Chapter 2 Constrained systems and d’Alembert principle

The chapter starts with a description of holonomic constraints on the trajectories of point particles. Real displacements are distinguished from virtual (instantaneous) displacements. D’Alembert’s theorem as an alternative equivalent formulation of Newtonian mechanics is stated and proved. For non-holonomic constraints, the same is achieved through d’Alembert’s theorem. A criterion to distinguish holonomic from non-holonomic constraints is provided. This chapter is based on [4], chapters 1.1, 1.2, 1.3.

2.1 Holonomic constraints

Consider \(\mathcal N\) particles moving in space \(\mathbb R^3\), with coordinates are denoted by \(x_i^\alpha \), \(i=1,\dots , \mathcal N\), \(\alpha =1,2,3\). Alternatively, we can label them by \(x^a\), with \(a=1,\dots ,3\mathcal N\). We assume that they are submitted to \(r\) constraints of the form

\begin{equation} \label {eq:54} G_m(x^\alpha _i,t)=0. \end{equation}

Such constraints that only involve the positions but not the velocitites of the particles are called holonomic. As before, we assume regularity conditions: the rank of matrix \(\frac {\partial G_m}{\partial x^\alpha _i}\) is maximal for all values of \(t\). When the constraints do not depend explicitly on time, they are called scleronomic, if they do, they are called rheonomic.

During a motion, the particles are assumed to remain on the surface defined by the constraints. If this motion \(dx^\alpha _i\) takes place during an infinitesimally small time interval \(dt\), this means that

\begin{equation} \label {eq:55} dG_m=dx^\alpha _i\frac {\partial G_m}{\partial x^\alpha _i}+dt\frac {\partial G_m}{\partial t}=0. \end{equation}

In other words, the constraints are constraints on the trajectories of the particles and give rise by differentiation with respect to time to constraints on the velocities,

\begin{equation} \label {eq:56} \dot x^\alpha _i\frac {\partial G_m}{\partial x^\alpha _i}+\frac {\partial G_m}{\partial t}=0, \end{equation}

where, for a trajectory \(x^\alpha _i(t)\), the notation \(\dot x^\alpha _i=\frac {dx^\alpha _i}{dt}\) and \(\ddot x^\alpha _i=\frac {d^2 x^\alpha _i}{(dt)^2}\) is used.

Consider two trajectories \(x^\alpha _i(t)\) and \(\tilde x^\alpha _i(t)\). What we will be interested in below is (instantaneous) virtual displacements i.e., the difference of these trajectories \(\Delta x^\alpha _i(t)=\tilde x^\alpha _i(t)-x^\alpha _i(t)\) at fixed times \(t\). It follows that

\begin{equation} \begin{split} \Delta \frac {d x^\alpha _i(t)}{dt}&=\Delta \lim _{\epsilon \to 0}\frac {x^\alpha _i(t+\epsilon )+x^\alpha _i(t)}{\epsilon }=\lim _{\epsilon \to 0}\frac {\tilde x^\alpha _i(t+\epsilon )-x^\alpha _i(t+\epsilon )-(\tilde x^\alpha _i(t)-x^\alpha _i(t))}{\epsilon }\\ &=\frac {d\tilde x^\alpha _i(t)}{dt}-\frac {dx^\alpha _i(t)}{dt}=\frac {d}{dt} \Delta x^\alpha _i(t)\label {eq:57}. \end {split} \end{equation}

More specifically we will be interested in infinitesimal displacements \(\delta x^\alpha _i\) of this type that are compatible with the constraints,

\begin{equation} \label {eq:58} \delta x^\alpha _i\frac {\partial G_m}{\partial x^\alpha _i}\approx 0. \end{equation}

Indeed, except for the interpretation in terms of work and forces, this follows directly from the previous lemma. The notation \(\cancel {\delta } W\) means that this elementary work is not necessarily the variation of a function. The geometric interpretation is that the reaction forces are orthogonal to the constraints. (For each \(m\), the gradient \(\frac {\partial G_m}{\partial x^\alpha _i}\) is a vector that is orthogonal to the tangent vectors of the surface defined by \(G_m=0\).)

In what follows, we limit ourselves (almost) exclusively to such constraints, i.e., to constraints whose reaction forces do not do any work during a virtual displacement. In particular this means that there can be no friction due to these constraints. Such constraints are called ideal.

2.2 d’Alembert’s theorem

The dynamics of the \(\mathcal N\) particles is governed by Newton’s equations,

\begin{equation} \label {eq:113} m_{(i)}\ddot x^\alpha _i=F^\alpha _i+R^\alpha _i, \end{equation}

where the parenthesis around the index \(i\) means that there is no summation, and where \(F^\alpha _i\) are the components of the applied forces acting on the \(i\)th particles, while \(R^\alpha _i\) are the reaction forces due to the constraints, assumed to be ideal. The latter keep the particles confined to the constraint surface.

The theorem states that Newton’s equations (2.7), together with the fact that the reaction forces associated to ideal constraints are of the form \(R^i_\alpha \approx \mu ^m\frac {\partial G_m}{\partial x^\alpha _i}\) on account of equation (2.6) of Lemma 3 are equivalent to (2.8) when the constraints (2.1) hold, and when limiting oneself to compatible infinitesimal virtual displacements, (2.5). That the condition is sufficient (\(\Longrightarrow \)) follows by contracting (2.7) with \(\delta x^\alpha _i\) and using (2.6) together with (2.5) to conclude that (2.8) holds. That the condition is necessary \((\Longleftarrow \)) follows because (2.1), (2.5) together with (2.8) imply, as in lemma 3, that \(F^\alpha _i-m_{(i)}\ddot x^\alpha _i\approx -\mu ^m\frac {\partial G_m}{\partial x^\alpha _i}\), for some \(\mu ^m\), which are Newton’s equation (2.7), together with (2.6).

2.3 Principle of virtual work

If one uses d’Alembert’s theorem in the particular case of a system at equilibrium, the accelerations vanish, \(\ddot x^\alpha _i=0\), and the theorem reduces to the

2.3.1 Worked Exercice 1: The Lever

Using the principal of virtual work, find the equilibrium position of a horizontal lever of fulcrum \(O\), with a downward vertical force of modulus \(A\) at a distance \(a\) of \(O\) to the left and a downward vertical force of modulus \(B\) at a distance \(b\) to the right of \(O\). Hint: The equilibrium condition links \(A,B,a,b\). Relate the vertical displacements of the extremities \(\delta s_A\) and \(\delta s_B\) to the rotation angle \(\delta \varphi \) (see figure).

Solution: the condition that the virtual work of the applied forces should vanish is

\begin{equation} \label {eq:123} \vec A\cdot \vec \delta s_A+\vec B\cdot \vec \delta s_B=0\iff A\delta s_A+B\delta s_B=0, \end{equation}

Furthermore \(\delta s_A=-a\delta \varphi \), \(\delta s_B=b\delta s_B\). Hence

\begin{equation} \label {eq:137} (-Aa+Bb)\delta \varphi =0,\forall \delta \varphi \Longrightarrow \boxed {Aa=Bb}. \end{equation}

2.3.2 Worked Exercice 2: The Block and Tackle

In a tackle with \(n\) pulleys at the fixed upper end and \(n\) pulleys at the moving lower end lower, let \(Q\) be the modulus of the weight force that one wishes to lift, and \(P\) the modulus of the lifting force applied at the to the other end of the rope. In a virtual displacement, let \(\delta p\) be the displacement of the end of the rope and \(-\delta q\) the displacement of the load (which is negative if \(\delta p\) is positive, see figure.

• 1. When taking the tackle as an ideal system without friction, what does the principle of virtual work imply ?

• 2. If you lift the load by \(\delta q\), each of the \(2n\) lengths of rope between the pulleys is shortened by \(\delta q\).

• 3. What is the relation between \(\delta p\) and \(\delta q\) ?

• 4. Find the link between \(Q,P\) and the number of pulleys \(n\) at the upper and lower end of the tackle to maintain the load suspended.

Solution:

Application of the principle:

\begin{equation} \label {eq:4} P\delta p-Q\delta q=0. \end{equation}

For 1 pulley at both the upper and the lower end, if one lifts by \(\delta q\), each of the 2 pieces of rope is shortened by \(\delta q\) so that \(\delta p=2\delta q\).

For 2 pulleys at both the upper and the lower end, if one lifts by \(\delta q\), each of the 4 pieces of rope is shortened by \(\delta q\) so that \(\delta p=4\delta q\).

For \(n\) pulleys at both the upper and the lower end, \(\delta p= 2n \delta q\) and

\begin{equation} \label {eq:3} (2nP-Q)\delta q=0,\forall \delta q\Longrightarrow \boxed {P=\frac {Q}{2n}}. \end{equation}

2.4 Non-holonomic constraints

Non-holonomic constraints are constraints on the velocities of point particles as in (2.3), but that do not originate from holonomic constraints, i.e., from constraints on positions, through differentiation with respect to time.

As an example, consider a vertical disk or coin rolling without sliding on a horizontal plane, \(z=0\) in standard Cartesian coordinates. Let \((x_C,y_C,z_C)\) the coodinates of the center of the disk. Only \(x_C,y_C\) are relevant since \(z_C=a\), with \(a\) the radius of the disk. Let \(\phi \) be the angle of rotation of the disk starting from a fixed initial direction and \(\theta \) the angle made by \(\vec 1_x\) and the axes of rotation of the disk. Since the velocity \(\vec v\) of the center of the disk is normal to this axes, and lies in a plane of constant \(z\), it follows that the angle between \(\vec 1_x\) and \(\vec v\) is \(\theta -\frac {\pi }{2}\), so that

\begin{equation} \label {eq:115} x_C=v\cos (\theta -\frac {\pi }{2})=v\sin \theta ,\quad y_C=v\sin (\theta -\frac {\pi }{2})=-v\cos \theta . \end{equation}

Futhermore, the velocity of a point on the edge of the disk is \(a\dot \phi \). This is the module of the velocity of the point of contact of the disk with the plane, and since there is no sliding this is also the velocity of the center of the disk, \(v=a\dot \phi \). By injecting into the previous relations, we thus get the following relations between velocitites for this motion

\begin{equation} \label {eq:116} \left \{\begin{array}{l}\dot x_C-a\sin \theta \dot \phi =0,\\ \dot y_C +a \cos \theta \dot \phi =0, \end {array}\right . \end{equation}

and the question is whether these constraints on velocities come from constraints on positions by differentiation with respect to time.

More generally, consider \(r\) constraints that are at most linear in velocities \(\dot x^a\),

\begin{equation} \label {eq:117} K^m_a \dot x^a +K^m_0=0, \end{equation}

where \(K^m_a(x^b,t),K^m_0(x^b,t)\) are functions that depend on the coordinates \(x^a\), \(a=1,\dots ,n\) (which may or may not simply be the positions \(x^\alpha _i\) in Cartesian space of the \(\mathcal N\) particles), and possibly an explicit dependence on time \(t\). The question then is whether the relation on the velocities (2.15), or any equivalent relation on the velocities

\begin{equation} \label {eq:118} {\Lambda ^m}_{m'}(K^{m'}_a \dot x^a +K^{m'}_0)=0, \end{equation}

with \({\Lambda ^m}_{m'}(x^b,t)\) an invertible matrix, can be written as

\begin{equation} \label {eq:119} {\Lambda ^m}_{m'}(K^{m'}_a \dot x^a +K^{m'}_0)=\frac {\partial G^m}{\partial x^a}\dot x^a+\frac {\partial G^m}{\partial t}. \end{equation}

for \(r\) functions \(G^m(x^a,t)\). Since the velocities are arbitrary, this relation can be decomposed as

\begin{equation} \label {eq:120} \left \{\begin{array}{l} \frac {\partial G^m}{\partial x^a}={\Lambda ^m}_{m'}K^{m'}_a ,\\ \frac {\partial G^m}{\partial t}={\Lambda ^m}_{m'}K^{m'}_0, \end {array}\right . \end{equation}

In order to answer this question, it is useful to introduce \(x^A=(t,x^a)\), \(A=0,\dots , n\), so that the above \(n+1\) equations can be written in a unified way as

\begin{equation} \label {eq:121} \frac {\partial G^m}{\partial x^A}={\Lambda ^m}_{m'}K^{m'}_A. \end{equation}

Necessary conditions¹ for the existence of such functions \(G^m\) is that mixed second derivatives of the right hand sides vanish,

\begin{equation} \label {eq:122} \frac {\partial ^2G_m}{\partial x^A\partial x^B}=\frac {\partial ^2G_m}{\partial x^B\partial x^A} \Longrightarrow \frac {\partial }{\partial x^A}({\Lambda ^m}_{m'}K^{m'}_B)=\frac {\partial }{\partial x^B}({\Lambda ^m}_{m'}K^{m'}_A). \end{equation}

This is however not the condition that we need because it still contains the unknown functions \({\Lambda ^m}_{m'}\). What we need is a condition on the functions \(K^m_A\) alone. A standard technical assumption on these functions, that we aasume to hold is that they are linearly independet in the following sense: if there exist some functions \(\lambda _m(x^B)\) such that \(\lambda _mK^m_A=0\) then necessarily \(\lambda _m=0\). The condition we need is then a consequence of the following theorem.

In equation (2.21), the square bracket denotes complete antisymmetrization of the indices \(A_1,\dots A_{r+2}\), divided by \((r+2)!\), the number of permutations of \(r+2\) objects. More explicitly,

\begin{equation} \label {eq:126} (\frac {}{\partial x^{[A_1}}{K^m_{A_2}})K^1_{A_3}\dots K^r_{A_{r+2}]}=\frac {1}{(r+2)!}\sum _{\sigma \in P_{r+2}}(1)^{|\sigma |}(\frac {}{\partial x^{A_{\sigma (1)}}}{K^m_{A_{\sigma (2)}}})K^1_{A_{\sigma (3)}} \dots K^r_{A_{\sigma {r+2}}}, \end{equation}

where \(P_{r+2}\) denotes a permutations of \(r+2\) elements and \(|\sigma |=0\) if the permutation is even, while \(|\sigma |=1\) if the permutation is odd.

Let us prove that these conditions are indeed necessary (\(\Longleftarrow \)). We start from

\begin{equation} \label {eq:127} 0=(\frac {\partial }{\partial x^{[A_1}}\frac {\partial G^m}{\partial x^{A_2}})K^1_{A_3}\dots K^r_{A_{r+2}]}, \end{equation}

which holds because the second order derivatives \(\frac {{\partial }^2G^m}{\partial x^{A_1} \partial x^{A_2}}\) are symmetric in \(A_1,A_2\), so that, when completely antisymmetrizing in the right hand side of equation (2.24), one finds always \(0\). Since we assume that (2.22) is true, we find by substitution

\begin{equation} \begin{split} \label {eq:128} 0&=(\frac {\partial }{\partial x^{[A_1}}({\Lambda ^m}_{m'} K^{m'}_{A_2}))K^1_{A_3}\dots K^r_{A_{r+2}]}\\&=(\frac {\partial }{\partial x^{[A_1}}{\Lambda ^m}_{m'}) K^{m'}_{A_2}K^1_{A_3}\dots K^r_{A_{r+2}]}+{\Lambda ^m}_{m'}(\frac {\partial }{\partial x^{[A_1}} K^{m'}_{A_2})K^1_{A_3}\dots K^r_{A_{r+2}]} \end {split} \end{equation}

The first term on the last line vanishes because the antisymmetry in \(A_2,\dots A_{r+2}\) implies the antisymmetry in \(m',1,\dots ,r\), but \(m'\) takes a value between \(1\) and \(r\) so that one of the values arises twice and a completely antisymmetric expression then vanishes. Equation (2.21) then follows because \({\Lambda ^m}_{m'}\) is invertible.

The proof that the condition is sufficient (\(\Longrightarrow \)) can be found in [3] or [10]. □

In the particular case of the disk, we have

\begin{equation} \label {eq:129} x^A=(t,x_C,y_C,\phi ,\theta ), \end{equation}

and we have two constraints, \(r=2\). More explicitly,

\begin{equation} \label {eq:130} \begin{split} K^1_0=0,\quad K^1_1=1,\quad K^1_2=0,\quad K^1_3=-a\sin \theta =-a\sin x^4,\quad K^1_4=0,\\ K^2_0=0,\quad K^2_1=0,\quad K^2_2=1,\quad K^1_3=a\cos \theta =a\cos x^4,\quad K^2_4=0. \end {split} \end{equation}

Let us show that the constraints are non-holonomic, i.e., that there is at least one of the conditions in (2.21) that is not satisfied. If we take \(m=1\), they become

\begin{equation} \label {eq:131} (\frac {\partial }{\partial x^{[A_1}}{K^1_{A_2}})K^1_{A_3} K^2_{A_{4}]}=0. \end{equation}

Choosing \(A_1=4,A_2=3,A_3=1,A_4=2\), one can check that only the first term in the complete antisymmetrization is non-vanishing, so that the result of the left hand side is given by \(\frac {1}{4!}(-a\cos x^4)\) which does not vanish.

2.5 d’Alembert’s principle

In the proof of d’Alembert theorem, nothing crucially depended on the fact that the constraints were holonomic. It can thus be extended to non-holonomic ones (by the replacement \(\frac {\partial G^m}{\partial x^a}\rightarrow K^m_a\) in the proof). Suppose then that there are \(r\) non holonomic constraints

\begin{equation} \label {eq:132} K^m_a \dot x^a+K^m_0=0, \end{equation}

and consider infinitesimal virtual displacements \(\delta x^a\) that are compatible,

\begin{equation} \label {eq:133} K^m_a\delta x^a=0. \end{equation}

d’Alembert’s principle then states that Newton’s equation

\begin{equation} \label {eq:134} m_{(a)} \ddot x^a=F^a+R^a, \end{equation}

together with the condition that the forces of reaction are of the form

\begin{equation} R_a\approx \lambda _m K^m_a\label {eq:135}, \end{equation}

are equivalent to the condition that the sum of the applied forces and of the forces of inertia produce no work during any compatible infinitesimal virtual displacement \(\delta x^a\),

\begin{equation} \label {eq:136} (F^a-m_{(a)}\ddot x^a)\delta x^a\approx 0. \end{equation}

Systems covered by d’Alembert’s theorem and principle are

• • systems of point particles that move on polished surfaces that are fixed or moving (holonomic constraints),

• • motions of solids with one point having a prescribed trajectory

• • sliding of solids on polished surfaces

• • rolling without sliding of solids on rough surfaces

• • fixed links between solids

What is not covered by the theorem and principle are constraints that involve friction or constraints that are expressed by inequalities.