In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) <math>\delta \gamma</math> shows how the mechanical system's trajectory can hypothetically (hence the term virtual) deviate very slightly from the actual trajectory <math>\gamma</math> of the system without violating the system's constraints. For every time instant <math> t,</math> <math>\delta \gamma(t)</math> is a vector tangential to the configuration space at the point <math>\gamma(t).</math> The vectors <math>\delta \gamma(t)</math> show the directions in which <math>\gamma(t)</math> 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 <math>\gamma</math> pass through the given point <math>\mathbf{q}</math> at the given time <math>\tau,</math> i.e. <math>\gamma(\tau) = \mathbf{q},</math> then <math>\delta\gamma (\tau) = 0.</math>
Notations
Let <math>M</math> be the configuration space of the mechanical system, <math>t_0,t_1 \in \mathbb{R}</math> be time instants, <math>q_0,q_1 \in M,</math> <math>C^\infty[t_0, t_1]</math> consists of smooth functions on <math>[t_0, t_1]</math>, and
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P(M) = \{\gamma \in C^\infty([t_0,t_1], M) \mid \gamma(t_0)=q_0,\ \gamma(t_1)=q_1\}.
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The constraints <math>\gamma(t_0)=q_0,</math> <math>\gamma(t_1)=q_1</math> are here for illustration only. In practice, for each individual system, an individual set of constraints is required.
Definition
For each path <math>\gamma \in P(M)</math> and <math>\epsilon_0 > 0,</math> a variation of <math>\gamma</math> is a smooth function <math>\Gamma : [t_0,t_1] \times [-\epsilon_0,\epsilon_0] \to M</math> such that, for every <math>\epsilon \in [-\epsilon_0,\epsilon_0],</math> <math>\Gamma(\cdot,\epsilon) \in P(M)</math> and <math>\Gamma(t,0) = \gamma(t).</math> The virtual displacement <math>\delta \gamma : [t_0,t_1] \to TM</math> <math>(TM</math> being the tangent bundle of <math>M)</math> corresponding to the variation <math>\Gamma</math> assigns