Page 1 : 1.5. CONSTRAINED MOTION, CONSTRAINTS AND DEGREES OF FREEDOM, Constraints :, A constrained motion is a motion which cannot proceed arbitrarily in any manner. Particle, motion is restricted to occur only, for example, along some specified path, or on a surface (plane or, curved) arbitrarily oriented in space. Motion along a specified path is the simplest example of a, constrained motion. Here, one co-ordinate is sufficient to describe the motion in contrast to the, situation where the particle is free to move in space and then three co-ordinates are needed to, describe its motion. Thus imposing constraints on a mechanical system is to simplifiy the, mathematical description. When a particle (bead) is made to slide on a wire, constraints require, that the position of the bead lie on the wire. Condition imposed on the system by the constraints, can, in most cases, be written down mathematically as a relation satisfied by the co-ordinates of the, particle at any time. This is the way in which constraints reduce the number of co-ordinates needed, to specify the configuration of a system., Example 1: Let us consider the motion of a simple pendulum confined to move in the, vertical plane. We would need only two co-ordinates (cartesian co-ordinates x and y or polar, co-ordinates r and 0 with respect to the point of suspension, O, as origin) to locate the position of, the bob in motion. However, motion of the bob is not free but takes place under a constraint that, the distance l of the bob is to remain the same from O all the ime. This condition imposed by the, constraint can be expressed in the form of an equation either between x and y or r and 0 :, x² + y² = 1²,, I|, r = l., ... (1), or, In plane polar co-ordinates, the equation looks simpler. Again one co-ordinate 0, in polar, co-ordinates, would suffice to describe the motion. Note that we have utilised eq. (1) to reduce the, number of co-ordinates which, otherwise, would have been two., three, Example 2 : Let us take up another example. A particle moving in space requir, co-ordinates to determine its position at any instant. If we restrict its movement on the surface of, a sphere, there exists a relation between these co-ordinates. Again we shall see that spherical polar, co-ordinates can be used here with advantage :, x² + y² + z² = a²,, +2² =, %3|, ... (2), or, r = a, in which a is the radius of the sphere. Each of eqs. (2) is the equation of the surface of the sphere, with its centre at origin. We may use the equation of constraints to eliminate one co-ordinate from, the set of three co-ordinates and we are left with two co-ordinates 0 and o to describe the position, of the particle completely. This can be performed conveniently by using spherical polar co-ordinates, Scanned by TapScanner

Page 2 : CLASSICAL MECHANICS, 14, when we select 0 and o as independent co-ordinates., In a vertical plane (which is a two-particle system connected by an inextensible light rod and, suspended by a similar rod fastened to one of the particles). We would require, four co-ordinates (two for each particle) to describe the system completely : but, two of them are eliminated by the equations of constraints viz. distances to, particle 1 should be constant when measured from point of suspension O and, from particle 2. Then two convenient co-ordinates would be the angles, 0; and 02 shown in the figure., Example 3: We take up a third example of double pendulum shown in fig. (1.5) moving, Example 4 : As a last and final example, we consider a rigid body. A, rigid body is defined as a system of particles in which the relative distances of, the constituent particles are fixed and cannot vary with time. In this case,, constraints are expressed by equation of the form, 8., 21, rij=Cij;, Fig. 1.5., in which c; ; are constants and r; ; denote the distance between i" and j", particles. In terms of co-ordinates r; (x; y; z;) and r; (x; y; z;) with respect to the, origin we have these conditions expressed as :, (x; – x;)² + (V; – y;)² + (z; – z;)² = (c; j)?., All these instances serve to demonstrate that each constraint, which can be expressed in the, form of an equation like (1), (2), (3) or in the general case of an N particle system as an equ, ... (3), %3D, |, connecting the co-ordinates of the particles having the form, (4), f (x1, Y1, 21; x2, Y2, 22 : Xn, Yn, Zn ; t) = 0, where time t may occur in case of constraints which may vary with time, and enables us to, eliminate one of the co-ordinates by choosing co-ordinate in such a manner that it is held constant, by the constraint. For a rigid body containing N particles, there are N (N – 1) pairs of particles., It is not difficult to show that it is sufficient to specify the mutual distances of the (3N - 6) pairs,, if N > 3. Hence we can replace the 3N cartesian co-ordinates originally needed, had the system, been free from constraints, by 3 co-ordinates of centre of mass and 3 co-ordinates describing the, orientation of the body (refer to chapter on rigid body motion). Now (3N – 6) distances are constant, and the problem is reduced to one of finding the motion in terms of only 3 plus 3 = 6 co-ordinates., .., |, Classification of constraints:, A constraint is, if constraint relations do not explicitly depend on time e.g., in case, of a rigid body., if constraint relations depend explicitly on time e.g., a bead, sliding on a moving wire., if constraint relations are or can be made independent of velocity, e.g., a cylinder rolling without sliding down an inclined plane., if constraint relations are not holonomic; that is, these relations, are irreducible functions of velocities eg., a sphere rolling, without sliding down an inclined plane., if the constraint relations are in the form of equations e.g., in case, (i) (a) Scleronomic, :, (b) Rheonomic, (i) (а) Нoloпоmiс, (b) Non-holonomic, :, (iii) (a) Bilateral, :, of rigid body., if the constraint relations are expressed in the form of, inequalities e.g., motion of molecules in a gas container, or motion, of a particle on the surface of a sphere under the action of, gravity-one time rolling on the surface of sphere and other time, leaving the surface (r² 2 a²)., (b) Unilateral, Scanned by TapScanner

Page 3 : MECHANICS OF A PARTICLE AND SYSTEM OF PARTICLES, 15, if forces of constraint do not do any work and total mechanical, energy of the system is conserved while performing the constraint, (iv) (a) Conservative, motion e.g., simple pendulum with rigid support., if the forces of constraint do work and the total mechanical, encrgy is not conserved c.g., pendulum with variable length., Now let us explain the case of a cylinder and a sphere rolling without sliding down an inclined, plane – a case of holonomic and a case of non-holonomic constraint respectively : Note that as the, point of contact is not sliding, frictional forces do not do any work. That means the total mechanical, of the rolling body is conserved and so the constraint is conservative. Let us take the case, of a cylinder of radius, r rolling down an inclined plane with its axis always horizontal. The position, of the cylinder can be located by two coordinates s and 0, where s is the distance of the axis of, cylinder measured along the inclined plane that the cylinder has moved and 0 is the angle that a, fixed radius in the cylinder has rotated from the radius to the instantaneous point of contact with, the plane. If the cylinder happens to roll without slipping, then the velocities s and 0 are such that, (b) Dissipative, energy, s = r0, de, dt, ds, .. (5), %3D, or, |, dt, which on integration yields a relation between coordinates. That is, s - re = à comstant,, and so the constraints are holonomic. That is to say, that sometimes the constraints are specified, by a restriction on the velocities rather than on coordinates b it velocity equation is integrable to, yield a relation of type of eqn. (4) and so constraints are still holonomic., However, in case of a spherical ball rolling down the inclined plane without slipping, the, angular velocity w can not simply be written as 0 (= d0/dt). That is, o is generally not expre ible, in the form of a total time derivative of any single coordinate. Consequently, equation of constr.int, can not be integrated and reduced to holonomic form. So in this case constraint is non-holonomic., Constraints in some cases :, System, Type of Constraint, Scleronomic, holonomic, bilateral, conservative, Rigid body, Deformable bodies, Simple pendulum with rigid support, 1, 2., Rheonomic, holonomic, bilateral, dissipative, Scleronomic, holonomic, bilateral, conservative, Rheonomic, holonomic, bilateral, dissipative, Scleronomic, holonomic, unilateral, conservative,, An expanding or contracting spherical Rheonomic, holonomic, unilateral, dissipative, 3., 4., Pendulum with variable length, 5., Gas filled hollow sphere, 6., container of gas, FORCES OF CONSTRAINTS :, There is one significant point to note. Constraints not only interfere with the solution of the, problems in that the co-ordinates are no longer independent but they are always associated with, the forces by virtue of which they restrict the motion of the system. Such forces are termed the, forces of constraints. We generally formulate the laws of mechanics in a way that the work done by, the forces of constraints is zero when the system is in motion. By this we do not mean that the, physical motion should be allowed to happen in a way not consistent with constraints; but we only, require our formulation to sidetrack the effect of forces of constraints without violating them., Scanned by TapScanner

Page 4 : CLASSICAL MECHANICS, 16, orces of constraints in the case of a bead sliding on the wire is the reaction by the wire, aerted on the bead at each point. Similarly the surface of sphere exerts a reaction force on the, particle normally at each point. In the case ef a rigid body the internal potential energy is constant, m time since this is a function of mutual distances of its particles which are held fixed and the, internal forces act along the line joining the particles, any displacement of the rigid body then can, cake place perpendicular to these distances and hence to these forces which then do no work and, S0 potential energy remains constant, Work done by the reaction forces likewise vanishes., These forces are, however, artificial, which are reactions of wire, walls, surface etc. In systems, Of physical interest such as atomic systems, constraints are introduced only as postulates and are, always simple to express., DEGREES OF FREEDOM, The number of independent ways in which a mechanical system can move without violating, any constraint which may be imposed, is called the number of degrees of freedom of the system It, is indicated by the least possible number of co-ordinates to describe the system completely. For, example, when a single particle moves in space, it has three degrees of freedom, but if it is, constrained to move along a certain space curve, it has only one. Similarly, a rigid body rotating, about an axis fixed in space has only one degree of freedom-that of rotation angle o about the, axis. We then conclude that imposing constraints is a way of simplifying the problems, mathematically in that the number of equations of motion are reduced to the same number as the, number of degrees of freedom. In a system of N particles subjected to k independent constraints, expressible in k equations of the form, g1 (x1, Y1, ... ZN, t) = a 1, g2 (x1, y1,, ZN, t) = a2,, %3D, ..., ... (6), ... ..., ..., .., ..., .., ..., gk (x1, y1,, ZN, t) = ak,, the number of degrees of freedom f are 3N – k ; that is, %3D, ..., f = 3N – k., (7), In eqs. (6), g1,82, ... gk are k specified functions of 3N co-ordinates and possibly of time if, constraints depend on time explicitly., 1.6. GENERALISED CO-ORDINATES, To describe the configuration of a system, we select the smallest possible number of variables., These are called the generalised co-ordinates of the system. We shall not restrict our choice only to, cartesian co-ordinates. In many cases these are not the most convenient co-ordinates in terms of, which we are to describe the motion of the system. A set of generalised co-ordinates is any set of, co-ordinates which describe the configuration. We wish, sometimes, to introduce not all, co-ordinates with respect to a fixed co-ordinate system but some of them may be selected with, respect to a new origin or to a moving co-ordinate system. For example, in dealing with rigid body, motion, we specify three cartesian co-ordinates to locate the centre of mass with respect to an, external origin and three angle co-ordinates relative to origin at the centre of mass. Thus the, generalised co-ordinates should all be chosen the conventional orthogonal position co-ordinates or, all may be angle co-ordinates. In fact all sorts of quantities may be impressed to serve as, generalised co-ordinates. Thus the amplitudes in a Fourier series expansion a, may be used as, generalised co-ordinates, two angles with the vertical in a double pendulum, the distance s along, the path of motion from the equilibrium position in case of a bob of pendulum moving in a vertical, plane, or we may find it convenient to employ quantities having dimensions of energy, angular, momentum or time., *Constraints in the case of a rigid body are not independent if we express them in terms of the constancy of, mutual distances (see chapter on rigid body motion)., Scanned by TapScanner

Page 5 : MECHANICS OF A PARTICLE AND SYSTEM OF PARTICLES, In doing so we must be guided by the following three principles :, (ii) They may be varied arbitrarily and independently of each other, without violating the, 17, Question arises how to choose a suitable set of generalised co.ordinates in a given situation ?, In doing so we must be guided by the following three princinles, (i) Their values determine the configuration of the system, (ii) They may be varied arbitrarıly and thndependently of each otver, without violating the, :, constraints on the system., (iii) There is no uniqueness in the choice of generalised co-ordinates. Then our choice should, fall on a set of co-ordinates that will give us a reasonable mathematical simplification of, по, us, the problem., Notation for generalised co-ordinates : Generansed co-ordinates are designated by letter, a with numerical subscripts; q1, 42, ·… In represent a set of n generaiised co-ordinates; or,, alternatively, by a letter subscript to q and specifying within brackets the numerical values that, the letter subscript is allowed to take, e.g., q; (j = 1; 2, ... n). When we switch over to describe a, specific problem, the symbols q1, 92 ... correspond to co-ordinates that we choose to describe the, motion. Thus when a particle moves in a plane, it may be described by cartesian co-ordinates, x, y o1 the polar co-ordinates, r, 0 and so on, and we write:, ..., 91 = x, 91 = r = v(x² + y²),, or, 92 = y, 92 = 0 = tan-1Y ., ... (8), When the problem involves some spherical symmetry, it is suitable. to, co-ordinates :, use spherical, 91 = r = (x² + y² + z²,1/2,, 92 = 0 = cot-1, (x² + y²j!/2, (9), 93 = 0 = tan-1 Y, If it is preferred to accept a co-ordinate system moving uniformly with velocity v in x- direction,, generalised co-ordinates are, 91 = x – xt., 92 = y, * = v = constant., ... (10), 93 = z., For a rod lying on a plane surface, capable of taking any orientation, the suitable choice of, co-ordinates to describe the configuration of the rod will be the cartesian co-ordinates 5 and n to locate, any point A of the rod and angle 0 indicating the orientation of the rod with respect to the co-ordinates, axes OX and OY. Then, 91 = 5,, 92 = N,, .. (11), 93 = 0., %3D, If x, y denote the cartesian co-ordinates of any other point B of the rod, distant r from the previous, fixed point A, we have the connection between x, y and 5, n, 0 as, y = n +r sin 0, = q2 +r sin q3, .. (12), x = 5 +r cos 0, = 91+r cos 93, relative to axes OXY. In the transformation relaions, r is simply a number. It cannot be treated as, distinct co-ordinate; for, r cannot be varied without violating the constraint that the distance of two, particles must remain constant in time. That is why we suppress explicit reference to such, numbers., Scanned by TapScanner