On Virtual Displacement and Virtual Work in Lagrangian Dynamics Subhankar Ray∗ Department of Physics, Jadavpur University, Calcutta 700 032, India and C. N. Yang Institute for Theoretical Physics, Stony Brook, NY 11794 J. Shamanna† Physics Department, Visva Bharati University, Santiniketan 731235, India (Dated: August 1, 2005) The confusion and ambiguity encountered by students, in understanding virtual displacement and virtual work, is discussed in this article. A definition of virtual displacement is presented that allows one to express them explicitly for holonomic (velocity independent), non-holonomic (veloc- ity dependent), scleronomous (time independent) and rheonomous (time dependent) constraints. It is observed that for holonomic, scleronomous constraints, 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 by the constraint forces on virtual displacements is zero. This motivates Lagrange’s extension of d’Alembert’s principle to system of particles in constrained motion. However a similar zero work principle does not hold for the allowed displacements. It is also demonstrated that d’Alembert’s principle 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 central to both Lagrange’s method of undetermined multipliers, and Lagrange’s equations in generalized 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 2. Hamilton’s principle of least action7,18 (1834), and variational approach to Lagrange’s equation. Almost all graduate level courses in classi- The two methods are logically and mathematically in- cal mechanics include a discussion of virtual dependent and individually self contained. The first displacement1,2,3,4,5,6,7,8,9,10,11 and Lagrangian method was historically proposed half a century earlier, dynamics1,2,3,4,5,6,7,8,9,10,11,12. From the concept of and it presents the motivation of introducing the La- zero work by constraint forces on virtual displacement, grangian as a new physical quantity. The second method the Lagrange’s equations of motion are derived. starts with the Lagrangian and the related action as However, the definition presented in most accessible quantities axiomatically describing the dynamics of the texts often seem vague and ambiguous to students. Even system. This method is applied without ambiguity in after studying the so called definition, it is rather com- some texts12,13 and courses19. However one also finds monplace that a student fails to identify, whether a sup- intermixing of the two approaches in the literature and plied vector is suitable as a virtual displacement, for a popular texts, often leading to circular definition and re- given constrained system. Though some of the more ad- lated confusion. A rational treatment demands an in- vanced and rigorous treatise13,14 present a more precise dependent presentation of the two methods, and then a arXiv:physics/0510204v2 [physics.ed-ph] 2 Jun 2006 and satisfactory treatment, they are often not easily com- demonstration of their interconnection. In the present prehensible to most students. In this article we attempt article we confine ourselves to the first method. a simple, systematic and precise definition of virtual dis- In this approach, due to Bernoulli, d’Alembert and La- placement, which clearly shows the connection between grange, one begins with a constrained system, defined by the constraints and the corresponding allowed and virtual equations of constraints connecting positions, time and displacements. This definition allows one to understand often velocities of the particles under consideration. The how far the virtual displacement is ‘arbitrary’ and how concept of virtual displacement is introduced in terms far it is ‘restricted’ by the constraint condition. of the constraint equations. The external forces alone There are two common logical pathways of arriving at cannot maintain the constrained motion. This requires Lagrange’s equation. the introduction of forces of constraints. The imposition of the principle of zero virtual work by constraint forces 1. Bernoulli’s principle of virtual velocity7 (1717), gives us a ‘special class of systems’, that are solvable. d’Alembert’s principle of zero virtual work7,15 A proper definition of virtual displacement is necessary (1743), Lagrange’s generalization of d’Alembert’s to make the said approach logically satisfactory. However principle to constrained system of moving particles, the various definitions found in popular texts are often and Lagrange’s equations of motion (1788)7,16,17. incomplete and contradictory with one another. These 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., pendulum with fixed or mov- φj (x1, x2,...,x3N ,t)=0 , j =1, 2,...,k. (1) ing support, particle sliding 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.