Depending on the settings, the displacement may be virtual, only modifying the surface normals to give the impression of displacement ( also known as bump mapping ) or a combination of real and virtual displacement.
12.
Lanczos presents this as the postulate : " The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints . " The argument is as follows.
13.
If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces ( which is not usually the case, so this derivation works only for special cases ), the constraint forces do no work.
14.
Which has a similar form to the total differential of " L ", but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements.
15.
Which has a similar form to the total differential of " L ", but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements.
16.
The principle states that the sum of the differences between the forces acting on a system of mass particles and the time derivatives of the momenta of the system itself along any virtual displacement consistent with the constraints of the system, is zero.
17.
And since these virtual displacements " ?q j " are independent and nonzero, the coefficients can be equated to zero, resulting in "'Lagrange's equations "'or the "'generalized equations of motion " ',
18.
In analytical mechanics, a branch of applied mathematics and physics, a "'virtual displacement "'? "'r " "'i " " is an assumed infinitesimal change of system coordinates occurring while time is held constant.
19.
Considering the virtual work, \ delta W, done by the total and inertial forces together through an arbitrary virtual displacement, \ delta \ mathbf r _ i, of the system leads to a zero identity, since the forces involved sum to zero for each particle.
20.
A special case of an infinitesimal displacement ( usually notated d "'r "'), a virtual displacement ( denoted ? "'r "') refers to an infinitesimal change in the position coordinates of a system such that the constraints remain satisfied.
