Virtual displacement

In <a href="/facts/Analytical_mechanics/aotB0n4t">analytical mechanics</a>, a branch of <a href="/facts/Applied_mathematics/qyqklXXm">applied mathematics</a> and <a href="/facts/Physics/a7aH2y08">physics</a>, a virtual displacement (or infinitesimal variation) 
 
 
 
 δ
 γ
 
 
 {\displaystyle \delta \gamma }
 
 shows how the mechanical system's trajectory can hypothetically (hence the term virtual) deviate very slightly from the actual trajectory 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 of the system without violating the system's constraints.: 263  For every time instant 
 
 
 
 t
 ,
 
 
 {\displaystyle t,}
 
 
 
 
 
 δ
 γ
 (
 t
 )
 
 
 {\displaystyle \delta \gamma (t)}
 
 is a vector <a href="/facts/Tangent_vector/UEq5yffm">tangential</a> to the <a href="/facts/Configuration_space_(physics)/LglVrWpq">configuration space</a> at the point 
 
 
 
 γ
 (
 t
 )
 .
 
 
 {\displaystyle \gamma (t).}
 
 The vectors 
 
 
 
 δ
 γ
 (
 t
 )
 
 
 {\displaystyle \delta \gamma (t)}
 
 show the directions in which 
 
 
 
 γ
 (
 t
 )
 
 
 {\displaystyle \gamma (t)}
 
 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 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 pass through the given point 
 
 
 
 
 q
 
 
 
 {\displaystyle \mathbf {q} }
 
 at the given time 
 
 
 
 τ
 ,
 
 
 {\displaystyle \tau ,}
 
 i.e. 
 
 
 
 γ
 (
 τ
 )
 =
 
 q
 
 ,
 
 
 {\displaystyle \gamma (\tau )=\mathbf {q} ,}
 
 then 
 
 
 
 δ
 γ
 (
 τ
 )
 =
 0.
 
 
 {\displaystyle \delta \gamma (\tau )=0.}

Virtual displacement open-in-new

Virtual displacement