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In
analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the ...
, a branch of
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, 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. For every time instant t, \delta \gamma(t) is a vector
tangential In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
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 \mathbf at the given time \tau, i.e. \gamma(\tau) = \mathbf, then \delta\gamma (\tau) = 0.


Notations

Let M be the configuration space of the mechanical system, t_0,t_1 \in \mathbb be time instants, q_0,q_1 \in M, C^\infty
_0, t_1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math> consists of
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s on
_0, t_1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math>, and P(M) = \. 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 : _0,t_1\times \epsilon_0,\epsilon_0\to M such that, for every \epsilon \in \epsilon_0,\epsilon_0 \Gamma(\cdot,\epsilon) \in P(M) and \Gamma(t,0) = \gamma(t). The ''virtual displacement'' \delta \gamma : _0,t_1\to TM (TM being the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of M) corresponding to the variation \Gamma assigns to every t \in _0,t_1/math> the
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
\delta \gamma(t) = \frac\Biggl, _ \in T_M. In terms of the
tangent map In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of ''φ, d\varphi_x,'' at a point ''x'' is, in some sense, the best ...
, \delta \gamma(t) = \Gamma^t_*\left(\frac\Biggl, _\right). Here \Gamma^t_*: T_0 \epsilon,\epsilon\to T_M = T_M is the tangent map of \Gamma^t : \epsilon,\epsilon\to M, where \Gamma^t(\epsilon) = \Gamma(t,\epsilon), and \textstyle \frac\Bigl, _ \in T_0 \epsilon,\epsilon


Properties

* ''Coordinate representation.'' If \^n_ are the coordinates in an arbitrary chart on M and n = \mathopM, then : \delta \gamma(t) = \sum^n_ \frac\Biggl, _ \cdot \frac\Biggl, _. * If, for some time instant \tau and every \gamma \in P(M), \gamma(\tau)=\text, then, for every \gamma \in P(M), \delta \gamma (\tau) = 0. * If \textstyle \gamma,\frac \in P(M), then \delta \frac = \frac\delta \gamma.


Examples


Free particle in R3

A single particle freely moving in \mathbb^3 has 3 degrees of freedom. The configuration space is M=\mathbb^3, and P(M)=C^\infty( _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_0\mathbb^3 such that \Gamma(t,\epsilon) = \gamma(t) + \sigma(t)\epsilon + o(\epsilon), as \epsilon \to 0. By the definition, \delta \gamma (t) = \left(\frac \Bigl(\gamma(t) + \sigma(t)\epsilon + o(\epsilon)\Bigr)\right)\Biggl, _ which leads to \delta \gamma (t) = \sigma(t) \in T_ \mathbb^3.


Free particles on a surface

N particles moving freely on a two-dimensional surface S \subset \mathbb^3 have 2N degree of freedom. The configuration space here is M= \, where \mathbf_i \in \mathbb^3 is the radius vector of the i^\text particle. It follows that T_ M = T_S \oplus \ldots \oplus T_S, and every path \gamma \in P(M) may be described using the radius vectors \mathbf_i of each individual particle, i.e. \gamma (t) = (\mathbf_1(t),\ldots, \mathbf_N(t)). This implies that, for every \delta \gamma(t) \in T_ M, \delta \gamma(t) = \delta \mathbf_1(t) \oplus \ldots \oplus \delta \mathbf_N(t), where \delta \mathbf_i(t) \in T_ S. Some authors express this as \delta \gamma = (\delta \mathbf_1, \ldots , \delta \mathbf_N).


Rigid body rotating around fixed point

A
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
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 In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
of dimension 3 (otherwise known as 3D rotation group), and P(M)=C^\infty( _0,t_1 M). We use the standard notation \mathfrak(3) to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map \exp : \mathfrak(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 _0,t_1 there is a unique path \Theta^t \in C^\infty( \epsilon_0, \epsilon_0 \mathfrak(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, \delta \gamma (t) = \left(\frac \Bigl(\gamma(t)\exp(\Theta^t(\epsilon))\Bigr)\right)\Biggl, _ = \gamma(t)\frac\Biggl, _. Since, for some function \sigma : _0,t_1to \mathfrak(3), \Theta^t(\epsilon) = \epsilon\sigma(t) + o(\epsilon), as \epsilon \to 0, \delta \gamma (t) = \gamma(t)\sigma(t) \in T_SO(3).


See also

*
D'Alembert principle D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. D'Alembert ...
*
Virtual work In mechanics, virtual work arises in the application of the ''principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for ...


References

{{DEFAULTSORT:Virtual Displacement Dynamical systems Mechanics Classical mechanics Lagrangian mechanics