archiv ray shamanna 2005

arXiv:physics/0510204v2 [physics.ed-ph] 2 Jun 2006On Virtual Displacement and Virtual Work in Lagrangian Dyna mics Subhankar Ray∗ Department of Physics, Jadavpur University, Calcutta 700 0 32, India and C. N. Yang Institute for Theoretical Physics, Stony Brook, N Y 11794 J. Shamanna† Physics Department, Visva Bharati University, Santiniket an 731235, India (Dated: August 1, 2005) The confusion and ambiguity encountered by students, in und erstanding virtual displacement and virtual work, is discussed in this article. A definition o f virtual displacement is presented that allows one to express them explicitly for holonomic (veloci ty independent), non-holonomic (veloc- ity dependent), scleronomous (time independent) and rheon omous (time dependent) constraints. It is observed that for holonomic, scleronomous constraint s, the virtual displacements are the dis- placements allowed by the constraints. However, this is not so for a general class of constraints. For simple physical systems, it is shown that, the work done b y the constraint forces on virtual displacements is zero. This motivates Lagrange’s extensio n of d’Alembert’s principle to system of particles in constrained motion. However a similar zero wor k principle does not hold for the allowed displacements. It is also demonstrated that d’Alembert’s p rinciple of zero virtual work is necessary for the solvability of a constrained mechanical problem. We identify this special class of constraints, physically realized and solvable, as the ideal constraints . The concept of virtual displacement and the principle of zero virtual work by constraint forces are c entral to both Lagrange’s method of undetermined multipliers, and Lagrange’s equations in gen eralized coordinates. PACS numbers: 45,45.20.Jj,01.40.Fk Keywords: d’Alembert’s principle, Lagrangian mechanics, Lagrange’s equations, Virtual work, Holonomic, Non-holonomic, Scleronomous, Rheonomous constraints I. INTRODUCTION Almost all graduate level courses in classi- cal mechanics include a discussion of virtual displacement1,2,3,4,5,6,7,8,9,10,11and Lagrangian dynamics1,2,3,4,5,6,7,8,9,10,11,12. From the concept of zero work by constraint forces on virtual displacement, the Lagrange’s equations of motion are derived. However, the definition presented in most accessible texts often seem vague and ambiguous to students. Even after studying the so called definition, it is rather com- monplace that a student fails to identify, whether a sup- plied vector is suitable as a virtual displacement, for a given constrained system. Though some of the more ad- vanced and rigorous treatise13,14present a more precise and satisfactory treatment, they are often not easily com- prehensible to most students. In this article we attempt a simple, systematic and precise definition of virtual dis- placement, which clearly shows the connection between the constraints and the corresponding allowed and virtual displacements. This definition allows one to understand how far the virtual displacement is ‘arbitrary’ and how far it is ‘restricted’ by the constraint condition. There are two common logical pathways of arriving at Lagrange’s equation. 1. Bernoulli’s principle of virtual velocity7(1717), d’Alembert’s principle of zero virtual work7,15 (1743), Lagrange’s generalization of d’Alembert’s principle to constrained system of moving particles, and Lagrange’s equations of motion (1788)7,16,17.2. Hamilton’s principle of least action7,18(1834), and variational approach to Lagrange’s equation. The two methods are logically and mathematically in- dependent and individually self contained. The first method was historically proposed half a century earlier, and it presents the motivation of introducing the La- grangian as a new physical quantity . The second method starts with the Lagrangian and the related action as quantities axiomatically describing the dynamics of the system. This method is applied without ambiguity in some texts12,13and courses19. However one also finds intermixing of the two approaches in the literature and popular texts, often leading to circular definition and re- lated confusion. A rational treatment demands an in- dependent presentation of the two methods, and then a demonstration of their interconnection. In the present article we confine ourselves to the first method. In this approach, due to Bernoulli, d’Alembert and La- grange, one begins with a constrained system, defined by equations of constraints connecting positions, time and often velocities of the particles under consideration. The concept of virtual displacement is introduced in terms of the constraint equations. The external forces alone cannot maintain the constrained motion. This requires the introduction of forces of constraints. The imposition of the principle of zero virtual work by constraint forces gives us a ‘special class of systems’, that are solvable. A proper definition of virtual displacement is necessary to make the said approach logically satisfactory. However the various definitions found in popular texts are often incomplete and contradictory with one another. These 2 ambiguities will be discussed in detail in the next section. In the literature, e.g., Greenwood2Eq.1.26 and Pars14 Eq.1.6.1, one encounters holonomic constraints of the form: φj(x1, x2, . . . , x 3N, t) = 0, j = 1,2, . . ., k. (1) The differential form of the above equations are sat- isfied by allowed infinitesimal displacements {dxi}, (Greenwood2Eq.1.27; Pars14Eq.1.6.3). 3N/summationdisplay i=1∂φj ∂xidxi+∂φj ∂tdt= 0, j = 1,2, . . ., k. (2) For a system under above constraints the virtual displace- ments {δxi}, satisfy the following equations (Greenwood2 Eq.1.28; Pars14Eq.1.6.5), 3N/summationdisplay i=1∂φj ∂xiδxi= 0, j = 1,2, . . ., k. (3) The differential equations satisfied by allowed and virtual displacements are different even for the non-holonomic case. Here, the equations satisfied by the allowed dis- placements {dxi}are (Goldstein1Eq.2.20, Greenwood2 Eq.1.29 and Pars14Eq.1.7.1), 3N/summationdisplay i=1ajidxi+ajtdt= 0, j = 1,2, . . ., m. (4) Whereas, the virtual displacements {δxi}satisfy (Goldstein1Eq.2.21, Greenwood2Eq.1.30; Pars14 Eq.1.7.2), 3N/summationdisplay i=1ajiδxi= 0, j = 1,2, . . ., m. (5) Thus, there appear in the literature certain equations, namely Eq.(3) and Eq.(5), which are always satisfied by thenot so precisely defined virtual displacements. It may be noted that these equations are connected to the con- straints but are not simply the infinitesimal forms of the constraint equations, i.e., Eqs.(2) and (4). This fact is well documented in the literature1,2,14. However, the na- ture of the difference between these sets of equations, i.e, Eqs.(3) and (5) on one hand and Eqs.(2) and (4) on the other, and their underlying connection are not ex- plained in most discussions. One may consider Eq.(3) or Eq.(5), as independent defining equation for virtual displacement. But it remains unclear as to how, the vir- tual displacements {δxi}defined by two different sets of equations for the holonomic and the non-holonomic cases, viz., Eq.(3) and Eq.(5), correspond to the same concept of virtual displacement. We try to give a physical connection between the defi- nitions of allowed and virtual displacements for any given set of constraints. The proposed definition of virtual dis- placement (Sec.IIA) as difference of two unequal alloweddisplacements (satisfying Eq.(2) or Eq.(4)) over the same time interval; automatically ensures that virtual displac e- ments satisfy Eq.(3) and Eq.(5) for holonomic and non- holonomic systems respectively. We show that in a num- ber of natural systems, e.g., pendulum with fixed or mov- ing support, particle sliding along stationary or moving frictionless inclined plane, the work done by the forces of constraint on virtual displacements is zero. We also demonstrate that this condition is necessary for the solv- ability of a constrained mechanical problem. Such sys- tems form an important class of natural systems. A. Ambiguity in virtual displacement In the literature certain statements appear in reference to virtual displacement, which seem confusing and mu- tually inconsistent, particularly to a student. In the fol- lowing we present few such statements found in common texts. 1. It is claimed that (i) a virtual displacement δris consistent with the forces and constraints imposed on the system at a given instant t1; (ii)a virtual dis- placement is an arbitrary, instantaneous, infinites- imal change of position of the system compatible with the conditions of constraint7; (iii)virtual dis- placements are, by definition, arbitrary displace- ments of the components of the system, satisfy- ing the constraint5; (iv) virtual displacement does not violate the constraints10; (v)we define a vir- tual displacement as one which does not violate the kinematic relations11; (vi)the virtual displacements obey the constraint on the motion4. These state- ments imply that the virtual displacements satisfy the constraint conditions, i.e., the constraint equa- tions. However this is true only for holonomic, scle- renomous constraints. We shall show that for non- holonomic constraints, or rheonomous constraints, e.g., a pendulum with moving support, this defini- tion violates the zero virtual work principle. 2. It is also stated that (i) virtual displacements do not necessarily conform to the constraints2; (ii)the virtual displacements δqhave nothing to do with actual motion. They are introduced, so to speak, as test quantities, whose function it is to make the sys- tem reveal something about its internal connections and about the forces acting on it7; (iii) the word “virtual” is used to signify that the displacements are arbitrary, in the sense that they need not corre- spond to any actual motion executed by the system5; (iv)it is not necessary that it (virtual displacement) represents any actual motion of the system9; (v) it is not intended to say that such a displacement (virtual) occurs during the motion of the particle considered, or even that it could occur3; (vi)virtual displacement is any arbitrary infinitesimal displace- ment not necessarily along the constrained path6. 3 From the above we understand that the virtual dis- placements do not necessarily satisfy the constraint equations, and they need not be the ones actually realized. We shall see that these statements are consistent with physical situations, but they cannot serve as a satisfactory definition of virtual displace- ment. Statements like: “not necessarily conform to the constraints” or “not necessarily along the con- strained path” only tell us what virtual displace- ment is not, they do not tell us what it really is. Reader should note that there is a conflict between the statements quoted under items 1 and 2. Thus it is not clear from the above, whether the virtual displacements satisfy the constraints, i.e., the constraint equations, or they do not. 3. It is also stated that (i) virtual displacement is to be distinguished from an actual displacement of the system occurring in a time interval dt1; (ii)it is an arbitrary, instantaneous, change of position of the system7; (iii)virtual displacement δrtakes place without any passage of time10. (iv)virtual displace- ment has no connection with the time - in contrast to a displacement which occurs during actual mo- tion, and which represents a portion of the actual path3; (v)one of the requirements on acceptable vir- tual displacement is that the time is held fixed4. We even notice equation like : “ δxi=dxifordt= 0”10. The above statements are puzzling to a student. If position is a continuous function of time, a change in position during zero time has to be zero. In other words, this definition implies that the vir- tual displacement cannot possibly be an infinitesi- mal (or differential) of any continuous function of time. In words of Arthur Haas: since its (virtual displacement) components are thus not functions of the time, we are not able to regard them as differ- entials, as we do for the components of the element of the actual path3. It will be shown later (Sec.II), that virtual displacement can be looked upon as a differential. It is indeed a differential change in po- sition or an infinitesimal displacement, consistent with virtual velocity /tildewidevk(t), taken over a time inter- valdt(see Eq.(13)). 4. Virtual displacement is variously described as: ar- bitrary ,virtual , and imaginary1,5,6,7,9. These ad- jectives make the definition more mysterious to a student. Together with the above ambiguities, students are of- ten unsure whether it is sufficient to discuss virtual dis- placement as an abstract concept or it is important to have a quantitative definition. Some students appreciate that the virtual displacement as a vector should not be ambiguous. The principle of zero virtual work is required to derive Lagrange’s equations. For a particle under con- straint this means that the virtual displacement is always orthogonal to the force of constraint.At this stage a student gets further puzzled. Should he take the forces of constraint as supplied, and the principle of zero virtual work as a definition of virtual displacement ? In that case the principle reduces merely to a defini- tion of a new concept, namely virtual displacement. Or should the virtual displacement be defined from the con- straint conditions independently ? The principle of zero virtual work may then be used to obtain the forces of constraint. These forces of constraint ensure that the constraint condition is maintained throughout the mo- tion. Hence it is natural to expect that they should be connected to and perhaps derivable from the constraint conditions. II. VIRTUAL DISPLACEMENT AND FORCES OF CONSTRAINT A. Constraints and Virtual displacement Let us consider a system of constraints that are ex- pressible as equations involving positions and time. They represent some geometric restrictions (holonomic) either independent of time (sclerenomous) or explicitly depen- dent on it (rheonomous). Hence for a system of Npar- ticles moving in three dimensions, a system of ( s) holo- nomic, rheonomous constraints are represented by func- tions of {rk}and (t), fi(r1,r2, . . . ,rN, t) = 0, i = 1,2, . . ., s. (6) The system may also be subjected to non-holonomic con- straints which are represented by equations connecting velocities {vk}, positions {rk}and time ( t). N/summationdisplay k=1Aik·vk+Ait= 0, i = 1,2, . . ., m, (7) where {Aik}and {Ait}are functions of positions {r1,r2, . . . ,rN}and time ( t). The equations for non- holonomic constraints impose restrictions on possible or allowed velocity vectors {v1,v2, . . . ,vN}, for given posi- tions {r1,r2, . . .,rN}and time ( t). The holonomic con- straints given by Eq.(6), are equivalent to the following equations imposing further restriction on the possible or allowed velocities. N/summationdisplay k=1/parenleftbigg∂fi ∂rk/parenrightbigg ·vk+∂fi ∂t= 0, i = 1,2, . . ., s. (8) For a system of Nparticles under ( s) holonomic and ( m) non-holonomic constraints, a set of vectors {v1,v2, . . . ,vN}satisfying Eq.(7) and Eq.(8) are called allowed velocities. It is worth noting at this stage that there are many, in fact infinitely many, allowed velocities, since we have imposed only ( s+m) number of scalar con- straints, Eq.(7) and Eq.(8), on (3 N) scalar components 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 calle d virtual velocities. /tildewidevk(t) =vk(t)−v′ k(t) k= 1,2, . . ., N. An 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/summationdisplay k=1/parenleftbigg∂fi ∂rk/parenrightbigg ·drk+∂fi ∂tdt= 0, i = 1,2, . . ., s, (10) N/summationdisplay k=1Aik·drk+Aitdt= 0, i = 1,2, . . ., m. (11) As there are many independent sets of allowed veloci- ties, we have many allowed sets of infinitesimal displace- ments. We propose to define virtual displacement as the difference between any two such (unequal) allowed dis- placements taken over the same time interval ( t, t+dt), δrk=drk−dr′ k, k = 1,2, . . ., N. (12) Thus virtual displacements are infinitesimal displace- ments over time interval dtdue to virtual velocity /tildewidevk(t), δrk=/tildewidevk(t)dt= (vk(t)−v′ k(t))dt, k = 1,2, . . ., N. (13) This definition is motivated by the possibility of (i) iden- tifying a special class of ‘ ideal constraints ’ (Sec.IIC), and (ii) verifying ‘ the principle of zero virtual work ’ in com- mon physical examples (Sec.III). It may be noted that, by this definition, virtual displacements {δrk}are not in- stantaneous changes in position in zero time. They are rather smooth, differentiable objects. The virtual displacements thus defined, satisfy the ho- mogeneous part of the constraint equations, i.e., Eq.(10) and Eq.(11) with ∂fi/∂t= 0 and Ait= 0. Hence, N/summationdisplay k=1∂fi ∂rk·δrk= 0, i = 1,2, . . ., s, (14) N/summationdisplay k=1Aik·δrk= 0, i = 1,2, . . ., m. (15) The logical connection between the equations of con- straint, equations for allowed displacements and equa- tions for virtual displacements are presented in FIG.1.FIG. 1: Virtual displacement defined as difference of allowed displacements The absence of the ( ∂fi/∂t) and Aitin the above equa- tions, Eq.(14) and Eq.(15), gives the precise meaning to the statement: “ virtual displacements are the allowed dis- placements in the case of frozen constraints ”. The con- straints are frozen in time in the sense that we make the (∂fi/∂t) and Aitterms zero, though the ∂fi/∂rk andAikterms still involve both position {r1,r2, . . .,rN}, and time ( t). In the case of stationary constraints, i.e., fi(r1, . . .,rN) = 0, and/summationtext kAik(r1, . . .,rN)·vk= 0, the virtual displacements are identical with allowed displace - ments as ( ∂fi/∂t) and Aitare identically zero. B. Existence of forces of constraints In the case of an unconstrained system of Nparticles described by position vectors {rk}and velocity vectors {vk}, the motion is governed by Newton’s Law, mkak=Fk(rl,vl, t), k, l = 1,2, . . ., N (16) where mkis the mass of the kthparticle, akis its accel- eration and Fkis the total external force acting on it. However, for a constrained system, the equations of con- straint, namely Eq.(6) and Eq.(7), impose the following restrictions on the allowed accelerations, N/summationdisplay k=1∂fi ∂rk·ak+N/summationdisplay k=1d dt/parenleftbigg∂fi ∂rk/parenrightbigg vk+d dt/parenleftbigg∂fi ∂t/parenrightbigg = 0, i= 1,2, . . ., s (17) N/summationdisplay k=1Aik·ak+d dtAik·vk+d dtAit= 0, i= 1,2, . . ., m. (18) 5 Given {rk},{vk}one is no longer free to choose all the accelerations {ak}independently. Therefore in general the accelerations {ak}allowed by Eq.(17), Eq.(18) are incompatible with Newton’s Law, i.e., Eq.(16). This implies that during the motion the constraint con- dition cannot be maintained by the external forces alone. Physically some additional forces, e.g., normal reaction from the surface of constraint, tension in the pendulum string, come into play to ensure that the constraints are satisfied throughout the motion. Hence one is compelled to introduce forces of constraints {Rk}and modify the equations of motion as, mkak=Fk+Rk, k = 1,2, . . ., N. (19) FIG. 2: Existence of constraint forces FIG.2 presents the connection between the equations of constraints and forces of constraints. Now the problem is to determine the motion of Npar- ticles, namely their positions {rk(t)}, velocities {vk(t)} and the forces of constraints {Rk}, for a given set of external forces {Fk}, constraint equations, Eq.(6) and Eq.(7), and initial conditions {rk(0),vk(0)}. It is im- portant that the initial conditions are also compatible with the constraints. There are a total of (6 N) scalar unknowns, namely the components of rk(t) andRk, con- nected by (3 N) scalar equations of motion, Eq.(19), and (s+m) equations of constraints, Eq.(6) and Eq.(7). For (6N >3N+s+m) we have an under-determined sys- tem. Hence to solve this problem we need (3 N−s−m) additional scalar relations. C. Solvability and ideal constraints In simple problems with stationary constraints, e.g., motion of a particle on a smooth stationary surface, we observe that the allowed displacements are tangential to the surface. The virtual displacement being a difference of two such allowed displacements, is also a vector tan- gential to it. For a frictionless surface, the force of con- straint, the so called ‘normal reaction’, is perpendicular to the surface. Hence the work done by the constraintforce is zero, on allowed as well as virtual displacement. N/summationdisplay k=1Rk·drk= 0,N/summationdisplay k=1Rk·δrk= 0. When the constraint surface is in motion, the allowed velocities, and hence the allowed displacements are no longer tangent to the surface (see Sec.III). The virtual displacement however remains tangent to the constraint surface. As the surface is frictionless, it is natural to assume that the force of constraint is still normal to the instantaneous position of the surface. Hence the work done by normal reaction on virtual displacement is zero. However the work done by constraint force on allowed displacements is no longer zero. N/summationdisplay k=1Rk·drk/ne}ationslash= 0,N/summationdisplay k=1Rk·δrk= 0. (20) 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 3 Nscalar components of the virtual displacements {δrk}are connected by ( s+m) equations, Eq.(14) and Eq.(15), only n= (3N−s−m) of these scalar components are independent. If the ( s+m) dependent quantities are expressed in terms of remaining (3N−s−m) independent objects we get, n/summationdisplay j=1/tildewideRj·δ/tildewidexj= 0. (21) where {/tildewidexj}are the independent components of {rk}. {/tildewideRj}are the coefficients of {δ/tildewidexj}, and are composed of different {Rk}. Since the above components of virtual displacements {δ/tildewidexj}are independent, one can equate each of their coefficients to zero ( /tildewideRj= 0). This brings in exactly (3 N−s−m) new scalar conditions or equations that are needed to make the system solvable (see FIG.3). 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 on virtual displacement, i.e.,/summationtext kRk·δrk= 0, the ideal constraint . Our interpretation of the principle of zero virtual work, as a definition of an ideal class of constraints, agrees with Sommerfeld. In his exact words, “ a general postulate of mechanics: in any mechanical systems the virtual work of the reactions equals zero. Far be it from us to want to give a general proof of this postulate, rather we regard it practically as definition of a mechanical system ”7. (Boldface is added by the authors). 6 FIG. 3: Solvability under ideal constraints III. EXAMPLES OF VIRTUAL DISPLACEMENTS A. Simple Pendulum with stationary support The motion of a pendulum is confined to a plane and its bob moves keeping a fixed distance from the point of suspension (see FIG.4). The equation of constraint therefore is, f(x, y, t).=x2+y2−r2 0= 0, where r0is the length of the pendulum. Whence ∂f ∂x= 2x,∂f ∂y= 2y,∂f ∂t= 0. The constraint equation for allowed velocities, Eq.(8), becomes, x·vx+y·vy= 0. Hence the allowed velocity ( vx,vy) is orthogonal to the instantaneous position ( x,y) of the bob relative to sta- tionary support. The same may also be verified taking a plane polar coordinate. The allowed velocities and the allowed displacements are perpendicular to the line of suspension. The virtual velocities and the virtual displacements, being the differ- ence of two unequal allowed velocities and displacements respectively, are also perpendicular to the line of suspen- sion. dr=v(t)dt, d r′=v′(t)dt, δr= (v(t)−v′(t))dt.FIG. 4: Allowed and virtual displacements of a pendulum with stationary support Although the virtual displacement is not uniquely speci- fied by the constraint, it is restricted to be in a plane per- pendicular to the instantaneous line of suspension. Hence it is not ‘completely arbitrary’. The ideal string of the pendulum provides a tension (T) along its length, but no shear. The work done by this tension on both allowed and virtual displacements is zero, T·dr= 0, T·δr= 0. B. Simple Pendulum with moving support Let us first consider the case when the support is mov- ing vertically with a velocity u. The motion of the pen- dulum is still confined to a plane. The bob moves keep- ing a fixed distance from the moving point of suspension (FIG.5). The equation of constraint is, f(x, y, t).=x2+ (y−ut)2−r2 0= 0, where uis the velocity of the point of suspension along a vertical direction. Whence ∂f ∂x= 2x,∂f ∂y= 2(y−ut),∂f ∂t=−2u(y−ut). Hence the constraint equation gives, x·vx+ (y−ut)·vy−u(y−ut) = 0, or, x·vx+ (y−ut)·(vy−u) = 0. The allowed velocities ( vx,vy) and the allowed displace- ments, are not orthogonal to the instantaneous position 7 FIG. 5: Allowed and virtual displacements of a pendulum with moving support 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 the difference of allowed velocities. Hence it is orthogonal to the instantaneous line of suspension. dr=v(t)dt=v⊥(t)dt+udt, dr′=v′(t)dt=v′ ⊥(t)dt+udt, δr= (v(t)−v′(t))dt= (v⊥(t)−v′ ⊥(t))dt. Hence none of these allowed or virtual vectors are ‘arbi- trary’. At any given instant, an ideal string provides a ten- sion along its length, with no shear. Hence the constraint force, namely tension T, does zero work on virtual dis- placement. T·dr=T·udt/ne}ationslash= 0, T·δr=T·v⊥dt= 0 For the support moving in a horizontal or any arbitrary direction, one can show that the allowed displacement is not normal to the instantaneous line of suspension. But the virtual displacement, as defined in this article, always remains perpendicular to the instantaneous line of support.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 θ0is the angle of the slope. Hence the constraint equation for allowed velocities, Eq.(8), gives, /parenleftbigg∂f ∂r/parenrightbigg ·v+∂f ∂t=/parenleftbigg∂f ∂r/parenrightbigg vr+/parenleftbigg∂f ∂θ/parenrightbigg vθ+∂f ∂t = 0 ·˙r+ 1·(r˙θ) + 0 = 0 Hence ˙θ= 0, implying that the allowed velocities are along the constant θplane. Allowed velocity, allowed and virtual displacements are, v= ˙r/hatwider, dr= ˙r/hatwiderdt, v′= ˙r′/hatwider, dr′= ˙r′/hatwiderdt, δr= (v−v′)dt= (˙r−˙r′)/hatwiderdt. where /hatwideris a unit vector along the slope. FIG. 6: Allowed and virtual displacements of a particle slid ing along a stationary slope As the inclined slope is frictionless, the constraint force Nis normal to the surface. The work done by this force on allowed as well as virtual displacement is zero. N·dr= 0, N·δr= 0 D. Motion along a moving inclined plane For an inclined plane moving along the horizontal side (FIG.7), the constraint is given by, (x−ut) y−cot(θ0) = 0 , f(x, y).= (x−ut)−cot(θ0)y= 0. Whence the constraint for allowed velocities Eq.(8) be- comes, (˙x−u)−cot(θ0) ˙y= 0. 8 FIG. 7: Allowed and virtual displacements of a particle slid ing along a moving slope Hence the allowed velocity (˙ x,˙y) is the sum of two vectors, one along the plane ( ˙ x−u,˙y), and the other equal to the velocity of the plane itself ( u,0). v(t) =v/bardbl(t) +u. Allowed displacements are vectors along the allowed ve- locities, however the virtual displacement is still a vecto r along the instantaneous position of the plane. dr= (v/bardbl(t) +u)dt, d r′= (v′ /bardbl(t) +u)dt, δr= (v(t)−v′(t))dt= (v/bardbl(t)−v′ /bardbl(t))dt. For the moving frictionless slope, the constraint force pro - vided by the surface is perpendicular to the plane. Hence the work done by the constraint force on virtual displace- ment remains zero. N·dr/ne}ationslash= 0, N·δr= 0. IV. LAGRANGE’S METHOD OF UNDETERMINED MULTIPLIERS A constrained system of particles obey the equations of motion given by, mkak=Fk+Rk, k = 1,2, . . ., N where mkis the mass of the kthparticle, akis its accel- eration. FkandRkare the total external force and force of constraint on the kthparticle. If the constraints are ideal, we can write, N/summationdisplay k=1Rk·δrk= 0, (22) whence we obtain, N/summationdisplay k=1(mkak−Fk)·δrk= 0. (23) If the components of {δrk}were independent, we could recover Newton’s Law for unconstrained system from thisequation. 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/summationdisplay k=1∂fi ∂rkδrk= 0, i = 1,2, . . ., s (14) N/summationdisplay k=1Ajk·δrk= 0, j = 1,2, . . ., m (15) We multiply Eq.(14) successively by ( s) scalar multi- pliers {λ1, λ2, . . .λ s}, Eq.(15) successively by ( m) scalar multipliers {µ1, µ2, . . .µ m}and then subtract them from the zero virtual work equation, namely Eq.(22). N/summationdisplay k=1 Rk−s/summationdisplay i=1λi∂fi ∂rk−m/summationdisplay j=1µjAjk δrk= 0. (24) These multipliers {λi}and{µj}are called the Lagrange’s multipliers. Explicitly in terms of components, N/summationdisplay k=1 Rk,x−s/summationdisplay i=1λi∂fi ∂xk−m/summationdisplay j=1µj(Ajk)x δxk +N/summationdisplay k=1[Y]kδyk+N/summationdisplay k=1[Z]kδzk= 0, (25) where [ Y]kand [Z]kdenote the coefficients of δykand δzkrespectively. The constraint equations, Eq.(14) and Eq.(15), allow 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, . . . , λ s}and {µ1, µ2, . . ., µ m}, such that the coefficients of ( s+m) dependent components of virtual displacement vanish. The remaining virtual displacements being independent, their coefficients must vanish as well. Thus it is possible to choose {λ1, λ2, . . . , λ s}and{µ1, µ2, . . . , µ m}such that all coefficients {[X]k,[Y]k,[Z]k}of virtual displacements {δxk,δyk,δzk}in Eq.(25) vanish. Hence we can express the forces of constraint in terms of the Lagrange’s multi- pliers. Rk=s/summationdisplay i=1λi∂fi ∂rk+m/summationdisplay j=1µjAjk, k = 1,2, . . ., N (26) Thus the problem reduces to finding a solution for the equations, mkak=Fk+s/summationdisplay i=1λi∂fi ∂rk+m/summationdisplay j=1µjAjk, k= 1,2, . . ., N (27) together with the equations of constraint, fi(r1,r2, . . . ,rN, t) = 0, i = 1,2, . . ., s (6) 9 FIG. 8: Lagrange’s method of undetermined multipliers: sol v- ability and N/summationdisplay k=1Aik·vk+Ait= 0, i = 1,2, . . ., m. (7) Here we have to solve (3 N+s+m) scalar equations involving (3 N+s+m) unknown scalar quantities, namely {xk, yk, zk, λi, µj}(see FIG.8). After solving this system of equations for {xk, yk, zk, λi, µj}, one can obtain the forces of constraint {Rk}using Eq.(26). V. LAGRANGE’S EQUATIONS IN GENERALIZED COORDINATES For the sake of completeness we discuss very briefly Lagrange’s equations in generalized coordinates. A more complete discussion can be found in most texts1,2,3,4,5,6,7,8,9,10,11,12,13,14. Consider a system of N particles under ( s) holonomic and ( m) non-holonomic constraints. In certain suitable cases, one can express (s+m) dependent coordinates in terms of the remaining (3N−s−m) independent ones. It may be noted that such a complete reduction is not possible for general cases of non-holonomic and time dependent constraints7,14. If we restrict our discussion to cases where this reduction is possible, one may express all the 3 Nscalar componentsof position {r1,r2, . . . ,rN}in terms of (3 N−s−m) in- dependent parameters {q1, q2, . . ., q n}and time ( t). rk=rk(q1, q2, . . ., q n, t), k = 1,2, . . ., N (28) The allowed and virtual displacements are given by, drk=n/summationdisplay j=1∂rk ∂qjδqj+∂rk ∂tdt, k = 1,2, . . ., N δrk=n/summationdisplay j=1∂rk ∂qjδqj, k = 1,2, . . ., N. (29) From Eq.(23) we obtain, N/summationdisplay k=1mkd˙rk dt n/summationdisplay j=1∂rk ∂qjδqj −N/summationdisplay k=1Fk n/summationdisplay j=1∂rk ∂qjδqj = 0 (30) Introduce the expression of kinetic energy, T=1 2N/summationdisplay k=1mk˙r2 k, and that of the generalized force, Qj=N/summationdisplay k=1Fk∂rk ∂qj, j = 1,2, . . ., n. (31) After some simple algebra one finds, n/summationdisplay j=1/parenleftbiggd dt∂T ∂˙qj−∂T ∂qj−Qj/parenrightbigg δqj= 0. (32) As{q1, q2, . . ., q n}are independent coordinates, coeffi- cient of each δqjmust be zero separately. Hence (see FIG.9), d dt∂T ∂˙qj−∂T ∂qj=Qj, j = 1,2, . . ., n. (33) 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}. In problems where forces {Fk}are derivable from a scalar potential /tildewideV(r1,r2, . . . ,rN), Fk=−∇k/tildewideV(r1,r2, . . . ,rN), k = 1,2, . . ., N. (34) One can obtain the generalized force as, Qj=−∇k/tildewideV·/parenleftbigg∂rk ∂qj/parenrightbigg =−∂V ∂qj, j = 1,2, . . ., n. (35) 10 FIG. 9: Lagrange’s equations in generalized coordinates Where Vis the potential /tildewideVexpressed as a function of {q1, q2, . . . , q n}. In addition as the potential Vis in- dependent of the generalized velocities, we obtain from Eq.(33), d dt∂(T−V) ∂˙qj−∂(T−V) ∂qj= 0, j = 1,2, . . ., n. (36) At this stage one introduces the Lagrangian function, L=T−V. In terms of the Lagrangian, the equations of motion take up the form, d dt∂L ∂˙qj−∂L ∂qj= 0, j = 1,2, . . ., n. (37) VI. CONCLUSION In this article we make an attempt to present a quan- titative definition of the virtual displacement. We show that for certain simple cases the work done by the forces of constraint on virtual displacement is zero. We also demonstrate that this zero virtual work principle gives us a solvable class of problems. Hence we define this special class of constraint as the ideal constraint . We demonstrate in brief how one can solve a general mechan- ical problem by: i) Lagrange’s method of undetermined multipliers and ii) Lagrange’s equations in generalized coordinates. In the usual presentations of Lagrange’s equation based on virtual displacement and d’Alembert’s princi- ple, Eq.(3) and Eq.(5) are satisfied by the virtual dis- placements. One may consider these equations as the definition of virtual displacements. However the situ- ation is far from satisfactory, as separate defining equa- tions are required for different classes of constraints. Thi s adhoc definition also fails to clarify the actual connection between the virtual displacements and the equations of constraints.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 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 external forces, would choose one of these various sets of displacements as actual displacement of the system. The definition of virtual displacement as difference of two un- equal allowed displacements over the same infinitesimal time interval ( t, t+dt), gives a unified definition of virtual displacement. This definition of virtual displacement sat- isfies the appropriate equations found in the literature, for both holonomic and non-holonomic systems. It is shown that Newton’s equation of motion with ex- ternal forces alone, is inconsistent with equations of con- straint. Hence the forces of constraint are introduced. Now there are 3 Nequations of motion and ( s+m) equations of constraint involving 6 Nunknown scalars {rk(t),Rk}. Without additional condition (d’Alembert principle), the problem is underspecified and unsolvable. It is verified that for simple physical systems, the virtual displacements, as defined in this article, sat- isfy d’Alembert principle for particles in motion. The rheonomous examples discussed in Sec.III show why the forces of constraint do zero work on virtual displace- ments and not on allowed displacements. The additional equations introduced by d’Alembert principle make the problem solvable. These justify (i) the peculiar defini- tion of virtual displacements and the equations they sat- isfy, (ii) Lagrange’s extension of d’Alembert’s principle to particles in motion, (iii) why the zero work princi- ple is related to virtual displacement and not to allowed displacement. Once the system is solvable, two meth- ods, originally proposed by Lagrange can be used, and are demonstrated. For Lagrange’s method of undeter- mined multipliers, one solves (3 N+s) equations to ob- tain the motion of the system {rk(t), . . . ,rN(t)}and La- grange’s multipliers {λi, . . ., λ s}. The forces of constraint {Rk, . . .,RN}are expressed in terms of these multipli- ers. For Lagrange’s equations in generalized coordinates one solves (3 N−s) equations to obtain the time evo- lution of the generalized coordinates {qj=qj(t), j= 1, . . .,3N−s}. This gives the complete description of 11 the motion {rk=rk(q1, q2, . . . , q n, t), k= 1, . . ., N }, ig- noring the calculation of the constraint forces. It may be noted that about a century later Appell’s equations7,14,20 were introduced for efficiently solving non-holonomic sys- tems. It is interesting to note that both the above mentioned methods require the principle of zero virtual work by con- straint forces as a crucial starting point. In the case of Lagrange’s method of undetermined multipliers we start with the ideal constraint condition Eq.(22). From there we obtain Eq.(23)-Eq.(27). Eq.(26) expresses the con- straint forces in terms of Lagrange’s multipliers. For La- grange’s equations in generalized coordinates, we start with the ideal constraint, Eq.(22). We work our way through Eq.(23), Eq.(30), Eq.(32) and finally obtain La- grange’s equations in generalized coordinates, Eq.(33) and Eq.(37). The last figure, FIG.(10), gives the com- plete logical flow of this article.Acknowledgment Authors wish to thank Professor John D. Jackson and Professor Leon A. Takhtajan for their remarks and sug- gestions. They also thank Professor Sidney Drell, Profes- sor Donald T. Greenwood and Professor Alfred S. Gold- haber for their valuable communications. The authors gratefully acknowledge the encourage- ment received from Professor Max Dresden at Stony Brook. Authors have greatly benefited from the books mentioned in this article, particularly those of Arnold13, Goldstein1, Greenwood2, Pars14and Sommerfeld7. The material presented in this article was used as part of Classical Mechanics courses at Jadavpur University during 2000-2003. SR would like to thank his students A. Chakraborty and B. Mal for meaningful discussions. ∗Electronic address: sray˙ju@rediffmail.com, subho@juphy s.ernet.in (S. Ray) †Electronic address: [email protected] (J. Shamanna) 1H. Goldstein, Classical Mechanics , Addison-Wesley Pub- lishing Co., Reading, Massachusetts, 1980. 2D. T. Greenwood, Classical Dynamics , Prentice Hall, New York, 1977. 3A. Haas, Introduction to Theoretical Physics, vol I , Con- stable and Company Ltd, London, 1924. 4L. N. Hand, J. D. Finch, Analytical Mechanics , Cambridge University Press, Cambridge, 1998. 5E. A. Hylleraas, Mathematical and Theoretical Physics , vol. I, Wiley Interscience, New York, 1970. 6D. A. Wells, Schaum Outline of Theory and Problems of Lagrangian Dynamics , McGraw-Hill Inc., New York, 1967. 7A. Sommerfeld, Mechanics, Lectures on Theoretical Physi- cs, vol. I, Academic Press, New York, 1952. 8J. L. Synge, and B. A. Griffith, Principles of Mechanics , McGraw-Hill Inc., New York, 1970. 9K. R. Symon, Mechanics , Addison-Wesley Publishing Co., Reading, Massachusetts, 1971. 10T. T. Taylor, Mechanics: Classical and Quantum , Perga- mon Press, Oxford, 1976.11D. Ter Haar, Elements of Hamiltonian Mechanics , North Holland Publishing Co., Amsterdam, 1961. 12L. D. Landau, E. M. Lifshitz, Mechanics , Pergamon Press, Oxford, 1976. 13V. I. Arnold, Mathematical Methods of Classical Mechan- ics, Springer Verlag, New York, 1989. 14L. A. Pars, A Treatise on Analytical Dynamics , Heine- mann, London, 1964. 15J. le R. d’Alembert, Trait´ e de dynamique, David ’Ame, Paris, 1743 (reprinted Gauthier-Villars, Paris, 1921). 16J. L. Lagrange, Miscell. Taurin., II, 1760. 17J. L. Lagrange, Mecanique Analytique, Imprimeur- Libraire pour les Mathematiques, Paris, 1788 (reprinted Gauthier-Villars, Paris, 1888). 18W. R. Hamilton, On a general method in Dynamics, 1834; Collected papers, II, Cambridge University Press, Cam- bridge, 1940, p 103-211. 19S. Drell, private communication, 2004. 20P. Appell, Trait´ e de M´ ecanique Rationelle, Vol I & II, Gauthier-Villars, Paris, 1896. 12 FIG. 10: Logical connection between constraint equations, virtual displacements, forces of constraints, d’ Alembert ’s principle and Lagrange’s equations.