SpaceTime Structure, Newton s Laws and Galilean Transformation


 Christopher Todd
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1 Chapte 1 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation What is motion? The phase change of position with time expesses the basic idea of motion. We say that an object is moving if it occupies diffeent positions in space at diffeent times. In contast, if an object does not change its position in space as time flows, we say that the object is at est. Inheent in this desciption of est and motion is the obseve (that is we ) and the concepts of a sepaate space and time. Ou peception tells us that space exists as a stationay, pemanent and absolute backgound in which we can place an object, o though which an object can move without any inteaction with it. Newton fomulated this idea by asseting, Absolute space, in its own natue, without elation to anything extenal, emains always simila and immovable. Time, accoding to Newton, is also absolute and flows on without elation to the pesence of any physical object o event. In his wods, Absolute, tue, and mathematical time, of itself, and fom its own natue, flows equably without elation to anything extenal, and by anothe name is called duation. In this Newtonian absolute space and time, motion of an object is chaacteized by change of its position in space as time evolves. Howeve, since this abstact absolute space cannot be seen and/o no pat of it can be distinguished fom anothe, how do we fix the position of an object in this space? Newton egaded the distant stas in the sky at est and defined the positions of othe objects in elation to them. Howeve, astonomes now tell us that none of these cosmological objects in the Univese ae at est. Thee is no object at est in this absolute space with espect to which we can define the position of anothe object. Theefoe, at any instant of time, the position of an object is always defined by the obseve with espect to himself. An obseve, theefoe, fist defines a local space, in which positions of othe objects ae then descibed. This local space, to which is attached the obseve, is called a fame of efeence. The concepts of est and motion ae theefoe elative, that is, these ae in elation to a fame of efeence o obseve. A house built on Eath does not change its position in the local space o fame of efeence associated with Eath; howeve, as obseved fom a fame of efeence attached to Moon fo example (o in othe wods, in the local space associated with Moon), its (house on Eath) position continuously changes. Thus, even within the Newtonian point of view, space does not exist on its own but is a elative space. Time, on
2 Mechanics the othe hand, is same and flows equably eveywhee, i.e., fo example its ate of flow is same both on Eath and Moon. 1.1 FRAMES OF REFERENCE AND COORDINATE SYSTEMS A fame of efeence is usually a collection of objects, togethe with the obseve, which ae at est elative to each othe*. Thus, fo example, we at est elative to Eath along with Eath and laboatoies built on Eath, etc., constitute a fame of efeence. It is with espect to this Eath s fame of efeence that we obseve and measue the changes of positions of othe objects. Simila fames of efeence can be associated with othe obseves (o objects) who may not be at est elative to us. Within a fame of efeence, we set up a coodinate system which is used to measue the position of an object. Measuements indicate that the space we inhabit is a theedimensional space. This means that we need thee numbes to specify the position of any (point) object in the space. The choice of numbes is detemined by the type of coodinate system that we use. The geneally used coodinate systems ae ectangula (Catesian) coodinates (x, y, z), spheicalpola coodinates (, θ, ), and cylindical coodinates (ρ,, z). A coodinate system may be set up anywhee in the fame of efeence, the choice of oigin has complete feedom. Given the oigin, a Catesian (ectangula) system is fomed by having thee mutually pependicula staight lines, the X, Y, and Z axes. Thee ae two independent ectangula systems possible in space, the lefthanded and the ighthanded, which cannot be made to coincide with each othe by any means of tanslation o otation. Catesian system, by convention, is chosen to be ighthanded, as shown in Fig The position of a paticle** P elative to oigin O is given by position vecto, chaacteized by its specific length and diection: = x i+ y j+ z k...(1.1) X Z O i k P (x, y, z) j Y whee i, j, kdenote constant unit vectos in the diections of X, Y, Z axes. We talk of a point, staight line, pependicula staight lines, length of line segment, etc. in the Euclidean sense. Ou expeience tells us that Euclidean geomety is applicable in ou odinay Newtonian space. This is valid if the objects ae not vey massive, o ae not moving with vey high speeds of the ode of speed of light. If the position of the paticle P changes, the diffeential change in its position vecto is, in geneal, given by, Fig. 1.1: Catesian coodinates. d = dx i+ dy j+ dz k...(1.) * The collection of such point objects constitute the concept of a igid body. ** We define paticle as a mass concentated at a point. The paticle is theefoe a point object.
3 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 3 The instantaneous velocity and acceleation of P, in Catesian system, theefoe ae d dx dy dz v = = i+ y+ k dt dt dt dt = v i+ v j+ v k...(1.3) x y z and dv dv dv x y dvz a = = i+ j+ k dt dt dt dt = a i+ a j+ a k...(1.4) x y z SpheicalPola Coodinates The position of paticle P elative to oigin O, in spheical pola coodinate system, is descibed by adial distance, pola angle θ, and azimuth, as shown in Fig. 1.a. Geomety of the figue shows the elations between Catesian and spheical pola coodinates: x = sin θcos = x + y + z...(1.5a) y = sinθsin tanθ = x + y z...(1.5b) z = cosθ tan = y x...(1.5c) Z Z S cos θ = z P (, θ, ) P e P e e θ θ θ X x O sin θ y Y X T Y (a) (b) Fig. 1.: Spheical coodinates. The unit vecto e points towads axis, i.e., in the diection of vecto along which only coodinate changes, θ and emain fixed. Hence, we can wite = e. Unit vecto e θ is tangent at P to cicle SPT; displacement along cicle SPT changes only coodinate θ, distance and angle emain fixed. Unit vecto e is tangent at P to cicle PP poduced
4 4 Mechanics by the otation of OP about Zaxis; displacement along this cicle changes only coodinate, distance OP = and angle θ emain fixed; see Fig. 1. (b). The geneal diffeential displacement of paticle P in spheical pola coodinates is given by, d = de + dθe + sin θde...(1.6) θ One can satisfy himself that the above elation is tue by consideing special cases: (i) if dθ = d = 0, d = d e gives displacement vecto when distance changes by d along line OP; similaly, (ii) if d = d = 0, and angle θ changes by dθ, displacement vecto d = ( dθ e θ ) and (iii) if d = dθ = 0, change of angle by d poduces a displacement d = (ρ d ) e, whee ρ = sin θ is the adius of the cicle as we otate P about Zaxis. The unit vectos e, e θ and e can be expessed in tems of i, j, k as follows (see Example 1.1 on p.6): e = sinθ cos i+ sinθsin j+ cosθ k...(1.7a) eθ = cosθ cos i+ cosθsin j sin θ k...(1.7b) e = sin i+ cos j...(1.7c) The unit vectos (e, e θ, e ), unlike (i, j, k), ae not constant vectos but change in diection as coodinates θ and change. Howeve, at each point, they constitute an othogonal ighthanded coodinate system, that is, we have e e = e e = eθ e = 0...(1.8a) e e = e, e e = e, e e = e...(1.8b) θ θ θ 1.1. Motion in Dimensions Quite often, we find a paticle constained to move in a plane, athe than in space in geneal. If we call this plane as XY plane, then position vecto of the paticle is given by, = x i + y j because z = 0 always fo the paticle duing motion. The spheical pola coodinates (, θ, ) then educe to cicula pola coodinates (, ); θ = 90 fo the paticle. That is, we define the position of the paticle in tems of adial distance fom oigin and angle such that, Y e e P (, ) O Fig. 1.3: Cicula pola coodinates. X = x + y x = cos...(1.9a) tan =y x y = sin...(1.9b) The displacement vecto d in cicula pola coodinate is obtained by putting θ = 90 and d θ = 0 in Eq. 1.6:
5 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 5 whee unit vectos e, e ae given by, d = de + de...(1.10) e = cos i + sin j...(1.11a) e = sini+ cos j...(1.11b) Velocity and Acceleation in Cicula Pola Coodinates The velocity and acceleation of a point object, moving in a plane, can be obtained in tems of pola coodinates (, ) by diffeentiating position vecto = e with espect to time: d d e v = = e + d...(1.1) dt dt dt Note that e is not a constant vecto; its time ate can be obtained by using Eqs. 1.11a, b: d e d d d = sin i+ cos j= e...(1.13a) dt dt dt dt Similaly, we obtain d e d = e...(1.13b) dt dt The velocity vecto, theefoe, is witten as v = ve + ve...(1.14) d whee v v d = dt =, = and dt = ae the adial and tangential (o azimuthal) components of velocity vecto v. To find the expession fo acceleation, we have F F HG F I HG K J F + v + HG d e dt d v dv e I e a = = e + v e HG d dv K J + + v d dt dt dt dt dt dv = v d dt dt o a = ae + ae...(1.15) whee a...(1.15a) = and a = +...(1.15b) ae the adial and tangential components of acceleation vecto a. dv dt I KJ e I KJ
6 6 Mechanics In paticula, if the object is moving in the plane, in a cicle of adius R (i.e., = R, constant), we find v = 0, v = R...(1.16a) and a = R, a = R...(1.16b) a denotes the centipetal acceleation (towads e ), and a is tangential acceleation esponsible fo incease in tangential speed v. Example 1.1: Deive Eqs. 1.7 a, b, c. Solution: We have, = xi+ y j+ zk Hence, and and Also, fom Eq. 1.6, we find = sinθcosi+ sinθsinj+ cosθk θ i θ j θk = sin cos + sin sin + cos b = cosθcos i+ cosθsinj sinθk θ b = sinθ sini+ cosj b = e θ, const. b = eθ, const. = sin θe, θ const. b g g Compaing above two sets of equations, we get Eqs. 1.7a, b, c Cylindical Coodinates In cylindical coodinates, the position of paticle P elative to oigin O is descibed by the two cicula pola coodinates (ρ, ) defined in the plane Z = 0, and the thid being the Zcoodinate itself. The cylindical coodinates (ρ,, z) ae elated to Catesian coodinates as, ρ = x + y y,tan =, = x z z...(1.17a) Convesely, x = ρcos, y = ρsin, z = z....(1.17b) The equation ρ = constant descibes a ight cicula cylinde of adius ρ about Zaxis. The equation = constant descibes the plane containing Zaxis and making an angle to g g g
7 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 7 the XZplane. The geneal diffeential displacement of paticle P in cylindical coodinates is given by d = dρe + ρde + dze Z ρ z whee e ρ and e ae elated to (i, j, k) as given by Eqs. 1.11a, b, and e z = k. The unit vectos (e ρ, e, e z ) constitute an othogonal (ighthanded) coodinate system, i.e., e e = e, e e = e, e e = e ρ z z ρ z ρ and e e = e e = e e = ρ ρ z z 0 In situations whee thee is an axis of symmety, cylindical coodinates become the ight choice e ρ X to descibe the system. A typical example is the Fig. 1.4: Cylindical coodinates. (plana) otation of a paticle about an axis. In geneal, if a paticle moves in a plane, we descibe its motion in pola coodinates (ρ, ), whee ρ gives the adial position of the paticle. (We can call the plane as Z = 0, o XYplane). The velocity and acceleation of the paticle ae then given by Eqs. 1.1 to 1.15, as discussed in last section. In paticula, if the paticle moves in a cicle of adius ρ, we have (Eq. 1.14): v= ρ e = ω ρ whee ρ = ρe ω ρ, and = ez. The vecto ω epesents angula velocity whose magnitude is and diection is about the axis of otation. The acceleation of the paticle otating about Zaxis can be ewitten as, d v d ω d ρ a = = ρ + ω dt dt dt = α ρ+ ω ω ρ whee, b g d i c ω ω ρ = ρ ρ = ρ ez ez e eρ is the centipetal acceleation, and d α ρ = ρ e e = ρ e z ρ is the tangential acceleation. Note that if the paticle is otating in the XYplane, ρ coesponds to position vecto of the paticle. Hence, one usually denotes ρ by and hoping no confusion, we wite v = ω...(1.18a) and a = ω ω + α i h c h...(1.18b) We shall fequently need above elations when descibing the motion in otating fame, and/o otating motion of an object. O ρ P ( ρ, θ, z) z e Y
8 8 Mechanics 1. NEWTON S LAWS OF MOTION In 1687, Isaac Newton aticulated the laws upon which all classical mechanics is based. In pesentday teminology these laws may be stated as follows: I Law: An object continues in its state of est, o of unifom motion in a staight line, unless it is compelled to change that state by extenal foces acting on it. II Law: The time ate of change of linea momentum of an object is popotional to the net extenal foce acting on it and occus in the diection of the esultant foce. That is, p F = k d = km a...(1.19) dt whee F is esultant extenal foce; m is inetial mass of the object, and p = m v is its linea momentum. k is constant of popotionality which is taken as 1 by defining appopiate units. III Law: When two objects inteact, the mutual foces of inteaction on each object ae equal in magnitude and opposite in diection. That is, to evey action thee is an equal and opposite eaction Comments on Newton s Laws I Law and fames of efeence: Newton s I law has two ingedients: fist, that of foce (o the absence of it) and second, that of measuement of est o unifom motion of the object. Foce is the expession fo inteaction between diffeent objects. Thee is a pioi, no way to know that thee is no foce acting on an object except by putting it so fa away fom all othe objects that we would believe that it is not acted by an extenal foce. Having such an object, Newton s I law assets that it would eithe emain at est o continue to move with constant velocity. But, with espect to whom? Obviously, if the obseve himself is sitting on an acceleated fame continuously changing his velocity, the object cannot always emain at est o in unifom motion with espect to him. A efeence fame with espect to which I law holds is called an inetial fame of efeence. Thus, Newton s fist law defines the citeion fo an inetial fame. The law says that thee exists cetain fames of efeence (called inetial fames) with espect to which an object, (believed to be) not acted upon by any extenal foce, stays eithe at est o in unifom motion in a staight line. Obviously, an inetial fame itself must be a fame moving with constant velocity, i.e., nonacceleating. The question whethe a given fame of efeence is inetial o not then becomes a matte of obsevation and expeiment. Fo most of the obsevations made on the suface of Eath, the fame of efeence attached to the suface of Eath behaves as an inetial fame to a vey good appoximation. Moe efined obsevations, which we shall discuss late, shows that it is not. Howeve, unless othewise stated, we shall efe to Eath s fame as the inetial fame. All othe fames moving unifomly in staight line elative to Eath theefoe also constitute inetial fames of efeence. Newton s I law is an independent law and is not contained in his II law: In fact, both II and III laws hold only in an inetial fame suitably chosen on the basis of I law. 1.. II Law Newton s II law is not a definition of foce: The equation, F = m a, is an equality
9 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 9 which says that the foce acting on a body detemines not the velocity but ate of change of velocity of the body. What is the natue and value of F is left unansweed in II law. Newton s II law is valid fo an object having constant inetial mass and moving with velocity small compaed to that of light. If the velocity of the object is high (i.e., a significant faction of velocity of light), we ente into the domain of special elativity. Thee we find inetial mass changes with velocity; diection of acceleation is geneally not same as that of foce; hence Newton s law needs to be evised in elativistic dynamics III Law and Pinciple of Momentum Consevation Newton s III law is the only statement made by Newton about the natue of foce. It says that no matte what is the exact fom of a foce (i.e., inteaction) between two objects, it behaves in such a way that at any instant of time, if an object 1 exets a foce F 1 on object, then object exets an equal and opposite foce F 1 on object 1. That is, at any instant which implies F + F = 0...(1.0) 1 1 dp1 dp dp + = = 0 dt dt dt o P = p 1 + p is constant in time....(1.1) Thus, when two objects inteact, the sum of thei momenta is constant. Newton s III law, altenatively, is a statement of momentum consevation fo a system of inteacting objects (in absence of any extenal foce). In its oiginal fom, Newton s III law has pofound limitations. Fo example, it is not satisfied by two sepaate chages inteacting though electomagnetic field. Popagation of electomagnetic inteaction takes place at a finite speed c, speed of light. The tansfe of momentum (i.e., tavel of foce field) fom one chage to anothe involves a time inteval equal to distance between the chages divided by c. Thus, at any instant, a change in momentum occus in fist chage without an equal and opposite change in momentum in the othe chage. That is, instant by instant, we find F F o p p Momentum consevation pinciple is eventually peseved by associating the instantaneous change in momentum of paticles with that of the electomagnetic field which caies the inteaction. In fact, no inteaction (whethe electomagnetic, o gavitational, o any othe) can popagate with speed geate than that of light. Hence, in geneal, momentum consevation pinciple fo an isolated, inteacting system of paticles is valid only when we conside paticles as well as fields that cay the inteactions as the complete system. Howeve, in this book, we shall not go into these complications and assume Newton s III law to be valid in its stong fom as given by Eq SPACETIME SYMMETRY AND CONSERVATION LAWS Fom ou expeience, we aticulate the concepts of space and time to descibe the motion of an object. It is fom the same expeience and obsevation, we ascibe cetain fundamental popeties to the concepts of space and time. One such popety is homogeneity and the othe is isotopy of space and time.
10 10 Mechanics Homogeneity of Space Space is homogeneous means one pat (o point) of space is identical o equivalent to any othe pat (o point) of space. Any physical pocess will occu the same way (o identically) unde identical conditions, no matte whee (i.e., in whateve pat of space) it occus; a paticula expeiment, whethe pefomed in Kanpu o New Yok, will yield same esult. (Incidentally Kanpu and New Yok epesent same inetial fame.) Let us apply the basic popety of homogeneity of space to an isolated system of two inteacting paticles: if both the paticles of the system ae displaced by same amount δ, thee should be no change in the state of the system o in its intenal motion. That is, the total wok done by intenal foces when the system is displaced by δ must be zeo: F δ+ F δ = 0 Since δ is abitay, we get 1 1 F1 + F1 = 0 This is Newton s III law which leads to law of momentum consevation fo the isolated system of two paticles. Thus, momentum consevation emeges fom the fundamental popety of homogeneity of space. [The above agument can be extended to an isolated system of ninteacting paticles. No wok done by intenal foces imply: b 1 1g b 13 31g... 0 δ F + F + δ F + F + = whee the seies includes all pais of two paticles out of npaticle system. That is, in shot, we can wite δ F + F =, i j =... n i j d ij jii 0 1 j Since δ is abitay, we find d F ij + F jii = 0, i j = 1,..., n; i j] i j The comments made in Sec on Newton s III law also apply to the above poof of momentum consevation (of paticles) fom homogeneity of space. If we conside the foces popagating with finite speed, homogeneity of space leads to consevation of momentum of complete system, viz, both paticles and the caie of foce. Since space is homogeneous, we can choose the oigin of ou coodinate system anywhee we wish. Shifting the oigin means displacing the system; and it does not affect the pocesses Homogeneity of Time Time is homogeneous means one instant of time (o duation) is identical to any othe instant (o duation) of time. An expeiment, whethe pefomed today o tomoow o a yea late will yield same esult unde othewise same conditions. Let us discuss the effect of concept of homogeneity of time on Newton s laws of motion. Homogeneity of time implies that the foces in natue should not depend explicitly on
11 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 11 time. [Fo example, the foce F of gavitation between two point masses m 1 and m sepaated by distance emains the same at time t (today) and t (tomoow).] That is, fo all kinds of foces, we have F t = 0 Now if a foce is consevative, we can define a potential enegy of inteaction U such that the foce is negative gadient of U, i.e., we have U F x F U y F U x =, y =, z = z This means that if F does not depend explicitly on t, then equivalently, potential enegy U also is not an explicit function of t, o, b U = U x, y, z whee x, y, z efe to coodinates of the paticle unde consideation in a given inetial fame. The explicit independence of U on t leads to consevation of mechanical enegy of the paticle. To show this, let us wite the total enegy of the paticle as, 1 E = mv + U Hence, we get de m d du = v v + dt dt dt F HG d g U dx U dy U dz = v F+ + + x dt y dt z dt = v F+ Fv Fv Fv x x y y z z = 0 O, E emains constant in time. Thus the law of consevation of enegy, in Newtonian mechanics, follows diectly as a consequence of homogeneity of time. Homogeneity of time implies that we can choose the zeo of time at any instant fo the obsevation of a physical pocess Isotopy of Space Space is isotopic means one diection in space is identical o equivalent to any othe diection. A paticula expeiment (i.e., physical pocess) will yield the same esult whethe ou laboatoy faces Noth o West. That is, an angula displacement of an isolated system does not change the intenal state of the system, no its intenal motion. When applied to an isolated system of two inteacting paticles, isotopy of space implies that the net wok done by intenal foces must be zeo when the system is otated by an angle d. i I KJ
12 1 Mechanics That is, d F + d F = whee d1 and d ae displacement vectos of paticles 1 and espectively. We know that v = ω ; that implies, d = d, and d = d 1 1 Thus, we find dd 1i F 1 + dd i F 1 = 0 o d F + d F = Since d is abitay, we get b 1 1g b 1g 0 1 F1 + F1 = 0 that is, the net toque poduced on the system by intenal foces is zeo. This implies that the angula momentum of the system emains constant. The above analysis can be extended to an npaticles system. Thus, consevation of angula momentum of an isolated system emeges as a consequence of a fundamental popety of space, viz., its isotopy. Isotopy of space implies we can oient ou coodinate axes in space in any way we wish Isotopy of Time Isotopy means equivalence of diections; isotopy of time means equivalence of diections of time. Howeve, we usually efe to only one diection of time, the fowad diection, fom pesent to futue. Nevetheless, we can conceive of a backwad diection of time, i.e., fom pesent to past. Theefoe, the question whethe time is isotopic o not means whethe time is evesible o not? Opeationally, it means to find out what happens if we change t by t? Newton s laws of motion do not change if we eplace t by t in it: m d d t b g m d = = F dt Since F does not explicitly contain t, it emains unaffected. Thus, the motion descibed by Newton s laws emains same when we eplace t by t. What it means is the following: conside a physical pocess govened by Newton s laws, as fo example, motion of a paticle thown vetically upwads. It goes up, stops, and then falls back unde gavity. Let us take a movie of the pocess as it poceeds fowad in time. If we now view the movie in evese, we look at the pocess occuing backwad in time. The entie pocess looks nomal: paticle goes up, stops, and falls back exactly accoding to Newton s laws. Thus, Newton s laws ae evesible in time. Pocesses in systems compising of lage numbe of paticles do not show time evesibility. As a typical example, conside a glass which falls fom a table and shattes into small pieces on the floo. If we un a motion pictue of the above pocess in evese, we see the pieces gatheing togethe on the floo into a whole glass, which then jumps onto the
13 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 13 table. This does not happen in natue. Thus, while each piece of glass follows Newton s laws which ae time evesible, the collective system beaks time evesibility. This is explained by II law of themodynamics which pohibits a macoscopic system to move fom less odeed to a moe odeed state (law of incease of entopy). Isotopy of time does not lead to any specific consevation pinciple in classical mechanics. To sum up, in this section, we descibed the spacetime symmeties and thei consequences in undestanding the natue of foces and consevation laws, within a given inetial fame. Now, we shall see how Newtonian spacetime stuctue connects motion as obseved fom two diffeent inetial fames. 1.4 PRINCIPLE OF RELATIVITY AND GALILEAN TRANSFORMATION It was fist obseved by Galileo, in multitude of expeiments, that mechanical phenomena occu identically in all inetial fames. (Late, this was expeimentally veified fo electomagnetic phenomena as well.) These obsevations of Galileo wee stated by Newton in the following wods: The motions of bodies included in a given space ae the same among themselves, whethe that space is at est o moves unifomly fowad in a staight line. The above statement, known as Galilean pinciple of special elativity, means that all phenomena in natue should be elated in same way in all inetial fames. In moden teminology, it says that the laws of natue must emain fominvaiant as we go fom one inetial fame to anothe. The measuements of the motion of a paticle P (i.e., its position, velocity, and acceleation at any instant) fom two diffeent inetial fames S and S leads to two sets of values. The elationships between these two sets of measuements ae called Galilean tansfomation equations. One of the efeence fames S is (abitaily) called stationay while the othe S is called moving. The two fames must be moving with a constant velocity V elative to each othe. Othewise, the two fames become same. Both the fames (o obseves) S and S may use Catesian coodinate systems fo measuements. Let us denote the coodinates of paticle P in fame S by = (x, y, z) and time t, and in fame S by = (x, y, z ) and time t ; t and t denote the time ead by clocks attached in fames S and S, at the instant when position coodinates of P ae espectively and. X Z O S Z R P Y S Y O X Z Y O R Z Y O P X Fig. 1.5 (a) Fig. 1.5 (b)
14 14 Mechanics In paticula, if we assume that oigins O and O of both the coodinate systems coincide at t = t = 0, then at any late time t = t, the oigin O of coodinate system S will be at a distance R = Vt fom the oigin O of coodinate system S, whee V is the constant velocity with which S moves elative to S. Hence, if and ae position vectos of P as obseved fom S and S at that instant, then we have (see Fig. 1.5a). = R+ = Vt +...(1.1a) o = Vt...(1.1b) The above elation, along with t = t, ae known as Galilean tansfomations. As a special case, if the fame S moves paallel elative to fame S with X axis coinciding with Xaxis, as shown in Fig. 1.5b, the Galilean tansfomations ae given in components as following: x = x Vt y = y...(1.) z = z t = t Galilean tansfomations ae not the tansfomation equations between two coodinate systems within a single inetial fame, like e.g., Eq. 1.5 which elate ectangula to spheical pola coodinates; these ae elations between two diffeent inetial fames. 1.5 INVARIANTS OF GALILEAN TRANSFORMATIONS The numeical values of vaious physical quantities geneally change unde a coodinate tansfomation. Tivial example is the value of position coodinates (x, y, z) of a paticle P, which change to (x, y, z ) unde the tansfomation. It means that the value of position coodinates itself has no objective eality, o in othe wods, it is not an objective popety of the paticle. It is simply the geometical position of the paticle elative to a paticula coodinate system. The physical quantities whose values do not change unde a tansfomation ae called the invaiants of the tansfomation. Such physical quantities eflect the objective popeties of the object (o phenomenon) unde consideation in the sense that these popeties emain independent of the choice of the coodinate system. In the following discussion, we look into the invaiants of Galilean tansfomation Invaiance of Acceleation Diffeentiating Eq. 1.1 with espect to time, we find that d dr d = + dt dt dt o v = V + v...(1.3) d d whee v = = is the velocity of P elative to S ; V is velocity of S elative to S, and dt dt v is velocity of P elative to S. Eq. 1. descibes well known elation of velocity addition in Newtonian mechanics.
15 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 15 Diffeentiating once moe, we get d v d v a = = = a...(1.4) dt dt because V is constant. That is, acceleation of paticle P is same when obseved fom two diffeent inetial fames. Acceleation of the paticle P, theefoe, emains invaiant unde Galilean tansfomation Invaiance of Space and Time Intevals Suppose thee ae two points P 1 and P whose position vectos ae ( 1, ) and b1, g as obseved fom two fames S and S espectively, at any given instant of time t = t. Then, accoding to Galilean tansfomation, we have 1 = 1 Vt and = Vt whee V is the constant velocity of S elative to S. Hence, we get 1 = 1 o l = 1 = 1 = l...(1.5) Eq. 1.5 means that, bx x1 g + by y1 g + bz z1 g b 1g b 1g b 1g = x x + y y + z z l (o ) denotes the space inteval, o spatial distance between the two points. It may also epesent the length of a igid od, fo example, whose end points ae P 1 and P. Thus, we find that space inteval, o length of a igid object in geneal, emains invaiant unde Galilean tansfomation. Similaly, if two events occu at times t1 = t1 and t = t, the time inteval between these events as obseved fom two inetial fames is same, that is, t t1 = t t1 In othe wods, time inteval is also an invaiant of Galilean tansfomation Galilean Invaiance of Newton s Laws and Natue of Foces Pinciple of (special) elativity demands that all laws of natue should be valid in all inetial fames. This is obviously satisfied by Newton s I law: if the velocity of a paticle is constant (v) in fame S, it emains constant (v ) in fame S as well, v = v V...(1.6) because elative velocity (V) between the two inetial fames is a constant. Let us now look into Newton s II law. Suppose a paticle of mass m moves with the acceleation a due to an extenal foce F, as obseved fom fame S.
16 16 Mechanics Then we have, F = m a...(1.7) If the II law has to be fominvaiant unde Galilean tansfomation, then in fame S, we should get the following elation: F = m a...(1.8) whee F and a ae the foce and acceleation as obseved fom S. It is assumed that inetial mass m, being a scala, is a fame independent quantity. We know that acceleation is an invaiant of Galilean tansfomation, i.e., a = a. Hence, if both the Eqs. 1.7 and 1.8 have to hold, then we must have, F = F...(1.9) The above condition imposes a sevee condition on the natue of foces; that is, foces appeaing in natue must be invaiants of Galilean tansfomation if Newton s II law has to be valid in all inetial fames. How do we know that F = F? What kind of function F (o F ) should be in ode to satisfy above equality? It tuns out that all the foces we encounte in Newtonian dynamics depend only on the elative position o elative velocity of two inteacting bodies (and not on thei absolute position o velocities in space). That is, as obseved fom fame S, a foce exeted by paticle on paticle 1 is found to have the following functional dependence, b g...(1.30) F = f, v v Hence, as seen fom S, the same foce would tansfom to, b F = f, v v Since, unde Galilean tansfomation, we have 1 = 1 and v v1 = v v1...(1.31) we find F = F 1 1 That is, the foces found in natue happen to be invaiants of Galilean tansfomation, thus making Newton s II law valid in all inetial fames. Futhe, if foces in natue have a functional dependence given by Eq. 1.30, then we find F = f, v v = f, v v = F g b g b g...(1.3) which is the statement of Newton s III law. Thus, the III law implicitly ensues that both II and III laws emain valid in all inetial fames. (Note that F1 = F1 = F1 = F1.) All the Newton s laws of motion theefoe emain fominvaiant unde Galilean tansfomation in a selfconsistent manne. Example 1.: Show that the consevation of total momentum of an isolated system of two colliding paticles in the pocess of collision follows fom the pinciple of Galilean invaiance and the law of consevation of enegy.
17 SpaceTime Stuctue, Newton s Laws and Galilean Tansfomation 17 Solution: To show this, let us conside the collision of two paticles 1 and as obseved fom an inetial fame S. Suppose initially 1 and, moving with velocities u 1 and u, ae so fa apat that they do not possess any potential enegy of inteaction. The total enegy of the system of both paticles is just thei kinetic enegy, 1 1 mu mu Afte they collide and sepaate fa apat, let thei velocities be v 1 and v 1. Accoding to law of consevation of enegy in fame S, we get mu mu = mv mv + E...(1.33) whee E is the change (loss o gain) in intenal enegy of the paticles duing collision. If E = 0, collision is elastic; othewise inelastic. Now let us view the pocess fom anothe inetial fame S moving with velocity V elative to S. Accoding to Galilean tansfomation, the velocities of paticles as obseved fom S ae, u = u V, u = u V (befoe)...(1.34a) 1 1 and v1 = v1 V, v = v V (afte)...(1.34b) Now we intoduce the idea that pinciple of consevation of enegy is fame independent. In fact, consevation of mechanical enegy being a consequence of homogeneity of time, is fundamentally an invaiant pinciple in Newtonian mechanics. As shown by expeiments, we futhe assume that intenal enegy change E also emains invaiant unde Galilean tansfomations. Hence, consevation of enegy in collision pocess as obseved in S, gives b g b g b g b g...(1.35) m1 u1 + m u = m1 v1 + m v + E Using Eq. 1.34, we find bg u = u V u1 V b 1g b g b 1g b g etc. Substituting fo u, u, v, v in Eq. 1.35, we get b g b g mu mu m1u1 + mu V = mv mv m1v1 + mv V + E Compaing above with Eq. 1.33, we get m u + m u = m v + m v which is the law of consevation of momentum in collision pocess. Thus, the pinciple of consevation of enegy and the concept of Galilean invaiance shows that momentum emains conseved duing collision, whethe elastic o inelastic. Of couse, momentum consevation in collision pocess follows athe staightfowad fom Newton s III law. If we conside the two colliding paticles as an isolated combined system on which no extenal foce acts, then the momentum of the system must emain
18 18 Mechanics conseved, no matte whethe they collide o do something else. Moeove, momentum consevation being a consequence of homogeneity of space must also be a fundamentally Galilean invaiant pinciple. That is, in fame S, we have m u + m u = m v + m v SOME BASIC FORCES IN NATURE A foce expesses the inteaction between two diffeent objects. We ae familia with a lage vaiety of foces such as gavitational foce, electic and magnetic foce, and intemolecula foces giving ise to dy fiction, viscosity, suface tension, tension in the sting, o elastic foce in a sping, etc. We ae also familia with nuclea foces esponsible fo binding of potons and neutons togethe in a nucleus, o the splitting of a neuton into a poton, electon and antineutino in betadecay. Among this lage divesity of tems, it tuns out that thee ae only fou types of basic foces (known at pesent) in natue, viz., 1. Gavitational foce. Electomagnetic foce 3. Stong nuclea foce 4. Weak foce. We shall now biefly outline the basic chaacteistic of these foces. 1. Gavitational foce: The gavitational inteaction is a univesal phenomenon which acts between all foms of matte and enegy. The quantitative natue of gavitational foce was fist given by Newton; accoding to Newton s law of gavitation, between any two mateial paticles thee exists an attactive foce of gavitation, given by F 1 G mm 1 = 1 whee m 1 and m ae the gavitational masses and 1 is the elative distance between the (inteacting) paticles. The value of gavitational constant G is about Nm /kg. The gavitational inteaction plays significant ole only when masses involved ae lage. It becomes the decisive foce when we study motions of planets, stas, o othe such cosmic objects. We shall study moe about the natue of gavitational inteaction in Ch 7. The gavitational inteaction between eath and othe elatively smalle objects nea the suface of eath poduces an almost unifom acceleation due to gavity g = 9.8 m/s (appoximately) in all these objects. Hence, in consideing the motion of any such pototype object, e.g., a block o a sphee of mass m on the suface of eath, it is implicit that gavity foce mg is acting on the object. On the othe hand, if we ae consideing the motion of micoscopic paticles like electons etc., the gavity foce is completely neglected because of the tiny masses of these paticles (as we discuss in Ch. 3; also see Poblem 5, set A, Ch. 1).. Electomagnetic foce: The second kind of fundamental foce is the electomagnetic foce that acts between chaged paticles. The fist law of electomagnetism is the Coulomb s law of electic foce between two chages q 1 and q fixed (i.e., kept at est) at elative distance 1. The Coulomb s law states that the electic foce between these static chages is given by 1 qq 1 F1 = 4 πε 0 1
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