The most general definition states that the number of degrees of freedom in a system is made up of the number of independent virtual displacements δξ (2.20).
For a holonomic system it also represents the number of coordinates (variables)
necessary and sufficient in order to define the position of a system. In accordance with this description every equation for holonomic constraints reduces the number of degrees of freedom by one (see 2.21, 2.22). This can be defined by the relation
h n
s= − (2.33) where:
s - the number of degrees of freedom of a holonomic system
n - the number of coordinates necessary for description of the position of an unconstrained system
h - the number of holonomic constraints
Under such assumptions regarding the number of degrees of freedom, the equation of nonholonomic constraints (2.11, 2.12) also leads to the reduction of the degrees of freedom despite the fact that the position of the system is not limited. This also means that the number of degrees of freedom of nonholonomic systems is lower than the number of coordinates necessary for the description of the position of such a system. For the time being we shall, however, focus on holonomic systems.
Generalized coordinates form the vector of q = (q1,q2,…,qs), and the compo- nents of this vector include any variables that fulfill three pre-requisites:
1° the number s of generalized coordinates is equal to the number of de- grees of freedom
2° generalized coordinates are selected in such a manner that they are com- patible with constraints present in the system, i.e. they fulfill the condition of identity with the equations of constraints
0 ))) ( (rq =
fj (2.34) 3° generalized coordinates need to be linearly independent, which means that the selection of them has to enable one to uniformly express Cartesian coordinates r = r(q), alternatively Ξ = Ξ(q), or coordinates of the primary description Χ = Χ(q), which gives
) , ,
( 1 s
j
j ξ q … q
ξ = or χj =χj(q1,…,qs) (2.35) Formally it means that the functional Jacobian matrix
⎥⎦
⎢ ⎤
⎣
⎡
∂
∂
k i
q
ξ or else ⎥
⎦
⎢ ⎤
⎣
⎡
∂
∂
k i
q
χ (2.36)
- is of s order in the entire area of the variation of coordinates.
The second of the equations (2.35) defines the so called primary description coordinates, which form an alternative to the Cartesian coordinate system, as they involve an arbitrary set of variables for the description of the position of a system, without an imposed limitation on the number of coordinates used in such a de- scription. The practical selection of generalized coordinates can be performed in a number of ways and tends to be much easier than it is implied from the study of formal requirements (2.34-2.36). Among Cartesian, polar, spherical or other
variables used in the description of a physical model of a system (which means all coordinates of the primary description) it is necessary to select such s of inde- pendent variables which are compatible with constraints and offer a comfortable source for the description of the position of a system. For the case of holonomic constraints the geometry of the constraints often suggests the selection of such variables. After an appropriate selection of the variables the resulting equations are succinct and short, while for other selection the resulting equations of motion might be complex and involve a lot of other components. However, the total num- ber of equations of motion remains constant (or the total order of a system of equations), which amounts to s equations of the second order for a holonomic sys- tem. The appropriate selection of the generalized coordinates in such a manner that simple and short forms of equations ensue can be found later in the text.
Transformational formulae – are functional relations which express the rela- tions between Cartesian coordinates of motion (r, Ξ) or coordinates of primary de- scription (Χ) and the vector of generalized coordinates. Similar transformational formulae account for the relations between velocities, which can be gained for holonomic constraints by differentiation of relations regarding position with re- spect to time. The transformational formulae which are expressed by equations (2.35) for position could be completed by explicit relation to time for the purposes of the general consideration. Such instances are non-isolated systems, e.g.
) , , , (q1q2 …qs t Ξ
=
Ξ or Χ=Χ(q1,q2,…qs,t) (2.37) From these relations transformational formula for velocity ensues in the form
q t q
i s
k
k k i
i ∂
+∂
∂
=∑∂
=
ξ ξ ξ
1
and
q t q
i s
k
k k
i
i ∂
+∂
∂
=∑∂
=
χ χ χ
1
(2.38) Similarly, during the calculation of the variation of variables (2.35), the result takes the form of virtual displacement of Cartesian coordinates (of the primary de- scription) expressed in terms of virtual displacements (variations) of generalized coordinates
∑= ∂
= s ∂
k
k k i
i q
1 q ξ δ
δξ or ∑
= ∂
= s ∂
k
k k
i
i q
1 q χ δ
δχ (2.39) One can note that the transformational formulae for virtual displacements (2.39) are the same as the ones resulting from the calculation of total differential of vari- ables for transformational formulae (2.35) not accounting for time. One also should note at this point that independence of virtual displacements for general- ized coordinates comes as a consequence of the fulfillment of constraint equations by the generalized coordinates
) , , ,
(δq1δq2 δqs
δq= … (2.40) and hence they can assume arbitrary values with the role of indefinite multipliers.