Virtual displacement explained

In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation)

\delta \gamma shows how the mechanical system's trajectory can hypothetically (hence the term virtual) deviate very slightly from the actual trajectory \gamma of the system without violating the system's constraints.^{[1] [2] [3]} For every time instant t, \delta \gamma(t) is a vector tangential to the configuration space at the point \gamma(t). The vectors \delta \gamma(t) show the directions in which \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

\gamma pass through the given point q at the given time \tau, i.e. \gamma(\tau) = q , then \delta\gamma (\tau) = 0 .

Notations

Let

M be the configuration space of the mechanical system, t _{0,t} _{1} \in R be time instants, q _{0,q} _{1} \in M, C ^{inf ty[t} _{0 ,} t _{1]} consists of smooth functions on [t _{0,} t _{1]} , and

$$P(M)\; =\; \backslash .$$

The constraints

\gamma(t _{0)=q} _{0,} \gamma(t _{1)=q} _{1} are here for illustration only. In practice, for each individual system, an individual set of constraints is required.

Definition

For each path

\gamma \in P(M) and \epsilon _{0} > 0 , a variation of \gamma is a function \Gamma : [t _{0,t} _{1]} x [ - \epsilon _{0,\epsilon} _{0]} \to M such that, for every \epsilon \in [ - \epsilon _{0,\epsilon} _{0],} \Gamma( ⋅ ,\epsilon) \in P(M) and \Gamma(t, 0 ) = \gamma(t). The virtual displacement \delta \gamma : [t _{0,t} _{1]} \to TM (TM being the tangent bundle of M) corresponding to the variation \Gamma assigns^[1] to every t \in [t _{0,t} _{1]} the tangent vector

$$\backslash delta\; \backslash gamma(t)\; =\; \backslash left.\backslash frac\backslash right|\_\; \backslash in\; T\_M.$$

In terms of the tangent map,

$$\backslash delta\; \backslash gamma(t)\; =\; \backslash Gamma^t\_*\backslash left(\backslash left.\backslash frac\backslash right|\_\backslash right).$$

Here

\Gamma ^{t} _{*:} T _{0[-\epsilon,\epsilon]} \to T _{\Gamma(t,0)} M = T _{\gamma(t)} M is the tangent map of \Gamma ^{t} : [ - \epsilon,\epsilon] \to M, where \Gamma ^{t(\epsilon)} = \Gamma(t,\epsilon), and style (d)/(d\epsilon) l | _{\epsilon} \in T _{0[-\epsilon,\epsilon].}

Properties

• Coordinate representation. If

\{q ^{n} _{i =1} are the coordinates in an arbitrary chart on M and n = \dim M, then $$\backslash delta\; \backslash gamma(t)\; =\; \backslash sum^n\_\; \backslash frac\backslash Biggl|\_\; \backslash cdot\; \backslash frac\backslash Biggl|\_.$$

• If, for some time instant

\tau and every \gamma \in P(M), \gamma(\tau) = const , then, for every \gamma \in P(M), \delta \gamma (\tau) = 0 .

• If

style \gamma, (d\gamma)/(dt) \in P(M), then \delta (d\gamma)/(dt) = (d)/(dt) \delta \gamma.

Examples

Free particle in R³

A single particle freely moving in

R ^{3} has 3 degrees of freedom. The configuration space is M = R ^{3,} and P(M) = C ^{inf ty([t} _{0 ,t} _{1],} M). For every path \gamma \in P(M) and a variation \Gamma(t,\epsilon) of \gamma, there exists a unique \sigma \in T ^{3} _{0 R} such that \Gamma(t,\epsilon) = \gamma(t) + \sigma(t) \epsilon + o(\epsilon), as \epsilon \to 0 . By the definition,

$$\backslash delta\; \backslash gamma\; (t)\; =\; \backslash left.\backslash left(\backslash frac\; \backslash Bigl(\backslash gamma(t)\; +\; \backslash sigma(t)\backslash epsilon\; +\; o(\backslash epsilon)\backslash Bigr)\backslash right)\backslash right|\_$$

which leads to

$$\backslash delta\; \backslash gamma\; (t)\; =\; \backslash sigma(t)\; \backslash in\; T\_\; \backslash mathbb^3.$$

Free particles on a surface

N particles moving freely on a two-dimensional surface S \subset R ^{3} have 2 N degree of freedom. The configuration space here is

$$M\; =\; \backslash ,$$

where

r _{i} \in R ^{3} is the radius vector of the i ^{th} particle. It follows that

$$T\_\; M\; =\; T\_S\; \backslash oplus\; \backslash ldots\; \backslash oplus\; T\_S,$$

and every path

\gamma \in P(M) may be described using the radius vectors r _{i} of each individual particle, i.e.

$$\backslash gamma\; (t)\; =\; (\backslash mathbf\_1(t),\backslash ldots,\; \backslash mathbf\_N(t)).$$

This implies that, for every

\delta \gamma(t) \in T _{( r _{1(t),} \ldots, r _{N(t))}} M,

$$\backslash delta\; \backslash gamma(t)\; =\; \backslash delta\; \backslash mathbf\_1(t)\; \backslash oplus\; \backslash ldots\; \backslash oplus\; \backslash delta\; \backslash mathbf\_N(t),$$

where

\delta r _{i(t)} \in T _{r _{i(t)}} S. Some authors express this as

$$\backslash delta\; \backslash gamma\; =\; (\backslash delta\; \backslash mathbf\_1,\; \backslash ldots,\; \backslash delta\; \backslash mathbf\_N).$$

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

M = SO( 3 ), the special orthogonal group of dimension 3 (otherwise known as 3D rotation group), and P(M) = C ^{inf ty([t} _{0 ,t} _{1],} M). We use the standard notation ak{so}(3) to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map \exp : ak{so}(3) \to SO( 3 ) guarantees the existence of \epsilon _{0} > 0 such that, for every path \gamma \in P(M), its variation \Gamma(t,\epsilon), and t \in [t _{0,t} _{1],} there is a unique path \Theta ^{t} \in C ^{inf ty([ - \epsilon} _{0 ,} \epsilon _{0],} ak{so}(3)) such that \Theta ^{t(0)} = 0 and, for every \epsilon \in [ - \epsilon _{0,\epsilon} _{0],} \Gamma(t,\epsilon) = \gamma(t)\exp(\Theta ^{t(\epsilon)).} By the definition,

$$\backslash delta\; \backslash gamma\; (t)\; =\; \backslash left.\backslash left(\backslash frac\; \backslash Bigl(\backslash gamma(t)\; \backslash exp(\backslash Theta^t(\backslash epsilon))\backslash Bigr)\backslash right)\backslash right|\_=\; \backslash gamma(t)\; \backslash left.\backslash frac\backslash right|\_.$$

Since, for some function

\sigma : [t _{0,t} _{1]\to} ak{so}(3), \Theta ^{t(\epsilon)} = \epsilon\sigma(t) + o(\epsilon) , as \epsilon \to 0 ,

$$\backslash delta\; \backslash gamma\; (t)\; =\; \backslash gamma(t)\backslash sigma(t)\; \backslash in\; T\_\backslash mathrm(3).$$

See also

• D'Alembert principle
• Virtual work

References

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