arXiv:physics/0510204v2 [physics.ed-ph] 2 Jun 2006 a ehnc nld icsino virtual of discussion a include cal arnesequation. Lagrange’s condition. how constraint the and by ‘arbitrary’ ‘restricted’ is is displacement it virtual far understand the to far one allows how definition This virtual and between allowed displacements. corresponding connection the and the constraints dis- shows the virtual clearly of attempt definition which we precise placement, article and this systematic In simple, a students. com- most easily to not often prehensible are they treatment, ad- a satisfactory more and treatise for the rigorous displacement, of and some virtual vanced Though a system. as sup- constrained a suitable com- given whether rather is identify, is to vector fails it plied student definition, a called that so Evenmonplace the students. to studying ambiguous after and vague seem often texts derived. displacement, are virtual of on equations Lagrange’s constraint the by zero lotalgaut ee ore nclassi- in courses level graduate all Almost hr r w omnlgclptwy farvn at arriving of pathways logical common two are There accessible most in presented definition the However, .Brolispicpeo ita virtual of principle Bernoulli’s 1. n arneseutoso oin(1788) motion of equations Lagrange’s particles, work moving and of system d’Alembert’s virtual constrained of to principle zero generalization Lagrange’s of (1743), principle d’Alembert’s 1,2,3,4,5,6,7,8,9,10,11,12 nVrulDslcmn n ita oki arninDyna Lagrangian in Work Virtual and Displacement Virtual On o-oooi,Slrnmu,Renmu constraints Rheonomous mechanics, Scleronomous, Lagrangian Non-holonomic, principle, d’Alembert’s Keywords: c 45,45.20.Jj,01.40.Fk are numbers: forces PACS gen constraint in by equations work Lagrange’s and virtual multipliers, zero undetermined of principle the hsclyraie n ovbe as We solvable, problem. and mechanical realized constrained p physically a d’Alembert’s of solvability that the demonstrated wor b for also zero is done similar It a work However extensio displacements. the motion. 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C. 13,14 rmtecnetof concept the From . rsn oeprecise more a present n Lagrangian and h da constraints ideal the 7 Dtd uut1 2005) 1, August (Dated: 7,16,17 (1717), uhna Ray Subhankar .Shamanna J. 7,15 . nopeeadcnrdcoywt n nte.These often another. are one texts with contradictory popular in and found incomplete definitions However various satisfactory. the logically approach solvable. said the are make that to systems’, forces of constraint class by ‘special work a virtual us zero imposition gives of The principle constraints. requires the of alone of This forces forces of motion. external introduction constrained the The terms the in maintain equations. introduced cannot constraint is the displacement of The virtual consideration. of under and particles concept time the of positions, connecting byoften defined constraints system, of constrained a equations with begins one grange, method. first present the the to ourselves In confine a we then interconnection. in- article and their an methods, of demands two demonstration the treatment of rational presentation re- A dependent and and definition confusion. circular literature to lated the leading in often approaches texts, two popular the of intermixing ytm hsmto sapidwtotabgiyin ambiguity without applied as is texts action method the some related This of dynamics the the system. and describing La- Lagrangian axiomatically the quantities the introducing with of starts motivation a first the as earlier, grangian The century presents a it half and contained. proposed in- historically self mathematically was method individually and logically and are dependent methods two The rprdfiiino ita ipaeeti necessary is displacement virtual of definition proper A La- and d’Alembert Bernoulli, to due approach, this In arneseutos ita ok Holonomic, work, Virtual equations, Lagrange’s .Hmlo’ rnil flataction least of principle Hamilton’s 2. h ocp fvruldslcmn and displacement virtual of concept The . † ∗ aitoa praht arnesequation. Lagrange’s to approach variational fdAebr’ rnil osse of system to principle d’Alembert’s of n rlzdcoordinates. eralized icpeo eovrulwr snecessary is work virtual zero of rinciple dis- the are displacements virtual the s, rnil osnthl o h allowed the for hold not does principle k yidpnet,nnhlnmc(veloc- non-holonomic independent), ty ita ipaeeti rsne that presented is displacement virtual f dniyti pca ls fconstraints, of class special this identify nrlt ohLgag’ ehdof method Lagrange’s both to entral mu tm eedn)constraints. dependent) (time omous ofragnrlcaso constraints. of class general a for so 12,13 h osritfre nvirtual on forces constraint the y e hsclquantity physical new rtnigvruldisplacement virtual erstanding n courses and n713,India 731235, an 2 ni and India 32, 11794 Y 19 oee n lofinds also one However . h eodmethod second The . mics 7,18 13) and (1834), 2 ambiguities will be discussed in detail in the next section. displacements (satisfying Eq.(2) or Eq.(4)) over the same In the literature, e.g., Greenwood2 Eq.1.26 and Pars14 time interval; automatically ensures that virtual displace- Eq.1.6.1, one encounters holonomic constraints of the ments satisfy Eq.(3) and Eq.(5) for holonomic and non- form: holonomic systems respectively. We show that in a num- ber of natural systems, e.g., with fixed or mov- φj (x1, x2,...,x3N ,t)=0 , j =1, 2,...,k. (1) ing support, particle along stationary or moving frictionless inclined plane, the work done by the forces The differential form of the above equations are sat- of constraint on virtual displacements is zero. We also isfied by allowed infinitesimal displacements {dx }, i demonstrate that this condition is necessary for the solv- (Greenwood2 Eq.1.27; Pars14 Eq.1.6.3). ability of a constrained mechanical problem. Such sys- 3N tems form an important class of natural systems. ∂φ ∂φ j dx + j dt =0 , j =1, 2,...,k. (2) ∂x i ∂t Xi=1 i A. Ambiguity in virtual displacement For a system under above constraints the virtual displace- 2 ments {δxi}, satisfy the following equations (Greenwood 14 In the literature certain statements appear in reference Eq.1.28; Pars Eq.1.6.5), to virtual displacement, which seem confusing and mu- 3N tually inconsistent, particularly to a student. In the fol- ∂φ j δx =0 , j =1, 2,...,k. (3) lowing we present few such statements found in common ∂x i Xi=1 i texts. The differential equations satisfied by allowed and virtual 1. It is claimed that (i)a virtual displacement δr is displacements are different even for the non-holonomic consistent with the forces and constraints imposed 1 case. Here, the equations satisfied by the allowed dis- on the system at a given instant t ; (ii) a virtual dis- 1 2 placements {dxi} are (Goldstein Eq.2.20, Greenwood placement is an arbitrary, instantaneous, infinites- Eq.1.29 and Pars14 Eq.1.7.1), imal change of position of the system compatible with the conditions of constraint7; (iii) virtual dis- 3N placements are, by definition, arbitrary displace- ajidxi + ajtdt =0 , j =1, 2,...,m. (4) ments of the components of the system, satisfy- Xi=1 ing the constraint5; (iv) virtual displacement does not violate the constraints10; (v) we define a vir- Whereas, the virtual displacements {δx } satisfy i tual displacement as one which does not violate the (Goldstein1 Eq.2.21, Greenwood2 Eq.1.30; Pars14 kinematic relations11; (vi) the virtual displacements Eq.1.7.2), obey the constraint on the motion4. These state- 3N ments imply that the virtual displacements satisfy ajiδxi =0 , j =1, 2,...,m. (5) the constraint conditions, i.e., the constraint equa- Xi=1 tions. However this is true only for holonomic, scle- renomous constraints. We shall show that for non- Thus, there appear in the literature certain equations, holonomic constraints, or rheonomous constraints, namely Eq.(3) and Eq.(5), which are always satisfied by e.g., a pendulum with moving support, this defini- the not so precisely defined virtual displacements. It may tion violates the zero virtual work principle. be noted that these equations are connected to the con- straints but are not simply the infinitesimal forms of the 2. It is also stated that (i) virtual displacements do constraint equations, i.e., Eqs.(2) and (4). This fact is not necessarily conform to the constraints2; (ii) the well documented in the literature1,2,14. However, the na- virtual displacements δq have nothing to do with ture of the difference between these sets of equations, actual motion. They are introduced, so to speak, as i.e, Eqs.(3) and (5) on one hand and Eqs.(2) and (4) on test quantities, whose function it is to make the sys- the other, and their underlying connection are not ex- tem reveal something about its internal connections plained in most discussions. One may consider Eq.(3) and about the forces acting on it7; (iii) the word or Eq.(5), as independent defining equation for virtual “virtual” is used to signify that the displacements displacement. But it remains unclear as to how, the vir- are arbitrary, in the sense that they need not corre- 5 tual displacements {δxi} defined by two different sets of spond to any actual motion executed by the system ; equations for the holonomic and the non-holonomic cases, (iv) it is not necessary that it (virtual displacement) viz., Eq.(3) and Eq.(5), correspond to the same concept represents any actual motion of the system9; (v) of virtual displacement. it is not intended to say that such a displacement We try to give a physical connection between the defi- (virtual) occurs during the motion of the particle nitions of allowed and virtual displacements for any given considered, or even that it could occur3; (vi) virtual set of constraints. The proposed definition of virtual dis- displacement is any arbitrary infinitesimal displace- placement (Sec.IIA) as difference of two unequal allowed ment not necessarily along the constrained path6. 3

From the above we understand that the virtual dis- At this stage a student gets further puzzled. Should he placements do not necessarily satisfy the constraint take the forces of constraint as supplied, and the principle equations, and they need not be the ones actually of zero virtual work as a definition of virtual displacement realized. We shall see that these statements are ? In that case the principle reduces merely to a defini- consistent with physical situations, but they cannot tion of a new concept, namely virtual displacement. Or serve as a satisfactory definition of virtual displace- should the virtual displacement be defined from the con- ment. Statements like: “not necessarily conform to straint conditions independently ? The principle of zero the constraints” or “not necessarily along the con- virtual work may then be used to obtain the forces of strained path” only tell us what virtual displace- constraint. These forces of constraint ensure that the ment is not, they do not tell us what it really is. constraint condition is maintained throughout the mo- Reader should note that there is a conflict between tion. Hence it is natural to expect that they should be the statements quoted under items 1 and 2. connected to and perhaps derivable from the constraint Thus it is not clear from the above, whether the conditions. virtual displacements satisfy the constraints, i.e., the constraint equations, or they do not. II. VIRTUAL DISPLACEMENT AND FORCES 3. It is also stated that (i)virtual displacement is to OF CONSTRAINT be distinguished from an actual displacement of the system occurring in a time interval dt1; (ii) it is A. Constraints and Virtual displacement an arbitrary, instantaneous, change of position of the system7; (iii) virtual displacement δr takes place Let us consider a system of constraints that are ex- without any passage of time10. (iv) virtual displace- pressible as equations involving positions and time. They ment has no connection with the time - in contrast represent some geometric restrictions (holonomic) either to a displacement which occurs during actual mo- independent of time (sclerenomous) or explicitly depen- tion, and which represents a portion of the actual dent on it (rheonomous). Hence for a system of N par- path3; (v) one of the requirements on acceptable vir- ticles moving in three dimensions, a system of (s) holo- tual displacement is that the time is held fixed4. We nomic, rheonomous constraints are represented by func- 10 even notice equation like : “δxi = dxi for dt = 0” . tions of {rk} and (t), The above statements are puzzling to a student. If position is a of time, a change fi(r1, r2,..., rN ,t)=0, i =1, 2,...,s. (6) in position during zero time has to be zero. In other words, this definition implies that the vir- The system may also be subjected to non-holonomic con- tual displacement cannot possibly be an infinitesi- straints which are represented by equations connecting mal (or differential) of any continuous function of velocities {vk}, positions {rk} and time (t). time. In words of Arthur Haas: since its (virtual N displacement) components are thus not functions of A · v + A =0 , i =1, 2,...,m, (7) the time, we are not able to regard them as differ- ik k it Xk=1 entials, as we do for the components of the element 3 of the actual path . It will be shown later (Sec.II), where {Aik} and {Ait} are functions of positions that virtual displacement can be looked upon as a {r1, r2,..., rN } and time (t). The equations for non- differential. It is indeed a differential change in po- holonomic constraints impose restrictions on possible or sition or an infinitesimal displacement, consistent allowed velocity vectors {v1, v2,..., vN }, for given posi- v with virtual velocity k(t), taken over a time inter- tions {r1, r2,..., rN } and time (t). The holonomic con- val dt (see Eq.(13)). straints given by Eq.(6), are equivalent to the following e equations imposing further restriction on the possible or 4. Virtual displacement is variously described as: ar- allowed velocities. bitrary, virtual, and imaginary1,5,6,7,9. These ad- jectives make the definition more mysterious to a N ∂fi ∂fi student. · vk + =0, i =1, 2,...,s. (8) ∂rk  ∂t Xk=1 Together with the above ambiguities, students are of- ten unsure whether it is sufficient to discuss virtual dis- For a system of N particles under (s) holonomic placement as an abstract concept or it is important to and (m) non-holonomic constraints, a set of vectors have a quantitative definition. Some students appreciate {v1, v2,..., vN } satisfying Eq.(7) and Eq.(8) are called that the virtual displacement as a vector should not be allowed velocities. It is worth noting at this stage that ambiguous. The principle of zero virtual work is required there are many, in fact infinitely many, allowed velocities, to derive Lagrange’s equations. For a particle under con- since we have imposed only (s+m) number of scalar con- straint this means that the virtual displacement is always straints, Eq.(7) and Eq.(8), on (3N) scalar components orthogonal to the of constraint. of the allowed velocity vectors. 4

At any given instant of time, the difference of any two such non-identical allowed sets of velocities, inde- pendently satisfying the constraint conditions, are called virtual velocities.

′ vk(t)= vk(t) − vk(t) k =1, 2,...,N. Ane infinitesimal displacement over time (t,t+dt), due to allowed velocities, will be called the allowed infinitesimal displacement or simply allowed displacement. drk = vk(t)dt, k =1, 2,...,N. (9)

Allowed displacements {drk} together with differential of time (dt) satisfy the infinitesimal form of the constraint equations. From Eq.(8) and Eq.(7), we obtain,

N ∂fi ∂fi · drk + dt =0, i =1, 2,...,s, (10) ∂rk  ∂t Xk=1 N FIG. 1: Virtual displacement defined as difference of allowed A · dr + A dt =0 , i =1, 2,...,m. (11) ik k it displacements Xk=1 As there are many independent sets of allowed veloci- ties, we have many allowed sets of infinitesimal displace- The absence of the (∂fi/∂t) and Ait in the above equa- ments. We propose to define virtual displacement as the tions, Eq.(14) and Eq.(15), gives the precise meaning to difference between any two such (unequal) allowed dis- the statement: “virtual displacements are the allowed dis- placements taken over the same time interval (t,t + dt), placements in the case of frozen constraints”. The con- straints are frozen in time in the sense that we make the (∂fi/∂t) and Ait terms zero, though the ∂fi/∂rk ′ A r r r δrk = drk − drk, k =1, 2,...,N. (12) and ik terms still involve both position { 1, 2,..., N }, and time (t). In the case of stationary constraints, i.e., Thus virtual displacements are infinitesimal displace- fi(r1,..., rN ) = 0, and k Aik(r1,..., rN ) · vk = 0, the ments over time interval dt due to virtual velocity vk(t), virtual displacements areP identical with allowed displace- ments as (∂fi/∂t) and Ait are identically zero. e ′ δrk = vk(t)dt = (vk(t) − vk(t))dt, k =1, 2,...,N. (13) B. Existence of forces of constraints This definitione is motivated by the possibility of (i) iden- ideal constraints tifying a special class of ‘ ’ (Sec.IIC), and In the case of an unconstrained system of N particles the principle of zero virtual work (ii) verifying ‘ ’ in com- described by position vectors {r } and velocity vectors mon physical examples (Sec.III). It may be noted that, k {vk}, the motion is governed by Newton’s Law, by this definition, virtual displacements {δrk} are not in- stantaneous changes in position in zero time. They are mkak = Fk(rl, vl,t), k,l =1, 2,...,N (16) rather smooth, differentiable objects. th The virtual displacements thus defined, satisfy the ho- where mk is the of the k particle, ak is its accel- mogeneous part of the constraint equations, i.e., Eq.(10) eration and Fk is the total external force acting on it. and Eq.(11) with ∂fi/∂t = 0 and Ait = 0. Hence, However, for a constrained system, the equations of con- straint, namely Eq.(6) and Eq.(7), impose the following N ∂fi restrictions on the allowed , · δrk =0, i =1, 2,...,s, (14) ∂rk Xk=1 N N ∂fi d ∂fi d ∂fi N · ak + vk + =0, ∂rk dt ∂rk  dt  ∂t  Xk=1 Xk=1 Aik · δrk =0, i =1, 2,...,m. (15) Xk=1 i =1, 2,...,s (17) N d d The logical connection between the equations of con- Aik · ak + Aik · vk + Ait =0, dt dt straint, equations for allowed displacements and equa- Xk=1 tions for virtual displacements are presented in FIG.1. i =1, 2,...,m. (18) 5

Given {rk}, {vk} one is no longer free to choose all the force is zero, on allowed as well as virtual displacement. accelerations {ak} independently. Therefore in general the accelerations {ak} allowed by Eq.(17), Eq.(18) are N N incompatible with Newton’s Law, i.e., Eq.(16). Rk · drk =0, Rk · δrk =0. This implies that during the motion the constraint con- Xk=1 kX=1 dition cannot be maintained by the external forces alone. Physically some additional forces, e.g., normal reaction When the constraint surface is in motion, the allowed from the surface of constraint, tension in the pendulum velocities, and hence the allowed displacements are no string, come into play to ensure that the constraints are longer tangent to the surface (see Sec.III). The virtual satisfied throughout the motion. Hence one is compelled displacement however remains tangent to the constraint to introduce forces of constraints {Rk} and modify the surface. As the surface is frictionless, it is natural to as, assume that the force of constraint is still normal to the instantaneous position of the surface. Hence the work mkak = Fk + Rk, k =1, 2,...,N. (19) done by normal reaction on virtual displacement is zero. However the work done by constraint force on allowed displacements is no longer zero.

N N Rk · drk 6=0, Rk · δrk =0. (20) Xk=1 kX=1

In a number of physically interesting simple problems, such as, motion of a pendulum with fixed or moving sup- port, motion of a particle along a stationary and moving slope, we observe that the above interesting relation be- tween the force of constraint and virtual displacement holds (see Sec.III). As the 3N scalar components of the virtual displacements {δrk} are connected by (s + m) equations, Eq.(14) and Eq.(15), only n = (3N −s−m) of FIG. 2: Existence of constraint forces these scalar components are independent. If the (s + m) dependent quantities are expressed in terms of remaining FIG.2 presents the connection between the equations (3N − s − m) independent objects we get, of constraints and forces of constraints. Now the problem is to determine the motion of N par- n ticles, namely their positions {rk(t)}, velocities {vk(t)} Rj · δxj =0. (21) and the forces of constraints {Rk}, for a given set of Xj=1 e external forces {Fk}, constraint equations, Eq.(6) and e Eq.(7), and initial conditions {rk(0), vk(0)}. It is im- where {xj } are the independent components of {rk}. portant that the initial conditions are also compatible {Rj} are the coefficients of {δxj }, and are composed of with the constraints. There are a total of (6N) scalar e different {Rk}. Since the above components of virtual unknowns, namely the components of rk(t) and Rk, con- e displacements {δxj } are independent,e one can equate nected by (3N) scalar equations of motion, Eq.(19), and each of their coefficients to zero (R = 0). This brings in (s + m) equations of constraints, Eq.(6) and Eq.(7). For j exactly (3N − s −em) new scalar conditions or equations (6N > 3N + s + m) we have an under-determined sys- that are needed to make the systeme solvable (see FIG.3). tem. Hence to solve this problem we need (3N − s − m) additional scalar relations. Thus we have found a special class of constraints, which is observed in nature (Sec.III) and which gives us a solvable mechanical system. We call this special class of constraints, where the forces of constraint do zero work C. Solvability and ideal constraints on virtual displacement, i.e., k Rk · δrk = 0, the ideal constraint. P In simple problems with stationary constraints, e.g., Our interpretation of the principle of zero virtual work, motion of a particle on a smooth stationary surface, we as a definition of an ideal class of constraints, agrees with observe that the allowed displacements are tangential to Sommerfeld. In his exact words, “a general postulate of the surface. The virtual displacement being a difference mechanics: in any mechanical systems the virtual work of two such allowed displacements, is also a vector tan- of the reactions equals zero. Far be it from us to want to gential to it. For a frictionless surface, the force of con- give a general proof of this postulate, rather we regard it straint, the so called ‘normal reaction’, is perpendicular practically as definition of a mechanical system”7. to the surface. Hence the work done by the constraint (Boldface is added by the authors). 6

FIG. 4: Allowed and virtual displacements of a pendulum with stationary support

Although the virtual displacement is not uniquely speci- FIG. 3: Solvability under ideal constraints fied by the constraint, it is restricted to be in a plane per- pendicular to the instantaneous line of suspension. Hence III. EXAMPLES OF VIRTUAL it is not ‘completely arbitrary’. DISPLACEMENTS The ideal string of the pendulum provides a tension (T) along its length, but no shear. The work done by A. Simple Pendulum with stationary support this tension on both allowed and virtual displacements is zero,

The motion of a pendulum is confined to a plane and T · dr =0, T · δr =0. its bob moves keeping a fixed distance from the point of suspension (see FIG.4). The equation of constraint therefore is, B. Simple Pendulum with moving support . 2 2 2 f(x,y,t) = x + y − r0 =0, Let us first consider the case when the support is mov- where r0 is the length of the pendulum. Whence ing vertically with a velocity u. The motion of the pen- dulum is still confined to a plane. The bob moves keep- ∂f ∂f ∂f =2x, =2y, =0. ing a fixed distance from the moving point of suspension ∂x ∂y ∂t (FIG.5). The equation of constraint is,

. 2 2 2 The constraint equation for allowed velocities, Eq.(8), f(x,y,t) = x + (y − ut) − r0 =0, becomes, where u is the velocity of the point of suspension along x · vx + y · vy =0. a vertical direction. Whence Hence the allowed velocity (vx, vy) is orthogonal to the instantaneous position (x,y) of the bob relative to sta- ∂f ∂f ∂f tionary support. The same may also be verified taking a =2x, = 2(y − ut), = −2u(y − ut). ∂x ∂y ∂t plane polar coordinate. The allowed velocities and the allowed displacements Hence the constraint equation gives, are perpendicular to the line of suspension. The virtual velocities and the virtual displacements, being the differ- x · vx + (y − ut) · vy − u(y − ut)=0, ence of two unequal allowed velocities and displacements respectively, are also perpendicular to the line of suspen- or, sion. x · vx + (y − ut) · (vy − u)=0. dr = v(t)dt, dr′ = v′(t)dt, The allowed velocities (vx, vy) and the allowed displace- δr = (v(t) − v′(t))dt. ments, are not orthogonal to the instantaneous position 7

C. Motion along a stationary inclined plane

Let us consider a particle sliding along a stationary inclined plane as shown in FIG.6. The constraint here is more conveniently expressed in polar coordinates. The constraint equation is, . f(r, θ) = θ − θ0 =0

where θ0 is the angle of the slope. Hence the constraint equation for allowed velocities, Eq.(8), gives, ∂f ∂f ∂f ∂f ∂f · v + = vr + vθ +  ∂r  ∂t  ∂r   ∂θ  ∂t = 0 · r˙ +1 · (rθ˙)+0=0 Hence θ˙ = 0, implying that the allowed velocities are along the constant θ plane. Allowed velocity, allowed and virtual displacements are, v =r ˙ r, dr =r ˙ r dt, v′ =r ˙′ r, dr′ =r ˙′ r dt, b b δr = (v − v′)dt = (r ˙ − r˙′) r dt. FIG. 5: Allowed and virtual displacements of a pendulum b b with moving support where r is a unit vector alongb the slope. b of the bob relative to the instantaneous point of suspen- sion (x, y − ut). It is easy to verify from the above equa- tion that the allowed velocity (vx, vy) is equal to the sum of a velocity vector (vx, vy − u) perpendicular to the position of the bob relative to the point of suspen- sion (x,y − ut), and the velocity of the support (0, u). If we denote v(t) = (vx, vy), v⊥(t) = (vx, vy − u) and u = (0,u), then, v(t)= v⊥(t)+ u. The allowed displacements are vectors collinear to al- lowed velocities. A virtual displacement being the differ- ence of two allowed displacements, is a vector collinear to FIG. 6: Allowed and virtual displacements of a particle sliding the difference of allowed velocities. Hence it is orthogonal along a stationary slope to the instantaneous line of suspension. As the inclined slope is frictionless, the constraint force dr = v(t)dt = v⊥(t)dt + udt, N is normal to the surface. The work done by this force ′ ′ ′ dr = v (t)dt = v⊥(t)dt + udt, on allowed as well as virtual displacement is zero. r v v′ v v′ δ = ( (t) − (t))dt = ( ⊥(t) − ⊥(t))dt. N · dr =0, N · δr =0 Hence none of these allowed or virtual vectors are ‘arbi- trary’. At any given instant, an ideal string provides a ten- D. Motion along a moving inclined plane sion along its length, with no shear. Hence the constraint force, namely tension T, does zero work on virtual dis- For an inclined plane moving along the horizontal side placement. (FIG.7), the constraint is given by,

T · dr = T · u dt 6=0, T · δr = T · v⊥ dt =0 (x − ut) − cot(θ0) = 0, y For the support moving in a horizontal or any arbitrary . direction, one can show that the allowed displacement f(x, y) = (x − ut) − cot(θ0)y = 0. is not normal to the instantaneous line of suspension. Whence the constraint for allowed velocities Eq.(8) be- But the virtual displacement, as defined in this article, comes, always remains perpendicular to the instantaneous line of support. (x ˙ − u) − cot(θ0)y ˙ =0. 8

equation. However for a constrained system {δrk} are de- pendent through the constraint equations, Eq.(14) and Eq.(15), for holonomic and non-holonomic systems re- spectively.

N ∂fi δrk =0, i =1, 2,...,s (14) ∂rk kX=1 N Ajk · δrk =0, j =1, 2,...,m (15) kX=1 We multiply Eq.(14) successively by (s) scalar multi- FIG. 7: Allowed and virtual displacements of a particle sliding pliers {λ1, λ2,...λs}, Eq.(15) successively by (m) scalar along a moving slope multipliers {µ1, µ2,...µm} and then subtract them from the zero virtual work equation, namely Eq.(22).

Hence the allowed velocity (x, ˙ y˙) is the sum of two N s m ∂fi vectors, one along the plane (x ˙ − u, y˙), and the other Rk − λi − µj Ajk δrk =0. (24) ∂rk equal to the velocity of the plane itself (u, 0). Xk=1 Xi=1 Xj=1   v v u (t)= k(t)+ . These multipliers {λi} and {µj} are called the Lagrange’s multipliers. Explicitly in terms of components, Allowed displacements are vectors along the allowed ve- locities, however the virtual displacement is still a vector N s m along the instantaneous position of the plane. ∂fi Rk,x − λi − µj (Ajk)x  δxk ∂xk r v u r′ v′ u Xk=1 Xi=1 Xj=1 d = ( k(t)+ )dt, d = ( k(t)+ )dt,   ′ ′ N N δr = (v(t) − v (t))dt = (vk(t) − vk(t))dt. + [Y ]kδyk + [Z]kδzk =0, (25) For the moving frictionless slope, the constraint force pro- Xk=1 kX=1 vided by the surface is perpendicular to the plane. Hence the work done by the constraint force on virtual displace- where [Y ]k and [Z]k denote the coefficients of δyk and ment remains zero. δzk respectively. The constraint equations, Eq.(14) and Eq.(15), allow N · dr 6=0, N · δr =0. us to write the (s + m) dependent virtual displacements in terms of the remaining n = (3N − s − m) independent ones. We choose (s + m) multipliers {λ1, λ2,...,λs} and IV. LAGRANGE’S METHOD OF {µ1, µ2,...,µm}, such that the coefficients of (s + m) UNDETERMINED MULTIPLIERS dependent components of virtual displacement vanish. The remaining virtual displacements being independent, A constrained system of particles obey the equations their coefficients must vanish as well. Thus it is possible of motion given by, to choose {λ1, λ2,...,λs} and {µ1, µ2,...,µm} such that all coefficients {[X]k,[Y ]k,[Z]k} of virtual displacements a F R mk k = k + k, k =1, 2,...,N {δxk,δyk,δzk} in Eq.(25) vanish. Hence we can express the forces of constraint in terms of the Lagrange’s multi- where m is the mass of the kth particle, a is its accel- k k pliers. eration. Fk and Rk are the total external force and force th of constraint on the k particle. If the constraints are s m ∂f ideal, we can write, R = λ i + µ A , k =1, 2,...,N (26) k i ∂r j jk i=1 k j=1 N X X Rk · δrk =0, (22) Thus the problem reduces to finding a solution for the Xk=1 equations, whence we obtain, s m ∂fi N mkak = Fk + λi + µj Ajk, k =1, 2,...,N (27) ∂rk (mkak − Fk) · δrk =0. (23) Xi=1 Xj=1 Xk=1 together with the equations of constraint, If the components of {δrk} were independent, we could recover Newton’s Law for unconstrained system from this fi(r1, r2,..., rN ,t)=0, i =1, 2,...,s (6) 9

of position {r1, r2,..., rN } in terms of (3N − s − m) in- dependent parameters {q1, q2,...,qn} and time (t).

rk = rk(q1, q2,...,qn,t), k =1, 2,...,N (28)

The allowed and virtual displacements are given by, n ∂r ∂r dr = k δq + k dt, k =1, 2,...,N k ∂q j ∂t Xj=1 j n ∂r δr = k δq , k =1, 2,...,N. (29) k ∂q j Xj=1 j From Eq.(23) we obtain,

N n N n dr˙ k ∂rk ∂rk mk  δqj  − Fk  δqj  =0 dt ∂qj ∂qj Xk=1 Xj=1 Xk=1 Xj=1     (30)

Introduce the expression of kinetic ,

N 1 T = m r˙ 2 , 2 k k Xk=1 and that of the generalized force,

N FIG. 8: Lagrange’s method of undetermined multipliers: solv- ∂rk ability Qj = Fk , j =1, 2,...,n. (31) ∂qj Xk=1 and After some simple algebra one finds, n N d ∂T ∂T Aik · vk + Ait =0 , i =1, 2,...,m. (7) − − Qj δqj =0. (32)  dt ∂q˙j ∂qj  Xk=1 Xj=1

Here we have to solve (3N + s + m) scalar equations As {q1, q2,...,qn} are independent coordinates, coeffi- involving (3N +s+m) unknown scalar quantities, namely cient of each δqj must be zero separately. Hence (see {xk,yk,zk, λi, µj } (see FIG.8). After solving this system FIG.9), of equations for {xk,yk,zk, λi, µj }, one can obtain the forces of constraint {R } using Eq.(26). d ∂T ∂T k − = Qj , j =1, 2,...,n. (33) dt ∂q˙j ∂qj

V. LAGRANGE’S EQUATIONS IN These are called Lagrange’s equations in generalized co- ordinates. To proceed further one has to impose addi- tional conditions on the nature of forces {Fk} or {Qj}. For the sake of completeness we discuss very briefly Lagrange’s equations in generalized coordinates. A more complete discussion can be found in most 1,2,3,4,5,6,7,8,9,10,11,12,13,14 texts . Consider a system of N In problems where forces {Fk} are derivable from a particles under (s) holonomic and (m) non-holonomic scalar potential V (r1, r2,..., rN ), constraints. In certain suitable cases, one can express e (s + m) dependent coordinates in terms of the remaining Fk = −∇kV (r1, r2,..., rN ), k =1, 2,...,N. (34) (3N − s − m) independent ones. It may be noted that such a complete reduction is not possible for general cases One can obtaine the generalized force as, of non-holonomic and time dependent constraints7,14. If we restrict our discussion to cases where this reduction is ∂rk ∂V Qj = −∇kV · = − , j =1, 2,...,n. (35) possible, one may express all the 3N scalar components  ∂qj  ∂qj e 10

At this stage one introduces d’Alembert’s principle of zero virtual work. Bernoulli7 (1717) and d’Alembert7,15 (1743) originally proposed this principle for a system in static equilibrium. The principle states that the forces of constraint do zero work on virtual displacement. For sys- tems in static equilibrium, virtual displacement meant an imaginary displacement of the system that keeps its stat- ical equilibrium unchanged. Lagrange generalized this principle to a constrained system of particles in motion. This principle is crucial in arriving at Lagrange’s equa- tion. However, most texts do not clearly address the questions, (i) why one needs to extend d’Alembert’s prin- ciple to particles in motion, and (ii) why the work done by constraint forces on virtual displacements, and not on allowed displacements, is zero ? In the present article the allowed infinitesimal displace- ments are defined as ones that satisfy the infinitesimal FIG. 9: Lagrange’s equations in generalized coordinates form of the constraint equations. They are the displace- ments that could have been possible if only the con- straints were present. Actual dynamics, under the given Where V is the potential V expressed as a function of external forces, would choose one of these various sets of {q1, q2,...,qn}. In addition as the potential V is in- displacements as actual displacement of the system. The dependent of the generalizede velocities, we obtain from definition of virtual displacement as difference of two un- Eq.(33), equal allowed displacements over the same infinitesimal time interval (t,t+dt), gives a unified definition of virtual d ∂(T − V ) ∂(T − V ) − =0, j =1, 2,...,n. (36) displacement. This definition of virtual displacement sat- dt ∂q˙j ∂qj isfies the appropriate equations found in the literature, for both holonomic and non-holonomic systems. At this stage one introduces the Lagrangian function, L = T − V . In terms of the Lagrangian, the equations of It is shown that Newton’s equation of motion with ex- motion take up the form, ternal forces alone, is inconsistent with equations of con- straint. Hence the forces of constraint are introduced. d ∂L ∂L Now there are 3N equations of motion and (s + m) − =0, j =1, 2,...,n. (37) dt ∂q˙j ∂qj equations of constraint involving 6N unknown scalars {rk(t), Rk}. Without additional condition (d’Alembert principle), the problem is underspecified and unsolvable. VI. CONCLUSION It is verified that for simple physical systems, the virtual displacements, as defined in this article, sat- In this article we make an attempt to present a quan- isfy d’Alembert principle for particles in motion. The titative definition of the virtual displacement. We show rheonomous examples discussed in Sec.III show why the that for certain simple cases the work done by the forces forces of constraint do zero work on virtual displace- of constraint on virtual displacement is zero. We also ments and not on allowed displacements. The additional demonstrate that this zero virtual work principle gives equations introduced by d’Alembert principle make the us a solvable class of problems. Hence we define this problem solvable. These justify (i) the peculiar defini- special class of constraint as the ideal constraint. We tion of virtual displacements and the equations they sat- demonstrate in brief how one can solve a general mechan- isfy, (ii) Lagrange’s extension of d’Alembert’s principle ical problem by: i) Lagrange’s method of undetermined to particles in motion, (iii) why the zero work princi- multipliers and ii) Lagrange’s equations in generalized ple is related to virtual displacement and not to allowed coordinates. displacement. Once the system is solvable, two meth- In the usual presentations of Lagrange’s equation ods, originally proposed by Lagrange can be used, and based on virtual displacement and d’Alembert’s princi- are demonstrated. For Lagrange’s method of undeter- ple, Eq.(3) and Eq.(5) are satisfied by the virtual dis- mined multipliers, one solves (3N + s) equations to ob- placements. One may consider these equations as the tain the motion of the system {rk(t),..., rN (t)} and La- definition of virtual displacements. However the situ- grange’s multipliers {λi,...,λs}. The forces of constraint ation is far from satisfactory, as separate defining equa- {Rk,..., RN } are expressed in terms of these multipli- tions are required for different classes of constraints. This ers. For Lagrange’s equations in generalized coordinates adhoc definition also fails to clarify the actual connection one solves (3N − s) equations to obtain the time evo- between the virtual displacements and the equations of lution of the generalized coordinates {qj = qj (t), j = constraints. 1,..., 3N − s}. This gives the complete description of 11 the motion {rk = rk(q1, q2,...,qn,t), k =1,...,N}, ig- Acknowledgment noring the calculation of the constraint forces. It may be 7,14,20 noted that about a century later Appell’s equations Authors wish to thank Professor John D. Jackson and were introduced for efficiently solving non-holonomic sys- Professor Leon A. Takhtajan for their remarks and sug- tems. gestions. They also thank Professor Sidney Drell, Profes- It is interesting to note that both the above mentioned sor Donald T. Greenwood and Professor Alfred S. Gold- methods require the principle of zero virtual work by con- haber for their valuable communications. straint forces as a crucial starting point. In the case of The authors gratefully acknowledge the encourage- Lagrange’s method of undetermined multipliers we start ment received from Professor Max Dresden at Stony with the ideal constraint condition Eq.(22). From there Brook. Authors have greatly benefited from the books we obtain Eq.(23)-Eq.(27). Eq.(26) expresses the con- mentioned in this article, particularly those of Arnold13, straint forces in terms of Lagrange’s multipliers. For La- Goldstein1, Greenwood2, Pars14 and Sommerfeld7. grange’s equations in generalized coordinates, we start The material presented in this article was used as part with the ideal constraint, Eq.(22). We work our way of courses at Jadavpur University through Eq.(23), Eq.(30), Eq.(32) and finally obtain La- during 2000-2003. SR would like to thank his students grange’s equations in generalized coordinates, Eq.(33) A. Chakraborty and B. Mal for meaningful discussions. and Eq.(37). The last figure, FIG.(10), gives the com- plete logical flow of this article.

∗ Electronic address: sray˙ju@rediffmail.com, [email protected] D. (S. Ter Ray) Haar, Elements of , North † Electronic address: [email protected] (J. Shamanna) Holland Publishing Co., Amsterdam, 1961. 1 H. Goldstein, Classical Mechanics, Addison-Wesley Pub- 12 L. D. Landau, E. M. Lifshitz, Mechanics, Pergamon Press, lishing Co., Reading, Massachusetts, 1980. Oxford, 1976. 2 D. T. Greenwood, Classical Dynamics, Prentice Hall, New 13 V. I. Arnold, Mathematical Methods of Classical Mechan- York, 1977. ics, Springer Verlag, New York, 1989. 3 A. Haas, Introduction to Theoretical Physics, vol I, Con- 14 L. A. Pars, A Treatise on , Heine- stable and Company Ltd, London, 1924. mann, London, 1964. 4 L. N. Hand, J. D. Finch, , Cambridge 15 J. le R. d’Alembert, Trait´ede dynamique, David ’Ame, University Press, Cambridge, 1998. Paris, 1743 (reprinted Gauthier-Villars, Paris, 1921). 5 E. A. Hylleraas, Mathematical and Theoretical Physics, 16 J. L. Lagrange, Miscell. Taurin., II, 1760. vol. I, Wiley Interscience, New York, 1970. 17 J. L. Lagrange, Mecanique Analytique, Imprimeur- 6 D. A. Wells, Schaum Outline of Theory and Problems of Libraire pour les Mathematiques, Paris, 1788 (reprinted Lagrangian Dynamics, McGraw-Hill Inc., New York, 1967. Gauthier-Villars, Paris, 1888). 7 A. Sommerfeld, Mechanics, Lectures on Theoretical Physi- 18 W. R. Hamilton, On a general method in Dynamics, 1834; cs, vol. I, Academic Press, New York, 1952. Collected papers, II, Cambridge University Press, Cam- 8 J. L. Synge, and B. A. Griffith, Principles of Mechanics, bridge, 1940, p 103-211. McGraw-Hill Inc., New York, 1970. 19 S. Drell, private communication, 2004. 9 K. R. Symon, Mechanics, Addison-Wesley Publishing Co., 20 P. Appell, Trait´ede M´ecanique Rationelle, Vol I & II, Reading, Massachusetts, 1971. Gauthier-Villars, Paris, 1896. 10 T. T. Taylor, Mechanics: Classical and Quantum, Perga- mon Press, Oxford, 1976. 12

FIG. 10: Logical connection between constraint equations, virtual displacements, forces of constraints, d’ Alembert’s principle and Lagrange’s equations.