3.4 Introduction to the Constrained Motion Theory
Elli avien cappe con cappucci bassi Dinanzi a li occhi, fatte della taglia Che in Clugn`ı per li monaci fassi. Di fuor dorate son, si ch’elli abbaglia; Ma dentro tutte piombo, e gravi tanto . . .3
The principle of least action inspires the following somewhat trivial con- siderations. Let x → A(x) be the action of Eqs. (3.3.4) and (3.3.7) defined on the motions inMt1,t2(ξ1, ξ2) of a system of N point masses subject to a
conservative force F.
Suppose a priori known that the force law is such that the motion x that develops under its influence from ξ1to ξ2, within times t1and t2, verifies some properties like |x(t)| ≤ S or |¨x(t)| ≤ P or |x(1)(t)
| = 0, etc. Then it is clear that the research of x inMt1,t2(ξ1, ξ2) can be restricted to the subsetM, of
the motions inMt1,t2(ξ1, ξ2) verifying the properties under consideration.
Very often it happens that a system of point masses is subject lo “con- straints”, i.e., to force laws that allow only a “few” motions among those a priori possible, at least for vast classes of initial data. Think of a point mass constrained to remain on a surface: in this case, the surface acts on the point with a force systematically such as to forbid the abandonment of the surface itself by the point, whenever the initial data (η, ξ) have ξ on the surface and η tangent to it.
Think, also, of a rigid system of N points. Now the i-th point will exert on the j-th point a force f(i→j) systematically such that the two points remain at a fixed distance from each other.
By taking into account the constraints, the allowable motions inMt1,t2(ξ1,
ξ2) will generally be parameterizable with ℓ coordinates, and often ℓ≪ Nd; consequently, it will be possible to imagine a description of the motions in terms of ℓ functions of time. Therefore, the Lagrangian and the action will also be expressible in terms of the same ℓ functions, and the action of a motion x allowed by the constraints will take the form
A(x) = Z t2
t1
dt eL( ˙a1(t), . . . , ˙aℓ(t), a1(t), . . . , aℓ(t), t) (3.4.1) if t→ (a1(t), . . . , aℓ(t)), t∈ [t1, t2], is the description of the motion x in the ℓ “essential coordinates”.
3In basic English:
They had capes with low hoods in front of the eyes, made in the fashion that in Cluny is used for the monks. Golden they are outside, so that they dazzle but inside they are all leaden and heavy a lot . . . (Dante, Inferno, Canto XXIII).
To be less vague, assume that there are NRd-valued functions in C∞(Rℓ): α= (α1, . . . , αℓ)→ X(i)(α) = X(i)(α1, . . . , αℓ), (3.4.2) i = 1, . . . , N , such that the set of the motions t→ x(t) = (x(1)(t), . . . , x(N )(t)) t ∈ [t1, t2], which are “constrained” or “allowed” by the constraints is sim- ply the set of the motions which is the image of the motions in Rℓ via the transformation (3.4.2). Thus, given a motion t→ a(t), t ∈ [t1, t2], inRℓ one describes, via Eq. (3.4.2), the constrained motion t→ x(t), t ∈ [t1, t2], where x(i)(t) = X(i)(a(t)), i = 1, 2, . . . , N (3.4.3) which we shorten as x(t) = X(a(t)).
In other words, let us admit that the conservative force law F for the system of N point masses under consideration is such that the motions in Mt1,t2(ξ1, ξ2) that can actually develop under its influence starting from a
given class of initial data are necessarily contained in the class of the motions having the form of Eq. (3.4.3) with a ∈ Mt1,t2(α1, α2), where α1, α2 ∈ R
ℓ and X(α1) = ξ1, X(α2) = ξ2.
If x is a constrained motion in the sense just discussed, its action, Eq. (3.3.4), with respect to the Lagrangian (3.3.7), where V is the potential energy of F, can be written as in Eq. (3.4.1) ifL ∈ C∞(R2ℓ+1) is the function
e L(β1, . . . , βℓ, α1, . . . , αℓ, t) = 1 2 N X i=1 mi N X j=1 ∂X(i) ∂αj βj 2 − V (X(α)), (3.4.4) because ˙x(i)(t) can be computed, by differentiating Eq. (3.4.3), as
˙x(i)(t) = N X j=1 ∂X(i) ∂αj (α(t)) ˙aj(t), j = 1, 2, . . . , N, (3.4.5) whenever x is the constrained motion image of a : x = X(a).
Hence, if x ∈ Mt1,t2(ξ1, ξ2) is the motion that actually develops under
the influence of the force F and if x is the image via Eq. (3.4.3) of a, then the action A with Lagrangian (3.3.7) is stationary in Mt1,t2(ξ1, ξ2) on x,
while the action eA with Lagrangian given by Eq. (3.4.4) is stationary on a in Mt1,t2(α1, α2). This property is an immediate consequence of the fact that
if A is stationary on a motion x in it is also stationary on x in any smaller set M′ ⊂ M
t1,t2(ξ1, ξ2) provided x∈ M′. In our case, through Eq. (3.4.3),
M′ would be the set of the motions which is the image of the motions in Mt1,t2(α1, α2).
By Proposition 4,§3.3, the stationarity condition for A, i.e., for the action onMt1,t2(α1, α2) with Lagrangian density (3.4.4), is
3.4 Introduction to the Constrained Motion Theory 155 d dt ∂ eL ∂βi ( ˙a(t), a(t), t) = ∂ eL ∂αi (3.4.6) i = 1, 2, . . . , ℓ,∀ t ∈ [t1, t2].
The importance of the above considerations is easily realized: Eq. (3.4.6) is already the equation of motion after the elimination of the parameters describing the system, necessary a priori but made “useless” or “redundant” by the presence of the constraints which allow one to reduce the number of the coordinates needed to describe the actually “possible” configurations, from N d down to ℓ via (3.4.2) and (3.4.3).
Therefore, the idea occurs that the mechanism for the elimination of the redundant coordinates in conservative systems subject to simple constraints, like Eqs. (3.4.2) and (3.4.3), might be particularly simple: it will be enough to rewrite the Lagrangian density of the action only in terms of the essential coordinates through Eq. (3.4.2) and, then, deduce Eq. (3.4.6).
However, the principle of conservation of difficulties makes it clear that there must be some serious obstacle to the actual applications of such a shining but simplistic vision.
The true constraints are, in fact, generated by forces that, as we shall see shortly, generally are neither simple nor conservative (in the sense of Definition 2, p.142, §3.1) but depend on the velocities of the points as well as on their positions.
In such situations, the above considerations become essentially useless since they are not applicable to the simplest and most interesting motions constrained in the sense that they are parameterizable as in Eqs. (3.4.2) and (3.4.3), by ℓ coordinates.
To understand better what has just been said, let us consider the case of a point constrained to stay on a curve Γ ⊂ R3 with intrinsic parametric equations given by
s→ ξ(s), s∈ R (3.4.7)
where s is the curvilinear abscissa on Γ (which will be supposed to be a simple curve, i.e., without double points and open). Assume that the curve Γ exerts a force on the point mass which keeps it on Γ for all motions starting from initial data (η, ξ) with ξ = ξ(s0), η = dξds(s0) ˙s0(i.e., with ξ∈ Γ and η tangent to it), with (s0, ˙s0)∈ R2.
If τ (s), n(s) denote, respectively, the tangent and the principal normal versors to Γ at the point with curvilinear abscissa s and if r(s) denotes the curvature radius at the same point, it is well known that
τ(s) = dξ(s) ds , n(s) r(s) = dτ (s) ds (3.4.8)
Then if t→ s(t), t ∈ R, is a motion on Γ described by the time variation of the curvilinear abscissa, we find
d dtξ(s(t)) = ˙s(t) τ (s(t)) (3.4.9) and d2 dt2ξ(s(t)) = ¨s(t) τ (s(t)) + ˙s(t)2 r(s(t))n(s(t)). (3.4.10) If the point is subject to a force which is the sum of the constraint reaction R( ˙s, s) and of an external force f (s), then
m ¨x = f + R (3.4.11)
if m > 0 is the mass and x(t) = ξ(s(t)) denotes the motion inR3. By Eq. (3.4.10), Eq. (3.4.11) becomes
m¨s = f· τ + R · τ , m˙s 2
r = f· n + R · n (3.4.12) and from the second equation, it follows that the normal component of the constraint reaction is
R· n = m ˙s 2
r(s)− f(s) · n(s) (3.4.13)
at the point of Γ with coordinate s when it is occupied by a mass m with speed along Γ given by ˙s.
From Eq. (3.4.13), one sees that R( ˙s, s) is necessarily ˙s dependent if 0 < r(s) < +∞, as will be supposed, and therefore the constraint reaction cannot be conservative in the very restrictive sense of§3.1.
Nevertheless, the essence of the idea which arose in connection with Eq. (3.4.6) will be saved: it will, however, be necessary to go through a long analysis which, as is to be expected, involves a deeper physico-mathematical discussion of the notion of constraint. Such a discussion will be aimed at clarifying the definition of constraint, i.e., the physical phenomenon mathe- matically modeled as a “constraint”.
In the next section a general mathematical definition of constraint will be presented, stressing its main mathematical properties and delaying until the later sections a deeper discussion showing how the empirical notion of a fric- tionless constraint is naturally schematized by the introduced mathematical structures.
3.4.1 Exercises
1.Let Γ be a circle inR3with radius r. Find r(s), n(s), τ (s) [see Eq. (3.4.8)].
2.Let Γ be an ellipse with equations z = 0, x2/a2+y2/b2= 1, a, b > 0. Find r(s), n(s), τ (s),
at the point (x, y, 0).
3.Show that the force law R( ˙x, x) =−m ˙x2
r2 x, ( ˙x, x)∈ R2× R2 produces a constraint for