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| [[File:Virtual Displacement 03.svg|right|thumb|300px|For the particle trajectory <math>x(t)</math> and its virtual tractory <math>x'(t)</math>, at position <math>x_1</math>, time <math>t_1</math>, the virtual displacement is <math>\delta x</math> 。 The starting and ending positions for both trajectories are at <math>x_0</math> and <math>x_2</math> respectively.]]
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| A '''virtual displacement''' <math>\delta \mathbf {r}_i\,</math> "is an assumed infinitesimal change of system coordinates occurring while time is held constant. It is called virtual rather than real since no actual displacement can take place without the passage of time."<ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref>{{rp|263}}
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| In modern terminology virtual displacement is a [[tangent space|tangent vector]] to the manifold representing the constraints at a fixed time. Unlike regular displacement which arises from differentiating with respect to time parameter <math>t\,</math> along the path of the motion (thus pointing in the direction of the motion), virtual displacement arises from differentiating with respect to the parameter <math>\epsilon\,</math> enumerating paths of the motion variated in a manner consistent with the constraints (thus pointing at a fixed time in the direction tangent to the constraining manifold). The symbol <math>\delta\,</math> is traditionally used to denote the corresponding derivative <math>\textstyle{\partial\over{\partial\epsilon}}\big|_{\epsilon=0}\,</math>.
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| The [[total differential]] of any set of system position vectors, <math>\mathbf {r}_i\,</math>, that are functions of other variables <math>\lbrace q_1, q_2, ..., q_m\rbrace\,</math>, and time <math>t\,</math>, may be expressed as follows:<ref name="Torby1984"/>{{rp|264}}
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| :<math>d \mathbf{r}_i = \frac {\partial \mathbf {r}_i}{\partial t} d t + \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} d q_j\,</math>
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| If, instead, we want the virtual displacement (virtual differential displacement), then<ref name="Torby1984"/>{{rp|265}}
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| :<math>\delta \mathbf{r}_i = \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j\,</math>
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| This equation is used in [[Lagrangian mechanics]] to relate [[generalized coordinates]], <math>q_j\,</math>, to [[virtual work]], <math>\delta W\,</math>, and [[generalized forces]], <math>Q_j\,</math>.
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| In [[analytical mechanics]] the concept of a virtual displacement, related to the concept of [[virtual work]], is meaningful only when discussing a physical system subject to constraints on its motion.{{Citation needed|date=September 2007}} A special case of an [[infinitesimal]] displacement (usually notated <math>d\mathbf{r}\,</math>), a virtual displacement (notated <math>\delta \mathbf{r}\,</math>) refers to an infinitesimal change in the position coordinates of a system such that the constraints remain satisfied.{{Citation needed|date=September 2007}}
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| For example, if a bead is constrained to move on a hoop, its position may be represented by the position coordinate <math>\theta\,</math>, which gives the [[angle]] at which the bead is situated. Say that the bead is at the top. Moving the bead straight upwards from its height <math>z\,</math> to a height <math>z + dz\,</math> would represent one possible [[infinitesimal]] [[displacement (vector)|displacement]], but would violate the constraint. The only possible virtual displacement would be a displacement from the bead's position, <math>\theta\,</math> to a new position <math>\theta + \delta\theta\,</math>{{Citation needed|date=September 2007}} (where <math>\delta\theta\,</math> could be [[positive number|positive]] or [[negative number|negative]]).
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| It is also worthwhile to note that virtual displacements are spatial displacements exclusively - [[time]] is fixed while they occur. When computing virtual differentials of quantities that are [[function (mathematics)|functions]] of [[space]] and [[time]] coordinates, no dependence on [[time]] is considered (formally equivalent to saying <math>\delta t = 0\,</math>).
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| ==See also==
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| *[[D'Alembert principle]]
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| *[[Virtual work]]
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Virtual Displacement}}
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| [[Category:Dynamical systems]]
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| [[Category:Mechanics]]
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| [[Category:Classical mechanics]]
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| [[Category:Lagrangian mechanics]]
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The author's title is Christy. The preferred hobby for him and his children is to play lacross and he would by no means give it up. Distributing manufacturing is how he tends to make a living. I've usually cherished living in Alaska.
Here is my homepage: clairvoyance (mouse click the next web page)