sub_sham paper

arXiv:physics/0410123v1 [physics.ed-ph] 18 Oct 2004Virtual Displacement in Lagrangian Dynamics Subhankar Ray∗ Dept of Physics, Jadavpur University, Calcutta 700 032, Ind ia 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: September 1, 2003) The confusion and ambiguity encountered by students, in und erstanding virtual displacement and virtual work, is addressed in this article. A definition o f virtual displacement is presented that allows one to express them explicitly for both time independ ent and time dependent constraints. It is observed that for time independent constraints the virtu al displacements are the displacements allowed by the constraints. However this is not so for a gener al time dependent case. For simple physical systems, it is shown that, the work done on virtual d isplacements by the constraint forces is zero in both the situations. For allowed displacements ho wever, this is not always true. It is also demonstrated that when constraint forces do zero work on vir tual displacement, as defined here, we have a solvable mechanical problem. We identify this specia l class of constraints, physically realized and solvable, as the ideal constraints . The concept of virtual displacement and the principle of ze ro virtual work by constraint forces are central to both Lagran ge’s method of undetermined multipliers, and Lagrange’s equations in generalized coordinates. 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 dynamics1,2,3,4,5,6,7,8,9,10,11,12,13. From the concept of zero work by virtual displacement the Lagrange’s equations of motion are derived. However, the definition of virtual displacement is rarely made precise and often seems vague and ambiguous to students. In this article we attempt a more systematic and precise definition, which not only gives one a qualitative idea of virtual displacement, but also allows one to quantitatively express the same for any given constrained system. We show that in a number of natural systems, e.g., particle moving on a frictionless slope, pendulum with moving point of suspension, the work done by the forces of constraint due to virtual displacement is zero. We also demostrate that this condition is necessary for the solvability of a mechanical problem. Hence we propose such systems as an important class of natural systems. A. Ambiguity in virtual displacement In the following we try to classify the difficulties faced by a student in understanding the definition of virtual displacement. 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 constraint2; (iii)virtual dis- placements are, by definition, arbitrary displace-ments of the components of the system, satisfy- ing the constraint3; (iv) virtual displacement does not violate the constraints8; (v)we define a vir- tual displacement as one which does not violate the kinematic relations10; (vi)the virtual displacements obey the constraint on the motion11. These state- ments imply that the virtual displacements satisfy the constraint conditions, i.e., the constraint equa- tions. However this is true only for time indepen- dent (sclerenomous) constraints. We shall show that for time dependent (rheonomous) constraints, such as a pendulum with moving support, this def- inition would violate the zero virtual work condi- tion. 2. 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 system2; (iii)virtual displacement δrtakes place without any passage of time8. (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 path9; (v) one of the requirements on acceptable virtual displacement is that the time is held fixed11. We even notice equation like ‘ δxi=dxifordt= 0’8. 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 fucntions of the time, we are not able to regard them as differ- entials, as we do for the components of the element 2 of the actual path9. We shall show that virtual dis- placement can be looked upon as a differential, it is indeed a differential increment in virtual velocity over a time dt, Eq.(6). 3. It is also stated that (i) virtual displacements do not necessarily conform to the constraints4; (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 it2; (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 system3; (iv)it is not necessary that it (virtual displacement) represents any actual motion of the system6; (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 occur9; (vi)virtual displacement is any arbitrary infinitesimal displace- ment not necessarily along the constrained path5. From the above we understand that the virtual displacemnts do not always 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 claims under items 1 and 3. It is not clear from the above, whether the virtual displacements sat- isfy the constraints, i.e., the constraint equations or not. 4. Virtual displacement is variously described as: ar- bitrary ,virtual , andimaginary1,2,3,5,6. These adjec- tives make the definition somewhat mysterious to a student. Together with the above ambiguities, students are of- ten confused as to whether it suffices to understand vir- tual displacement as an abstract concept, or they need 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 used 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 may get further puzzled. Should he take the forces of constraint as supplied, and then the principle of zero virtual work as a definition of virtual displacement ? In that case the principle re- duces merely to a definition of a new concept, namely virtual displacement. Or should the virtual displacement be defined from the constraint conditions independently? The principle of zero virtual work may then be used to calculate the forces of constraint that ensure constraint condition throughout the motion. 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 rkand (t), fi(r1,r2, . . . ,rN, t) = 0, i = 1,2, . . ., s (1) Each constraint of this form imposes a restriction on the possible or allowed velocities , which must satisfy, N/summationdisplay k=1/parenleftbigg∂fi ∂rk/parenrightbigg ·vk+∂fi ∂t= 0, i = 1,2, . . ., s (2) It is worth noting at this stage that there are many, in fact infinitely many, allowed velocities, since we have im- posed only ( s) number of constraints on (3 N) scalar com- ponents of the allowed velocity vectors. An infinitesimal displacement over time ( dt) due to allowed velocities will be called the allowed infinitesimal displacement or simply allowed displacement. drk=vkdt k = 1,2, . . ., N (3) Allowed displacements drktogether with differential of time ( dt) satisfy constraint equations similar to Eq.(2). N/summationdisplay k=1/parenleftbigg∂fi ∂rk/parenrightbigg ·drk+∂fi ∂tdt= 0, i = 1,2, . . ., s (4) As there are many allowed velocities we have many al- lowed infinitesimal displacements. We propose to define virtual displacement as the difference between any two such (unequal) allowed displacements, δrk=drk−dr′ k, k = 1,2, . . ., N (5) This definition is motivated by the possibility of identi- fying a special class of ‘ ideal constraints ’ (sec. IIc), and verifying ‘ the principle of zero virtual work ’ in common physical examples (sec. III). It may be noted that, by this definition, virtual displacement δrkis not a change in position in zero time. It is rather the difference of any two allowed displacements during a time dt. δrk= (vk−v′ k)dt, k = 1,2, . . ., N (6) 3 The difference of two allowed velocities /tildewidevk=vk−v′ kmay be defined as the virtual velocity. The virtual displacements thus defined satisfy the ho- mogeneous part of the constraint equation Eq.(4) (i.e., with∂fi/∂t= 0). N/summationdisplay k=1∂fi ∂rk·δrk= 0, i = 1,2, . . ., s (7) The absence of the ( ∂fi/∂t) in the above equation, Eq.(7), gives the precise meaning to the statement that virtual displacements are the allowed displacements in the case of frozen constraints . The constraints are frozen in time in the sense that we make the ( ∂fi/∂t) term zero, though the ∂fi/∂rkterm still involves time. In the case of stationary constraints, i.e., f(r1, . . .,rN) = 0, the virtual displacements are identical with allowed displacements as (∂fi/∂t) is 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 (8) where mkis the mass of the kth particle, akis its ac- celeration and Fkis the total external force acting on it. However, for a constrained system, the equations of con- straint, namely Eq.(1), 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 (9) Given rk,vkone is no longer free to choose all the ac- celerations akindependently. Therefore in general the accelerations akallowed by Eq.(9) are incompatible with Newton’s Law, mkak=Fk, k = 1,2, . . ., N 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, Hence one is compelled to introduce forces of constraints Rkand modify the equations of motion as, mkak=Fk+Rk, k = 1,2, . . ., N (10) Now the problem is to determine the motion of Nparti- cles, namely their positions ( rk(t)), velocities ( vk(t)) and the forces of constraints ( Rk), for a given set of externalforcesFk, constraint equations ( fi(r1,r2, . . . ,rN, t) = 0, i= 1,2, . . ., s ) and initial conditions ( rk(0),vk(0)). It is important 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, connected by (3 N) equa- tions of motion, Eq.(10), and ( s) equations of constraints, Eq.(1). For (6 N >3N+s) we have an under-determined system. Hence to solve this problem we need (3 N−s) additional scalar relations. C. Solvability and ideal constraints In simple problems with stationary constraints, e.g., motion 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 tangential to it. The force of constraint, so called ‘normal reaction’, is perpendicular to the surface. Hence the work done by the constraint forces on allowed as well as virtual dis- placement is zero, 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 remains tangent to the constraint surface. If the forces of constraint can still be assumed normal to the instantaneous position of the surface, we have zero virtual work. However note that the work by constraint forces on allowed displacements is not zero. N/summationdisplay k=1Rk·drk/negationslash= 0,N/summationdisplay k=1Rk·δrk= 0 (11) In a number of physically interesting simple problems, such as, motion of a pendulum with fixed and moving support, motion of a particle along a stationary and mov- ing slope, we observe that the above interesting relation between the force of constraint and virtual displacement holds (see sec. III). Out of the above 3 Nvirtual dis- placements, only n= 3N−sare independent. If the ( s) dependent quantities are expressed in terms of remaining n= 3N−sindependent objects we get n/summationdisplay j=1/tildewideRj·δ/tildewidexj= 0 (12) where/tildewidexjare the independent components of rk./tildewideRjare 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 (3 N−s) new scalar 4 conditions or equations and the system is solvable (not under-determined) again. Thus we find a special class of constraints which is observed in nature (see sec. III) and which gives us a solvable system. We call this special class of constraints, satisfying zero virtual work principle by constraint force s, 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, find sup- port in Sommerfeld. In his 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 ”. III. EXAMPLES OF VIRTUAL DISPLACEMENTS A. Simple Pendulum with stationary support The motion of the pendulum is confined to a plane and the bob moves at a fixed distance from the point of sus- pension. The equation of constraint by Eq.(1) therefore is, f(x, y, t).=x2+y2−r2 0= 0 Whence ∂f ∂x= 2x,∂f ∂y= 2y,∂f ∂t= 0 r=v vδ ’( - ) dt r=θ dtvddtdr’v’= FIG. 1: Allowed and virtual displacements for a pendulum with stationary support Hence the constraint equation for allowed velocities (compare Eq.(2)) is, 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 displacements are always collinear to al- lowed velocities. Virtual displacement being difference of two allowed displacements, is also a vector collinear to the allowed velocities, hence tangential to the line of suspension. dr=vdt, d r′=v′dt δr= (v−v′)dt We may assume that the string of the pendulum provides a tension ( T) but no shear (ideal string). We get zero work by tension due to allowed and virtual displacements, 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 keeping a fixed distance from point of suspension. 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. udt dtdr’v’= r= dtvdvtdtru =v vδ ’( - ) dtdtv’tthe point of(velocity of suspension ) θ FIG. 2: Allowed and virtual displacements for a pendulum with moving support Whence ∂f ∂x= 2x,∂f ∂y= 2(y−ut),∂f ∂t=−2u(y−ut) 5 Hence the constraint equation gives, x·vx+ (y−ut)·vy−u(y−ut) = 0 or, x·vx+ (y−ut)·(vy−u) = 0 Hence the allowed velocities ( vx,vy) and hence the al- lowed displacements, are not orthogonal to the instanta- neous position ( x,y−ut) of the bob relative to the instan- taneous position of the support. It is easy to verify from the above equation that the allowed velocity ( vx,vy) is equal to the sum of a velocity vector ( vx,vy−u) perpen- dicular to the relative position of the bob with respect to the point of suspension ( x,y−ut), and the velocity of the support (0, u). v=vt+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 tangential to the instantaneous line of suspension. dr=vdt=vtdt+udt δr= (v−v′)dt= (vt−v′ t)dt At any given instant string provides a tension along its length, with no shear (ideal string). Hence the constraint force, tension, still does zero work on virtual displace- ment. T·dr/negationslash= 0, T·δr= 0 If one considers the support moving in a horizontal (or in 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 instan- taneous line of support. C. Motion along a fixed inclined plane The constraint is more conveniently expressed in the polar coordinate. The constraint equation is, f(r, θ).=θ−θ0= 0 where θ0is a constant. Hence the constraint equation for allowed velocities, Eq.(2), gives, N/summationdisplay k=1/parenleftbigg∂f ∂rk/parenrightbigg ·vk+∂f ∂t.=˙θ+ 0 = 0 Thus the allowed velocities are along the constant θ plane. Allowed velocity, allowed and virtual displace- ments are, v= ˙r/hatwider, dr= ˙r/hatwiderdt, δ r= (˙r−˙r′)/hatwiderdt0r= dtvdr=v vδ ’( - ) dtdt θdr’v’= FIG. 3: Allowed and virtual displacements for a particle on a stationary slope If the inclined slope is frictionless (ideal), the constrai nt force provided by the surface is the normal reaction; which is perpendicular to the plane. Hence 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, 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.(2) be- come, ( ˙x+u)−cot(θ0) ˙y= 0 Hence the allowed velocity ( ˙ x,˙y) is the sum of two vec- tors, one along the plane ( ˙ x+u,˙y), and the other equal to the velocity of the plane itself ( −u,0). v=vt+u Allowed displacements are vectors along the allowed velocities, however the virtual displacement is still a vec - tor along the instantaneous position of the plane. dr= (vt+u)dt, d r′= (v′ t+u)dt δr= (v−v′)dt= (vt−v′ t)dt For the moving frictionless (ideal) slope, the constraint force provided by the surface is perpendicular to the plane. Hence the work done by the constraint force on virtual displacement is remains zero. N·dr/negationslash= 0, N·δr= 0 6 dtr= dtvdr=v vδ ’( - ) dtdtdr’v θ’ u= (velocity of the wedge)0u FIG. 4: Allowed and virtual displacements for a particle on a moving slope IV. LAGRANGE’S METHOD OF UNDETERMINED MULTIPLIERS A constrained system of particles follow the equation of motion given by, mkak=Fk+Rk, k = 1,2, . . ., N where mkis the mass of the kth particle, akis its accel- eration. FkandRkare the total external force and force of constraint on the particle. If the constraints are ideal, we can write N/summationdisplay k=1Rk·δrk= 0 (13) whence we obtain, N/summationdisplay k=1(mkak−Fk)·δrk= 0 (14) If the components of δrkwere independent, we could re- cover Newton’s Law for unconstrained system from this equation. However for a constrained system δrkare de- pendent through the constraint equations, fi(r1,r2, . . .,rN, t) = 0, i = 1,2, . . ., s (15) or, δfi=N/summationdisplay k=1∂fi ∂rkδrk= 0, i = 1,2, . . ., s (16) We multiply the above equations, Eq.(16), successively bysscalar multipliers ( λ1, λ2, . . . λ s), called the La- grange’s multipliers, and subtract them from the zero virtual work equation, Eq.(13). N/summationdisplay k=1/parenleftBigg Rk−s/summationdisplay i=1λi∂fi ∂rk/parenrightBigg δrk= 0 (17)Explicitly in terms of components, N/summationdisplay k=1/parenleftBigg/bracketleftBigg Rk,x−s/summationdisplay i=1λi∂fi ∂xk/bracketrightBigg δxk+ [Y]kδyk+ [Z]kδzk/parenrightBigg = 0 (18) where [ Y]kand [Z]kdenote the coefficients of δykand δzkrespectively. The constraint equations Eq.(15) allows us to write the (s) dependent virtual displacements in terms of the re- maining n= 3N−sindependent ones. We choose ( s) multipliers ( λ1, λ2, . . . , λ s) such that the coefficients of (s) dependent components of virtual displacement van- ish. The remaining virtual displacements being indepen- dent, their coefficients must vanish as well. Thus it is pos- sible to choose ( λ1, λ2, . . ., λ s) such that all coefficients ([X]k,[Y]k,[Z]k) of virtual displacements ( δxk,δyk,δzk) in Eq.(18) vanish. Hence we have the forces of constraint in terms of the Lagrange’s multipliers. Rk=s/summationdisplay i=1λi∂fi ∂rk, k = 1,2, . . ., N (19) Thus the problem of mechanics reduces to finding solu- tion of equations of motion, mkak=Fk+s/summationdisplay i=1λi∂fi ∂rk, k = 1,2, . . ., N (20) with the constraints, fi(r1,r2, . . .,rN, t) = 0, i = 1,2, . . ., s (21) Thus we have to solve 3 N+sscalar equations in 3 N+s unknown scalar quantities ( xk, yk, zk, λi). After solving this system we can obtain the forces of constraint Rk from Eq.(19). V. GENERALIZED COORDINATES AND LAGRANGE’S EQUATIONS OF MOTION For the sake of completeness we discuss very briefly Lagrange’s equations in generalized coordinates (for de- tail see1,2,3,4,5,6,7,8,9,10,11,12,13). Consider a system of N particles under sholonomic, rheonomous constraints of the form given by Eq.(1). We can in principle express sof these coordinates in terms of the remaining 3 N−s independent ones. Or we may express all the 3 Nscalar components of position in terms of n= 3N−sindepen- dent parameters q1, q2, . . . , q nand time ( t). rk=rk(q1, q2, . . ., q n, t), k = 1,2, . . ., N (22) The allowed and virtual displacements are given by, drk=n/summationdisplay j=1∂rk ∂qjδqj+∂rk ∂tdt, δrk=n/summationdisplay j=1∂rk ∂qjδqj, k = 1,2, . . ., N (23) 7 From the Eq.(14) 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 (24) Introducing 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 ∂qjj= 1,2, . . ., n (25) After some simple algebra one finds, n/summationdisplay j=1/parenleftbiggd dt∂T ∂˙qj−∂T ∂qj−Qj/parenrightbigg δqj= 0 (26) since the qjare independent coordinates, coefficient of eachδqjmust be zero separately. d dt∂T ∂˙qj−∂T ∂qj=Qj, j = 1,2, . . ., n (27) In problems where forces Fkare derivable from a scalar potential/tildewideV(r1,r2, . . .,rN), Fk=−∇k/tildewideV(r1,r2, . . . ,rN), k = 1,2, . . ., N (28) we can write the generalized force as, Qj=−∇k/tildewideV·/parenleftbigg∂rk ∂qj/parenrightbigg =−∂V ∂qj, j = 1,2, . . ., n (29) Where Vis the potential /tildewideVexpressed as a function of (q1, q2, . . . , q n). In addition if the potential Vdoes not depend on the generalized velocities, we obtain from Eq.(27), d dt∂(T−V) ∂˙qj−∂(T−V) ∂qj= 0, j = 1,2, . . ., n (30) At this stage one introduces the Lagrangian function L= T−Vand in terms of the Lagrangian, the equations of motion Eq.(30) take up the form d dt∂L ∂˙qj−∂L ∂qj= 0, j = 1,2, . . ., n (31)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 virtual displacement does zero work on forces of constraint. We also demon- strate that this zero work principle allows us to have a solvable class of problems. Hence we define this special class of constraint, the ideal constraint . We demonstrate in brief how one can solve a general mechanical problem by: i) Lagrange’s method of undetermined multiplier and ii) Lagrange’s equations in generalized coordinates. In Lagrange’s method of undetermined multipliers we have to solve a larger number (3 N+s) of equations, than in the case of Lagrange’s equations (3 N−s) in general- ized coordinates. However we can immediately derive the forces of (ideal) constraints in the former case. It is interesting to note that both the abovementioned methods require the zero virtual work by constraint forces as a crucial starting point. In the case of La- grange’s method of undetermined multipliers we start with the ideal constraint condition Eq.(13). From there we write down Eq.(14), Eq.(17), Eq.(18) and express the constraint forces in terms of Lagrange’s multipliers, Eq.(19). For Lagrange’s equations in generalized coor- dinates we start with the ideal constraint, Eq.(13). We work our way through Eq.(14), Eq.(24), Eq.(26) and fi- nally obtain Lagrange’s equations in generalized coordi- nates, Eq.(27) and Eq.(31). Acknowledgement The authors gratefully acknowledge their teachers in related graduate courses at Stony Brook, Prof. Max Dresden, Prof. A. S. Goldhaber and Prof. Leon A. Takhtajan. Authors also acknowledge the encourage- ment received from Prof. Shyamal SenGupta of Pres- idency College, Calcutta. The material presented here was used in graduate level classical mechanics courses at Jadavpur University during 1998 −2001. SR would like to thank his students, in particular, A. Chakraborty (J.U.) for pointing out the difficulty in understanding the concept of virtual displacement in its usual presenta- tion. Authors have greatly benefited from the books men- tioned in this article, particularly those of Sommerfeld2, Hylleraas3and Arnold13. ∗Electronic address: [email protected] †Electronic address: jshamanna@rediffmail.com1H. Goldstein, Classical Mechanics , Addison-Wesley Pub- lishing Co., Reading, Massachusetts, 1980. 8 2A. Sommerfeld, Mechanics, Lectures on Theoretical Physi- cs, vol. I, Academic Press, New York, 1952. 3E. A. Hylleraas, Mathematical and Theoretical Physics , vol. I, Wiley Interscience, New York, 1970. 4D. T. Greenwood, Classical Dynamics , Prentice Hall, New York, 1977. 5D. A. Wells, Schaum Outline of Theory and Problems of Lagrangian Dynamics , McGraw-Hill Inc., New York, 1967. 6K. R. Symon, Mechanics , Addison-Wesley Publishing Co., Reading, Massachusetts, 1971. 7J. L. Synge, and B. A. Griffith, Principles of Mechanics , McGraw-Hill Inc., New York, 1970. 8T. T. Taylor, Mechanics: Classical and Quantum , Perga-mon Press, Oxford, 1976. 9A. Haas, Introduction to Theoretical Physics, vol I , Con- stable and Company Ltd, London, 1924. 10D. Ter Haar, Elements of Hamiltonian Mechanics , North Holland Publishing Co., Amsterdam, 1961. 11L. N. Hand, J. D. Finch, Analytical Mechanics , Cambridge University Press, Cambridge, 1998. 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.