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- 1. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics What is mass?Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics Wang Xiong 2012-08-11 at http://www.duobei.com/room/6758541343Wang Xiong Email:wangxiong8686@gmail.com 1
- 2. Thanks for all classmates! Thanks to duobei.com! 2
- 3. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsContents1 Introduction 42 Mass in Newtonian Mechanics 10 2.1 The amount of matter . . . . . . . . . . . . . . . . . . . 16 2.2 The inertia mass . . . . . . . . . . . . . . . . . . . . . . 19 2.3 The gravitational mass . . . . . . . . . . . . . . . . . . . 21 2.4 The additive of mass . . . . . . . . . . . . . . . . . . . . 28 2.5 Mass is conserved quantity . . . . . . . . . . . . . . . . . 29 2.6 Mass is invariant . . . . . . . . . . . . . . . . . . . . . . 303 Mass in Lagrangian Mechanics 31Wang Xiong Email:wangxiong8686@gmail.com 3
- 4. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics1 IntroductionWhat is physics?Physics is the study of matter and its motion through space and time. Sothe fundamental concepts of physics are matter, space-time, and motion.Wang Xiong Email:wangxiong8686@gmail.com 4
- 5. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsWhat are the most important questions?The fundamental questions of physics are: 1. What is the physical world made of? 2. What is the structure of space-time? 3. How do things change through space-time? The deeper understanding of these fundamental concepts and moreprofound answers to these fundamental questions, indicate the develop-ment of physical theory.Wang Xiong Email:wangxiong8686@gmail.com 5
- 6. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsMatter and MassOf these fundamental concepts, mass is the modeling of matter. On the one hand, diﬀerent properties of matter may lead to diﬀerentconcepts of mass. So we may discover many diﬀerent physical attributeof matter, and use many diﬀerent concepts of mass. On the other hand, some concepts may appear very diﬀerent at ﬁrst,but as we see deeper and deeper, we ﬁnd a single and uniﬁed origin ofthese diﬀerent concepts of mass.Wang Xiong Email:wangxiong8686@gmail.com 6
- 7. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics This kind of development of physic concept is very interesting andprofound. So one may distinguish conceptually between many diﬀerent attributesof mass or diﬀerent physical phenomena that can be explained using theconcept of mass There is no doubt that the problem of mass is one of the key problemsof modern physics.Wang Xiong Email:wangxiong8686@gmail.com 7
- 8. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsOverview of these series of talks“To see a World in a Grain of Sand, And a Heaven in a Wild” We will try to see the development and the whole picture of theoreticalphysic through the evolution of the very fundamental concept of mass. 1. The inertial mass in Newtonian mechanics 2. The Newtonian gravitational mass 3. Mass in Lagrangian formulism 4. Mass in the special theory of relativity 5. E = M C 2Wang Xiong Email:wangxiong8686@gmail.com 8
- 9. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics 6. Mass in quantum mechanics 7. Principle of equivalence and general relativity 8. The energy momentum tensor in general relativity 9. Mass in the standard model of particle physics 10. The higgs mechanismWang Xiong Email:wangxiong8686@gmail.com 9
- 10. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics2 Mass in Newtonian MechanicsWang Xiong Email:wangxiong8686@gmail.com 10
- 11. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsNewton’s three laws of motionNewton’s laws of motion are three physical laws that form the basis forclassical mechanics. They describe the relationship between the forcesacting on a body and its motion due to those forces. They have beenexpressed in several diﬀerent ways over nearly three centuries, and canbe summarized as follows: 1. First law: Every object continues in its state of rest, or of uniform motion in a straight line, unless compelled to change that state by external forces acted upon it. 2. Second law: The acceleration a of a body is parallel and directly proportional to the net force F acting on the body, is in the directionWang Xiong Email:wangxiong8686@gmail.com 11
- 12. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics of the net force, and is inversely proportional to the mass m of the body, i.e., F = ma. 3. Third law: When two bodies interact by exerting force on each other, these forces (termed the action and the reaction) are equal in magnitude, but opposite in direction.Wang Xiong Email:wangxiong8686@gmail.com 12
- 13. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsNewton’s law of universal gravitationNewton’s law of universal gravitation states that the gravitational forcebetween two bodies of mass m and M separated by a distance r is mM F = −G r2.Wang Xiong Email:wangxiong8686@gmail.com 13
- 14. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsThe MASSThe term “mass’ was introduced into mechanics by Newton in 1687 inhis “Principia”. There are the following properties of mass in Newtonianmechanics: • Mass is a measure of the amount of matter. • Mass of a body is a measure of its inertia. • Masses of bodies are sources of their gravitational attraction to each other. • Mass of a composite body is equal to the sum of masses of theWang Xiong Email:wangxiong8686@gmail.com 14
- 15. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics bodies that constitute it; mathematically that means that mass is additive. • Mass of an isolated body or isolated system of bodies is conserved: it does not change with time. • Mass of a body does not change in the transition from one reference frame to another.Wang Xiong Email:wangxiong8686@gmail.com 15
- 16. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics2.1 The amount of matterNewton deﬁned mass as the amount of matter. Macroscopically, mass isassociated with matter. the mass of an object is somehow relate with thenumber and type of atoms or molecules it contains, and with the energyinvolved in binding it together (which contributes a negative ”missingmass,” or mass deﬁcit). But matter, unlike mass, is poorly deﬁned in science. The generallyaccepted deﬁnition of matter does not exist even today. This is partiallybecause there are so many diﬀerent kinds of matters with quite diﬀerentproperties.Wang Xiong Email:wangxiong8686@gmail.com 16
- 17. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsCan matter have zero mass?Some may claim that matter is anything that occupies space and has restmass. Under this deﬁnition one will not consider photons – particles oflight – as particles of matter, because they are massless. For the samereason they do not consider as matter the electromagnetic ﬁeld. It is notquite clear whether they consider as matter almost massless neutrinos,which usually move with velocity close to that of light.Wang Xiong Email:wangxiong8686@gmail.com 17
- 18. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsWe only known so little about matterEven more strangely, the ordinary matter, in the quarks and leptonsdeﬁnition, constitutes about 4% of the energy of the observable universe.The remaining energy is theorized to be due to exotic forms, of which 23%is dark matter and 73% is dark energy. We have very little knowledgeabout the majority matter of the universe. Anyway roughly speaking, the mass is somehow the measure of amoun-t of matter.Wang Xiong Email:wangxiong8686@gmail.com 18
- 19. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics2.2 The inertia massInertial mass is a measure of an object’s resistance to changing its stateof motion when a force is applied, which arises naturally from Newton’ssecond law of motion. It is determined by applying a force to an object and measuring theacceleration that results from that force. An object with small inertialmass will accelerate more than an object with large inertial mass whenacted upon by the same force. One says the body of greater mass hasgreater inertia.Wang Xiong Email:wangxiong8686@gmail.com 19
- 20. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsHow to measure the inertiaWe can measure its acceleration (assuming only that we can readily mea-sure length and time) and hence the force acting on it in terms of its mass.The relation between mass and force is given byF = mi r. ¨ For any arbitrary body, the inertial mass is deﬁned as follows: we takethe body and let it interact, somehow, with a standard inertial mass (onekilogram). Both the body and the standard will accelerate towards or away fromone another. Designating the acceleration of the body and of the stan-dard by aand as , respectively, we can deﬁne the inertial mass by as mi = . aWang Xiong Email:wangxiong8686@gmail.com 20
- 21. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics2.3 The gravitational massThe gravitational mass can be described as a measure of magnitude ofthe gravitational force which is • exerted by an object (active gravitational mass), and • experienced by an object (passive gravitational force)when interacting with a second object. Newton’s law of universal gravitation states that the gravitationalforce between two bodies of mass mG and MG separated by a distance ris mG MG F = −G . r2Wang Xiong Email:wangxiong8686@gmail.com 21
- 22. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsHow to measure the gravitational massThe gravitational mass can be deﬁned in a similar manner. We takea standard object and deﬁne its gravitational mass to be one unit; forconvenience we will use the standard kilogram as a standard for bothinertial and gravitational mass. We now place the body at some distance r and let it interact grav-itationally with the standard. The body will accelerate towards the s-tandard. We deﬁne the gravitational mass of the body in terms of thisacceleration and the distance between the objects: ami r2 mG ≡ lim . r→∞ GWang Xiong Email:wangxiong8686@gmail.com 22
- 23. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics as Alternatively, using mi = a we can write this as as r 2 mG ≡ lim . r→∞ GThis gives the gravitational mass in kilograms. The limiting procedure r → ∞ is needed in order to eliminate theeﬀects of multipole ﬁelds, which depend on the mass distribution of thetwo bodies. Also, proceeding to the limit r → ∞ eliminates the eﬀect ofshorter range forces (nuclear force, Van der Waals force etc.). At large distances only the gravitational and electrostatic forces willremain, but the latter can be eliminated by taking the precaution ofkeeping the standard body neutral.Wang Xiong Email:wangxiong8686@gmail.com 23
- 24. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics If we take two identical copies of the standard mass and let them falltowards each other, the acceleration of each serves to deﬁne the constantG: G = lim ass r2 . r→∞Wang Xiong Email:wangxiong8686@gmail.com 24
- 25. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsPut togetherWith these precise deﬁnitions of mi and mG it is clear that the gravita-tional force between two particles is mG MG F = −G , r2and the equation of motion is d2 r mG MG mi 2 = −G . dt r2Whether all particles fall in the gravitational ﬁeld of the particle of massMG with the same acceleration depends on whether all particles have theWang Xiong Email:wangxiong8686@gmail.com 25
- 26. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicssame value of mi mGWang Xiong Email:wangxiong8686@gmail.com 26
- 27. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsTwo conceputure diﬀernet massIf this ratio is a universal constant, it must have the value one (thestandard body has this value by deﬁnition). The question is then, is theequation mi = mG ,satisﬁed for all bodies? This is a question which can be answered only by experimental means.Wang Xiong Email:wangxiong8686@gmail.com 27
- 28. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics2.4 The additive of massIn Newtonian Mechanics, mass of a composite body is equal to the sumof masses of the bodies that constitute it; mathematically that meansthat mass is additive. Together with the believe that matter cannot be destroyed or elimi-nated, the mass is believe to be always a positive real number.Wang Xiong Email:wangxiong8686@gmail.com 28
- 29. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics2.5 Mass is conserved quantityMass of an isolated body or isolated system of bodies is conserved: itdoes not change with time. This is the reﬂection of the believe that matter cannot be destroyedor created. Things will not be so obviously in relativity.Wang Xiong Email:wangxiong8686@gmail.com 29
- 30. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics2.6 Mass is invariantMass of a body does not change in the transition from one inertial ref-erence frame to another in Newtonian Mechanics. Things will not be soobviously in relativity.Wang Xiong Email:wangxiong8686@gmail.com 30
- 31. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics3 Mass in Lagrangian MechanicsWang Xiong Email:wangxiong8686@gmail.com 31
- 32. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsElementary Introduction of Lagrangian DynamicsLet’s consider the Newton equation F = main the one-dimensional case, in which all the physical parameters dependon the variable xonly. Assume that the force is conservative, which meansthat it is given as the spatial derivative of the potential energyU (x): dU F =− . dxThus for this case Newton’s equation can be re-written as dU m¨ = − x , (1) dxWang Xiong Email:wangxiong8686@gmail.com 32
- 33. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicswhere dx x= ˙ =v dtand d2 x x = 2 = a. ¨ dtDeﬁne the Lagrangian L (x, x)as a function of two variables, the posi- ˙tion x and the speed x, ˙ 1 L (x, x) = T (x) − U (x) = mx2 − U (x) , ˙ ˙ ˙ (2) 2where the kinetic energy T (x) = 1 mx2 is a function of the speed variable ˙ 2 ˙only and the potential energy is a function of the position only.Wang Xiong Email:wangxiong8686@gmail.com 33
- 34. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics From Eq. (2) one can easily see that ∂L dU ∂L =− , = mx = p. ˙ (3) ∂x dx ∂x ˙Then obviously d ∂L = m¨. x (4) dt ∂x ˙With the use of Eq. (3) Newton’s Eq. (1) becomes d ∂L ∂L = , dt ∂ x ˙ ∂xwhich is called the Euler-Lagrange equation in one dimension.Wang Xiong Email:wangxiong8686@gmail.com 34
- 35. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics To ﬁnd the Hamiltonian formulation of dynamics, we deﬁne ﬁrst theHamiltonian H (x, p) as a function of two new variables, the momentump and the position x: H (x, p) = px − L (x, x) , ˙ ˙which is just the total energy T + U as 1 H (x, p) = px − L (x, x) = mx2 − ˙ ˙ ˙ mx2 − U (x) ˙ = T + U. (5) 2The Hamilton equations, which replace Newton’s equation of motion Eq.(1) are given by ∂H ∂H x= ˙ ,p = − ˙ . (6) ∂p ∂xWang Xiong Email:wangxiong8686@gmail.com 35
- 36. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsThe principle of least actionIn physics, action is an attribute of the dynamics of a physical system. Itis a mathematical functional which takes the trajectory, also called pathor history, of the system as its argument and has a real number as itsresult. Generally, the action takes diﬀerent values for diﬀerent paths Physical laws are frequently expressed as diﬀerential equations, whichdescribe how physical quantities such as position and momentum changecontinuously with time. Given the initial and boundary conditions forthe situation, the solution to the equation is a function describing thebehavior of the system (positions and momenta of the particles) at alltimes and all positions within the set boundaries. There is an alternative approach to ﬁnding equations of motion. Clas-Wang Xiong Email:wangxiong8686@gmail.com 36
- 37. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicssical mechanics postulates that the path actually followed by a physicalsystem is that for which the action is minimized, or, more strictly, isstationary. That is to say, the action satisﬁes a variational principle:the principle of stationary action (see also below). The action is deﬁnedby an integral, and the classical equations of motion of a system canbe derived from minimizing the value of the action integral, rather thansolving diﬀerential equations. The formal deﬁnition of the principle of least action is that it is ”theprinciple stating that the actual motion of a conservative dynamical sys-tem between two points takes place in such a way that a function of thecoordinates and velocities, called the action, has a minimum value withreference to all other paths between the points which correspond to theWang Xiong Email:wangxiong8686@gmail.com 37
- 38. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicssame energy.” The action Sof a system, which is a scalar quantity, is deﬁned as thetime integral of the Lagrangian function, t2 S= L (x(t), x(t)) dt. ˙ t1Therefore the problem of the motion of a mechanical system can be s-tated as that of ﬁnding the path x(t), t1 ≤ t ≤ t2 , such that the actionSis minimal. From a mathematical point of view this class of problem-s belongs to the ﬁeld of mathematics called variational calculus and aquantity deﬁned like the action is named functional2 . A functional willhave its extremal value when its ”variation” is equal to zero. (This isWang Xiong Email:wangxiong8686@gmail.com 38
- 39. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicslike a function having an extremal value when its derivative is equal tozero). Consider two points A andB. There are many trajectories joiningthe points, but a mechanical system which is evolving between them ischoosing the trajectory that makes the action functional extremal. Let’s assume that x(t)is the real trajectory of the body. Let’s usalso imagine a second trajectory, which is very near to the ﬁrst, givenbyx(t) + εh(t), where h(t) is an arbitrary time dependent function and εis a constant satisfying the condition ε << 1. Obviously at the ends ofall varied paths h(t) satisﬁes the conditions h (t1 ) = h (t2 ) = 0. (7)The value of the action integral, which is a scalar, will be necessarilyWang Xiong Email:wangxiong8686@gmail.com 39
- 40. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsdiﬀerent according to the path taken by the particle to go from A toB, S [x(t)] = S [x + εh(t)] .To ﬁnd the curve or curves and the path or paths that make the actionextremal, we shall use the variation of the trajectory and evaluate theaction for the extremal and for the varied trajectory. For the Lagrangianalong the varied path we obtain, by using a Taylor series expansion, ˙ ∂L ∂L ˙ L x + εh, x + εh = L (x, x) + ε ˙ ˙ h+ h . ∂x ∂x ˙Therefore the variation of the action along the two paths is given by t2 ∂L ∂L ˙ S [x + εh(t)] − S [x(t)] = ε h+ h dt. t1 ∂x ∂x ˙Wang Xiong Email:wangxiong8686@gmail.com 40
- 41. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsThe second term in the integrand can be transformed by using partialintegration, t2 t2 t2 t2 ∂L ˙ ∂L d ∂L d ∂L hdt = h(t) − h(t)dt = − h(t)dt, t1 ∂x ˙ ∂x ˙ t1 t1 dt ∂ x ˙ t1 dt ∂ x ˙since h (t1 ) = h (t2 ) = 0. Then the variation of the action integral for a very small ε is t2 ∂S S [x + εh(t)] − S [x(t)] ∂L d ∂L = lim = − h(t)dt. ∂ε ε→0 ε t1 ∂x dt ∂ x ˙Since h(t) = 0for t1 < t < t2 , for the path x(t)followed by the particlebetween the two ﬁxed points Aand Bbe an extremal of the actionS, itWang Xiong Email:wangxiong8686@gmail.com 41
- 42. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsis necessary and suﬃcient that the quantity between the brackets in theintegral be zero. Then this condition gives the Euler-Lagrange Equations: d ∂L ∂L = . dt ∂ x ˙ ∂xWang Xiong Email:wangxiong8686@gmail.com 42
- 43. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsAll are in LagrangianTherefore the equation of motion of a particle under the action of aconservative force can be derived from the Principle of the Least Action.So all information about the dynamics of the system is contained in the 1 L (x, x) = T (x) − U (x) = mx2 − U (x) ˙ ˙ ˙ 2Wang Xiong Email:wangxiong8686@gmail.com 43
- 44. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsWhy such Lagrangian?Lagrangian of free particle can be determined by space-time symmetries. In the above, we ﬁnd the Lagrangian from the newton’s law of motion.Also in reverse, if we known the right Lagrangian, we could derive theequation of motion through the Euler-Lagrange Equations. If we want do it in reverse, how could we ﬁnd the right Lagrangian? One might wonder whether one must simply guess at the form of theLagrangian, such as the choice x2 , or if there is a more basic principle for ﬁnding the right ˙Lagrangian.Wang Xiong Email:wangxiong8686@gmail.com 44
- 45. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsSymmetry is fearful!One way of doing so, is by thinking the symmetry of space-time.The remarkable thing is that certain symmetries are so powerfuland so restrictive that they entirely determine the functionalform of the Lagrangian, and therefore the equations of motion.Wang Xiong Email:wangxiong8686@gmail.com 45
- 46. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsNewtonian picture of the worldThe very least we can say about our world according to Newtonian me-chanics is: the Universe is made of particles, space and time. The posi-tions of particles change with time, according to Newton’s law of motion. In other words, we try to describe the Lagrangian as a function ofposition x, time t and velocity x= dx . ˙ dt The basic assumption about the Lagrangian of Newtonianmechanics is then it has the form L 0 (x, x, t) ˙.Wang Xiong Email:wangxiong8686@gmail.com 46
- 47. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsProperties of LagrangianThe choice of a least action principle as the basis of a theory of dynamicsimmediately implies that the Lagrangian has certain properties. First, we get the same equations of motion if we multiply that functionwith an arbitrary constant. Second, we get the same motion if we add to the integrand the totaltime derivative of an arbitrary function of space-time: df (x,t) L =L + dtthen df (x,t) S = L dt= Ldt + dt=S + fB -fA dtWang Xiong Email:wangxiong8686@gmail.com 47
- 48. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics The additional terms are ﬁxed, so that they have no inﬂuence on theequations of motion. The above two properties of L are mathematical consequences of ourdecision to use a least action principle.Wang Xiong Email:wangxiong8686@gmail.com 48
- 49. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsCreate Lagrangian from Symmetry principleBut further restrictions on L must be built in by using clues from Nature.These may be of any kind, but in practice the most powerful and generalhave proven to be symmetries of space-time. To begin with, consider the general form Lagrangian of a free particle L0 (x, x, t) ˙.Wang Xiong Email:wangxiong8686@gmail.com 49
- 50. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsInvariance under Spatial and Time Translations:The ﬁrst symmetry is the homogeneity of space and time. That is to say, the motion of a free particle does not depend on theplace from which we measure its progress, nor does it depend on the timeat which we start our measurement. Accordingly, L0 cannot explicitlydepend on x or t, so that L 0 (x, x, t) =L 0 (x) ˙ ˙Wang Xiong Email:wangxiong8686@gmail.com 50
- 51. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsInvariance under Spatial Rotations:The second symmetry is that of the isotropy of space. That is to say, the motion of a free particle does not depend on theorientation of our coordinate system. We conclude immediately that thismeans that L0 cannot depend on the direction ofx, but only depend on ˙the magnitude. It must be invariant under any rotations: L 0 (R x) = L 0 (x) = L 0 (|x|) ˙ ˙ ˙ √where R is a rotation operator and |x| = x · x ˙ ˙ ˙Wang Xiong Email:wangxiong8686@gmail.com 51
- 52. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsInvariance under Galilean Transformations:There is yet another symmetry essential to the Newtonian mechanics:Galilean relativity. Coordinates of S and S , moving with relative velocity v, are relatedby x = x − vt and t = t If S is an inertial system in which Newton’s 1st law holds, so is S . Physics law should be invariant under Galilean transforma-tions, so does the Lagrangian.Wang Xiong Email:wangxiong8686@gmail.com 52
- 53. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics How will this symmetry restrict the form of Lagrangian? We will seethrough the following trick. We denote X = 1 |x|2 , using this new variable to express the free 2 ˙particle with Lagrangian L 0 (|x|) = L 0 (X). ˙ Thus the Euler-Lagrange equations are d ∂ L0 d ∂ X dL 0 d dL 0 0= = = x ˙ dt ∂x˙ dt ∂ x dX ˙ dt dX ∂X ∂ 1where ∂x ˙ = ∂x˙ 2 x ˙ ·x =x ˙ ˙ d dL 0 ∂ dL 0 d2 L 0 =x· ¨ =x·x ¨ ˙ dt dX ∂x˙ dX dX 2Wang Xiong Email:wangxiong8686@gmail.com 53
- 54. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanics dL 0 d dL 0 dL 0 d2 L 0 0=x ¨ +x ˙ =x ¨ + x (¨ · x) ˙ x ˙ dX dt dX dX dX 2 dL 0 d2 L 0 x ¨+ x (¨ · x) ˙ x ˙ =0 dX dX 2Under the Galilean transformation x→x =x−v x→x =x ˙ ˙ ˙ ¨ ¨ ¨ 1 1 X→X = (x − v) · (x − v) = X − x · v + v2 ˙ ˙ ˙ 2 2The Euler-Lagrange equation can remain unchanged iﬀ d2 L 0 =0 dX 2Wang Xiong Email:wangxiong8686@gmail.com 54
- 55. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicswhich imply that dL 0 = const ≡ m dX So 1 L 0 = m x2 + const ˙ 2 And the mass is nothing but just a integral constant in the La-grangian.Wang Xiong Email:wangxiong8686@gmail.com 55
- 56. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsSymmetry determine the dynamicThe conclusion is that Lagrangian of free particle is determined by s-pacetime symmetries. Note that the above mechanism of using symmetries to nail down thefunctional form of L works wonderfully in other cases, too. In fact, allknown forces of Nature can be derived from a symmetry principle which,remarkably enough, not only prescribes the Lagrangian of a free particle(like the way in which we obtained above), but the interactions as well.Wang Xiong Email:wangxiong8686@gmail.com 56
- 57. What is mass? Lecture one: Mass in Newtonian Mechanics and Lagrangian mechanicsSummary: what is mass then? • a quantitative measure of an object’s resistance to acceleration • or a measure of magnitude of the gravitational force • or just a parameter in the Lagrangian?Wang Xiong Email:wangxiong8686@gmail.com 57

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