,

 

See also

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References

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  1. ^ a b Takhtajan, Leon A. (2017). "Part 1. Classical Mechanics". Classical Field Theory (PDF). Department of Mathematics, Stony Brook University, Stony Brook, NY.
  2. ^ Goldstein, H.; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (3rd ed.). Addison-Wesley. p. 16. ISBN 978-0-201-65702-9.
  3. ^ Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0-03-063366-4.