ABSTRACT
EXAMPLE 24.1. Constraints For the motion on a given surface the equation of this surface will define a constraint. For instance, along the surface of a sphere of constant radius R, the constraining equations are
In a rigid body the distances between the N component particles are all constant, and the
constraints are given by the equations ( ) 0or 22 =−−= ijjiijij arrar GG Such constraints, which are independent of time, are said to be scleronomous. If the given surface or body is moving, the time will enter the previous equations as an explicit parameter, and the constraints become rheonomous. The independent coordinates of a system of N particles, free from constraints, represent 3N degrees of freedom of their motion. The l independent equations (24.1) enable us to express l of these coordinates in term of the remaining
such that the system is left with a reduced number of
iii zyx ,,
lN −3 3f N l= − degrees of freedom. However the constraints necessarily imply unknown reaction forces or forces of constraint iR
G which are only specified in terms of their effect on the motion. Their
components must be considered as new degrees of freedom in place of the eliminated coordinates, so that the set of equations of motion (2.15) must be written in the form iiii RFrm
GGG += (24.2)
where the vector sum iR G
of forces of constraint experienced by the ith particle is
iki RR GG
In Eqs.(24.2) we know the applied forces iF G
only, given in terms of andei ijF F G G
by Eq.(2.15), and it is appropriate to reformulate these equations so that the unknown reaction forces iR
G disappear. The motion of particles with given positions ir
G consistent with the imposed constraints, at a given instant t, must obey the equations obtained by differentiation with respect to time of the conditions of constraint (24.1), which gives
G G Gr G r r t t
∂ ∂ ∂⋅ + = ∇ ⋅ + = =∂ ∂ ∂∑ ∑G G …G k l
where ir G are the velocities of the various particles. The actual displacements in a time
interval dt are and we have ,i idr r dt=G G
( ) 0=∂ ∂+⋅∇∑ dttGrdG kii ki
G During the time interval dt the constraints might change, so that we need to consider what that equation would reduce to if the explicit time dependence is eliminated. This can be achieved by equating the difference between possible displacements, in a given time interval, with a virtual displacement irδ G defined as
i ir dr d riδ ′= −G G G
where It follows that the displacements and .i i i idr r dt d r r dt′ ′=G G G G = irδ G are consistent with the conditions of constraint, as given by the time independent equation
( ) lkrG i i
ki ,,1,0 … G ==⋅∇∑ δ
The virtual displacements ir
Gδ express all possible motions of the system allowed under given constraints, having a geometrical rather than a physical significance.
