Newtonian mechanics Lecture notes

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Preprint typeset in JHEP style - HYPER VERSION Lent Term, 2013 Dynamics and Relativity University of Cambridge Part IA Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK 1 Recommended Books and Resources  Tom Kibble and Frank Berkshire, \Classical Mechanics"  Douglas Gregory, \Classical Mechanics" Both of these books are well written and do an excellent job of explaining the funda- mentals of classical mechanics. If you're struggling to understand some of the basic concepts, these are both good places to turn.  S. Chandrasekhar, \Newton's Principia (for the common reader)" Want to hear about Newtonian mechanics straight from the horse's mouth? This is an annotated version of the Principia with commentary by the Nobel prize winning astrophysicist Chandrasekhar who walks you through Newton's geometrical proofs. Although, in fairness, Newton is sometimes easier to understand than Chandra.  A.P. French, \Special Relativity" A clear introduction, covering the theory in some detail.  Wolfgang Pauli, \Theory of Relativity" Pauli was one of the founders of quantum mechanics and one of the great physicists of the last century. Much of this book was written when he was just 21. It remains one of the most authoritative and scholarly accounts of special relativity. It's not for the faint of heart. (But it is cheap). A number of excellent lecture notes are available on the web. Links can be found on the course webpage: 1. Newtonian Mechanics 1 1.1 Newton's Laws of Motion 2 1.1.1 Newton's Laws 3 1.2 Inertial Frames and Newton's First Law 4 1.2.1 Galilean Relativity 5 1.3 Newton's Second Law 8 1.4 Looking Forwards: The Validity of Newtonian Mechanics 9 2. Forces 11 2.1 Potentials in One Dimension 11 2.1.1 Moving in a Potential 13 2.1.2 Equilibrium: Why (Almost) Everything is a Harmonic Oscillator 16 2.2 Potentials in Three Dimensions 18 2.2.1 Central Forces 20 2.2.2 Angular Momentum 21 2.3 Gravity 22 2.3.1 The Gravitational Field 22 2.3.2 Escape Velocity 24 2.3.3 Inertial vs Gravitational Mass 25 2.4 Electromagnetism 26 2.4.1 The Electric Field of a Point Charge 27 2.4.2 Circles in a Constant Magnetic Field 28 2.4.3 An Aside: Maxwell's Equations 31 2.5 Friction 31 2.5.1 Dry Friction 31 2.5.2 Fluid Drag 32 2.5.3 An Example: The Damped Harmonic Oscillator 33 2.5.4 Terminal Velocity with Quadratic Friction 34 3. Interlude: Dimensional Analysis 40 1 4. Central Forces 48 4.1 Polar Coordinates in the Plane 48 4.2 Back to Central Forces 50 4.2.1 The E ective Potential: Getting a Feel for Orbits 52 4.2.2 The Stability of Circular Orbits 53 4.3 The Orbit Equation 55 4.3.1 The Kepler Problem 56 4.3.2 Kepler's Laws of Planetary Motion 60 4.3.3 Orbital Precession 62 4.4 Scattering: Throwing Stu at Other Stu 63 4.4.1 Rutherford Scattering 64 5. Systems of Particles 67 5.1 Centre of Mass Motion 67 5.1.1 Conservation of Momentum 68 5.1.2 Angular Momentum 68 5.1.3 Energy 69 5.1.4 In Praise of Conservation Laws 70 5.1.5 Why the Two Body Problem is Really a One Body Problem 71 5.2 Collisions 72 5.2.1 Bouncing Balls 73 5.2.2 More Bouncing Balls and the Digits of  74 5.3 Variable Mass Problems 76 5.3.1 Rockets: Things Fall Apart 77 5.3.2 Avalanches: Stu Gathering Other Stu 80 5.4 Rigid Bodies 81 5.4.1 Angular Velocity 82 5.4.2 The Moment of Inertia 82 5.4.3 Parallel Axis Theorem 85 5.4.4 The Inertia Tensor 87 5.4.5 Motion of Rigid Bodies 88 6. Non-Inertial Frames 93 6.1 Rotating Frames 93 6.1.1 Velocity and Acceleration in a Rotating Frame 94 6.2 Newton's Equation of Motion in a Rotating Frame 95 6.3 Centrifugal Force 97 6.3.1 An Example: Apparent Gravity 97 2 6.4 Coriolis Force 99 6.4.1 Particles, Baths and Hurricanes 100 6.4.2 Balls and Towers 102 6.4.3 Foucault's Pendulum 103 6.4.4 Larmor Precession 105 7. Special Relativity 107 7.1 Lorentz Transformations 108 7.1.1 Lorentz Transformations in Three Spatial Dimensions 111 7.1.2 Spacetime Diagrams 112 7.1.3 A History of Light Speed 113 7.2 Relativistic Physics 115 7.2.1 Simultaneity 115 7.2.2 Causality 117 7.2.3 Time Dilation 118 7.2.4 Length Contraction 122 7.2.5 Addition of Velocities 124 7.3 The Geometry of Spacetime 125 7.3.1 The Invariant Interval 125 7.3.2 The Lorentz Group 128 7.3.3 A Rant: Why c = 1 131 7.4 Relativistic Kinematics 132 7.4.1 Proper Time 133 7.4.2 4-Velocity 134 7.4.3 4-Momentum 137 7.4.4 Massless Particles 139 7.4.5 Newton's Laws of Motion 141 7.4.6 Acceleration 142 7.4.7 Indices Up, Indices Down 145 7.5 Particle Physics 146 7.5.1 Particle Decay 147 7.5.2 Particle Collisions 148 7.6 Spinors 151 7.6.1 The Lorentz Group and SL(2; C) 152 7.6.2 What the Observer Actually Observes 155 7.6.3 Spinors 160 3 Acknowledgements I inherited this course from Stephen Siklos. His excellent set of printed lecture notes form the backbone of these notes and can be found at: I'm grateful to the students, and especially Henry Mak, for pointing out typos and corrections. My thanks to Alex Considine for putting up with the lost weekends while these lectures were written. 4 1. Newtonian Mechanics Classical mechanics is an ambitious theory. Its purpose is to predict the future and reconstruct the past, to determine the history of every particle in the Universe. In this course, we will cover the basics of classical mechanics as formulated by Galileo and Newton. Starting from a few simple axioms, Newton constructed a mathematical framework which is powerful enough to explain a broad range of phenomena, from the orbits of the planets, to the motion of the tides, to the scattering of elementary particles. Before it can be applied to any speci c problem, the framework needs just a single input: a force. With this in place, it is merely a matter of turning a mathematical handle to reveal what happens next. We start this course by exploring the framework of Newtonian mechanics, under- standing the axioms and what they have to tell us about the way the Universe works. We then move on to look at a number of forces that are at play in the world. Nature is kind and the list is surprisingly short. Moreover, many of forces that arise have special properties, from which we will see new concepts emerging such as energy and conserva- tion principles. Finally, for each of these forces, we turn the mathematical handle. We turn this handle many many times. In doing so, we will see how classical mechanics is able to explain large swathes of what we see around us. Despite its wild success, Newtonian mechanics is not the last word in theoretical physics. It struggles in extremes: the realm of the very small, the very heavy or the very fast. We nish these lectures with an introduction to special relativity, the theory which replaces Newtonian mechanics when the speed of particles is comparable to the speed of light. We will see how our common sense ideas of space and time are replaced by something more intricate and more beautiful, with surprising consequences. Time goes slow for those on the move; lengths get smaller; mass is merely another form of energy. Ultimately, the framework of classical mechanics falls short of its ambitious goal to tell the story of every particle in the Universe. Yet it provides the basis for all that follows. Some of the Newtonian ideas do not survive to later, more sophisticated, theories of physics. Even the seemingly primary idea of force will fall by the wayside. Instead other concepts that we will meet along the way, most notably energy, step to the fore. But all subsequent theories are built on the Newtonian foundation. Moreover, developments in the past 300 years have con rmed what is perhaps the most important legacy of Newton: the laws of Nature are written in the language of 1 mathematics. This is one of the great insights of human civilisation. It has ushered in scienti c, industrial and technological revolutions. It has given us a new way to look at the Universe. And, most crucially of all, it means that the power to predict the future lies in hands of mathematicians rather than, say, gypsy astrologers. In this course, we take the rst steps towards grasping this power. 1.1 Newton's Laws of Motion Classical mechanics is all about the motion of particles. We start with a de nition. De nition: A particle is an object of insigni cant size. This means that if you want to say what a particle looks like at a given time, the only information you have to specify is its position. During this course, we will treat electrons, tennis balls, falling cats and planets as particles. In all of these cases, this means that we only care about the position of the object and our analysis will not, for example, be able to say anything about the look on the cat's face as it falls. However, it's not immediately obvious that we can meaningfully assign a single position to a complicated object such as a spinning, mewing cat. Should we describe its position as the end of its tail or the tip of its nose? We will not provide an immediate answer to this question, but we will return to it in Section 5 where we will show that any object can be treated as a point-like particle if we look at the motion of its centre of mass. To describe the position of a particle we need a reference z frame. This is a choice of origin, together with a set of axes which, for now, we pick to be Cartesian. With respect to this frame, the x position of a particle is speci ed by a vector x, which we denote using bold font. Since the particle moves, the position depends on y time, resulting in a trajectory of the particle described by Figure 1: x = x(t) In these notes we will also use both the notation x(t) and r(t) to describe the trajectory of a particle. The velocity of a particle is de ned to be dx(t) v dt 2 Throughout these notes, we will often denote di erentiation with respect to time by a _ \dot" above the variable. So we will also write v = x. The acceleration of the particle is de ned to be 2 d x(t) a x  = 2 dt A Comment on Vector Di erentiation The derivative of a vector is de ned by di erentiating each of the components. So, if x = (x ;x ;x ) then 1 2 3   dx dx dx dx 1 2 3 = ; ; dt dt dt dt Geometrically, the derivative of a path x(t) lies tangent to the path (a fact which you will see in the Vector Calculus course). In this course, we will be working with vector di erential equations. These should be viewed as three, coupled di erential equations one for each component. We will frequently come across situations where we need to di erentiate vector dot-products and cross-products. The meaning of these is easy to see if we use the chain rule on each component. For example, given two vector functions of time, f(t) and g(t), we have d df dg (f g) =  g + f dt dt dt and d df dg (f g) =  g + f dt dt dt As usual, it doesn't matter what order we write the terms in the dot product, but we have to be more careful with the cross product because, for example, df=dt g = gdf=dt. 1.1.1 Newton's Laws Newtonian mechanics is a framework which allows us to determine the trajectory x(t) of a particle in any given situation. This framework is usually presented as three axioms known as Newton's laws of motion. They look something like:  N1: Left alone, a particle moves with constant velocity.  N2: The acceleration (or, more precisely, the rate of change of momentum) of a particle is proportional to the force acting upon it. 3  N3: Every action has an equal and opposite reaction. While it is worthy to try to construct axioms on which the laws of physics rest, the trite, minimalistic attempt above falls somewhat short. For example, on rst glance, it appears that the rst law is nothing more than a special case of the second law. (If the force vanishes, the acceleration vanishes which is the same thing as saying that the velocity is constant). But the truth is somewhat more subtle. In what follows we will take a closer look at what really underlies Newtonian mechanics. 1.2 Inertial Frames and Newton's First Law Placed in the historical context, it is understandable that Newton wished to stress the rst law. It is a rebuttal to the Aristotelian idea that, left alone, an object will naturally come to rest. Instead, as Galileo had previously realised, the natural state of an object is to travel with constant speed. This is the essence of the law of inertia. However, these days we're not bound to any Aristotelian dogma. Do we really need the rst law? The answer is yes, but it has a somewhat di erent meaning. We've already introduced the idea of a frame of reference: a Cartesian coordinate system in which you measure the position of the particle. But for most reference frames you can think of, Newton's rst law is obviously incorrect. For example, suppose the coordinate system that I'm measuring from is rotating. Then, everything will appear to be spinning around me. If I measure a particle's trajectory in my coordinates as 2 2 x(t), then I certainly won't nd that d x=dt = 0, even if I leave the particle alone. In rotating frames, particles do not travel at constant velocity. We see that if we want Newton's rst law to y at all, we must be more careful about the kind of reference frames we're talking about. We de ne an inertial reference frame to be one in which particles do indeed travel at constant velocity when the force acting on it vanishes. In other words, in an inertial frame  x = 0 when F = 0 The true content of Newton's rst law can then be better stated as  N1 Revisited: Inertial frames exist. These inertial frames provide the setting for all that follows. For example, the second law which we shall discuss shortly should be formulated in inertial frames. 4 One way to ensure that you are in an inertial frame is to insist that you are left alone yourself: y out into deep space, far from the e ects of gravity and other in uences, turn o your engines and sit there. This is an inertial frame. However, for most purposes it will suce to treat axes of the room you're sitting in as an inertial frame. Of course, these axes are stationary with respect to the Earth and the Earth is rotating, both about its own axis and about the Sun. This means that the Earth does not quite provide an inertial frame and we will study the consequences of this in Section 6. 1.2.1 Galilean Relativity Inertial frames are not unique. Given one inertial frame, S, in which a particle has 0 coordinates x(t), we can always construct another inertial frameS in which the particle 0 has coordinates x (t) by any combination of the following transformations, 0  Translations: x = x + a, for constant a. 0 T  Rotations: x =Rx, for a 33 matrixR obeyingR R = 1. (This also allows for re ections if detR =1, although our interest will primarily be on continuous transformations). 0  Boosts: x = x + vt, for constant velocity v. It is simple to prove that all of these transformations map one inertial frame to another. Suppose that a particle moves with constant velocity with respect to frame S, so that 2 2 2 0 2 d x=dt = 0. Then, for each of the transformations above, we also have d x =dt = 0 0 which tells us that the particle also moves at constant velocity in S . Or, in other 0 words, ifS is an inertial frame then so too isS . The three transformations generate a group known as the Galilean group. The three transformations above are not quite the unique transformations that map between inertial frames. But, for most purposes, they are the only interesting ones 0 The others are transformations of the form x = x for some 2 R. This is just a trivial rescaling of the coordinates. For example, we may choose to measure distances 0 in S in units of meters and distances in S in units of parsecs. We have already mentioned that Newton's second law is to be formulated in an inertial frame. But, importantly, it doesn't matter which inertial frame. In fact, this is true for all laws of physics: they are the same in any inertial frame. This is known as the principle of relativity. The three types of transformation laws that make up the Galilean group map from one inertial frame to another. Combined with the principle of relativity, each is telling us something important about the Universe 5  Translations: There is no special point in the Universe.  Rotations: There is no special direction in the Universe.  Boosts: There is no special velocity in the Universe The rst two are fairly unsurprising: position is relative; direction is relative. The third perhaps needs more explanation. Firstly, it is telling us that there is no such thing as \absolutely stationary". You can only be stationary with respect to something else. Although this is true (and continues to hold in subsequent laws of physics) it is not true that there is no special speed in the Universe. The speed of light is special. We will see how this changes the principle of relativity in Section 7. So position, direction and velocity are relative. But acceleration is not. You do not have to accelerate relative to something else. It makes perfect sense to simply say that you are accelerating or you are not accelerating. In fact, this brings us back to Newton's rst law: if you are not accelerating, you are sitting in an inertial frame. The principle of relativity is usually associated to Einstein, but in fact dates back at least as far as Galileo. In his book, \Dialogue Concerning the Two Chief World Systems", Galileo has the character Salviati talk about the relativity of boosts, Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some ies, butter ies, and other small ying animals. Have a large bowl of water with some sh in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals y with equal speed to all sides of the cabin. The sh swim indi erently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not uctuating this way and that. You will discover not the least change in all the e ects named, nor could you tell from any of them whether the ship was moving or standing still. Galileo Galilei, 1632 6 Absolute Time There is one last issue that we have left implicit in the discussion above: the choice of 0 time coordinatet. If observers in two inertial frames,S andS , x the units seconds, minutes, hours in which to measure the duration time then the only remaining choice they can make is when to start the clock. In other words, the time variable in S and 0 S di er only by 0 t =t +t 0 This is sometimes included among the transformations that make up the Galilean group. The existence of a uniform time, measured equally in all inertial reference frames, is referred to as absolute time. It is something that we will have to revisit when we discuss special relativity. As with the other Galilean transformations, the ability to shift the origin of time is re ected in an important property of the laws of physics. The fundamental laws don't care when you start the clock. All evidence suggests that the laws of physics are the same today as they were yesterday. They are time translationally invariant. Cosmology Notably, the Universe itself breaks several of the Galilean transformations. There was a very special time in the Universe, around 13.7 billion years ago. This is the time of the Big Bang (which, loosely translated, means \we don't know what happened here"). Similarly, there is one inertial frame in which the background Universe is stationary. The \background" here refers to the sea of photons at a temperature of 2:7 K which lls the Universe, known as the Cosmic Microwave Background Radiation. This is the afterglow of the reball that lled all of space when the Universe was much younger. Di erent inertial frames are moving relative to this background and measure the radi- ation di erently: the radiation looks more blue in the direction that you're travelling, redder in the direction that you've come from. There is an inertial frame in which this background radiation is uniform, meaning that it is the same colour in all directions. To the best of our knowledge however, the Universe de nes neither a special point, nor a special direction. It is, to very good approximation, homogeneous and isotropic. However, it's worth stressing that this discussion of cosmology in no way invalidates the principle of relativity. All laws of physics are the same regardless of which inertial frame you are in. Overwhelming evidence suggests that the laws of physics are the 7 same in far ung reaches of the Universe. They were the same in rst few microseconds after the Big Bang as they are now. 1.3 Newton's Second Law The second law is the meat of the Newtonian framework. It is the famous \F =ma", which tells us how a particle's motion is a ected when subjected to a force F. The correct form of the second law is d (mx _ ) = F(x; x _ ) (1.1) dt This is usually referred to as the equation of motion. The quantity in brackets is called the momentum, _ pmx Herem is the mass of the particle or, more precisely, the inertial mass. It is a measure of the reluctance of the particle to change its motion when subjected to a given force F. In most situations, the mass of the particle does not change with time. In this case, we can write the second law in the more familiar form, mx  = F(x; x _ ) (1.2) For much of this course, we will use the form (1.2) of the equation of motion. However, in Section 5.3, we will brie y look at a few cases where masses are time dependent and we need the more general form (1.1). Newton's second law doesn't actually tell us anything until someone else tells us what the force F is in any given situation. We will describe several examples in the next section. In general, the force can depend on the position x and the velocity x _ of the particle, but does not depend on any higher derivatives. We could also, in principle, consider forces which include an explicit time dependence, F(x; x _;t), although we won't do so in these lectures. Finally, if more than one (independent) force is acting on the particle, then we simply take their sum on the right-hand side of (1.2). The single most important fact about Newton's equation is that it is a second order di erential equation. This means that we will have a unique solution only if we specify two initial conditions. These are usually taken to be the position x(t ) and the velocity 0 x _ (t ) at some initial time t . However, exactly what boundary conditions you must 0 0 choose in order to gure out the trajectory depends on the problem you are trying to solve. It is not unusual, for example, to have to specify the position at an initial time t and nal time t to determine the trajectory. 0 f 8 The fact that the equation of motion is second order is a deep statement about the Universe. It carries over, in essence, to all other laws of physics, from quantum mechanics to general relativity to particle physics. Indeed, the fact that all initial conditions must come in pairs two for each \degree of freedom" in the problem has important rami cations for later formulations of both classical and quantum mechanics. For now, the fact that the equations of motion are second order means the following: if you are given a snapshot of some situation and asked \what happens next?" then there is no way of knowing the answer. It's not enough just to know the positions of the particles at some point of time; you need to know their velocities too. However, once both of these are speci ed, the future evolution of the system is fully determined for all time. 1.4 Looking Forwards: The Validity of Newtonian Mechanics Although Newton's laws of motion provide an excellent approximation to many phe- nomena, when pushed to extreme situation they are found wanting. Broadly speaking, there are three directions in which Newtonian physics needs replacing with a di erent framework: they are 8 1  When particles travel at speeds close to the speed light, c  3 10 ms , the Newtonian concept of absolute time breaks down and Newton's laws need modi cation. The resulting theory is called special relativity and will be described in Section 7. As we will see, although the relationship between space and time is dramatically altered in special relativity, much of the framework of Newtonian mechanics survives unscathed.  On very small scales, much more radical change is needed. Here the whole frame- work of classical mechanics breaks down so that even the most basic concepts, such as the trajectory of a particle, become ill-de ned. The new framework that holds on these small scales is called quantum mechanics. Nonetheless, there are quantities which carry over from the classical world to the quantum, in particular energy and momentum.  When we try to describe the forces at play between particles, we need to introduce a new concept: the eld. This is a function of both space and time. Familiar examples are the electric and magnetic elds of electromagnetism. We won't have too much to say about elds in this course. For now, we mention only that the equations which govern the dynamics of elds are always second order di erential 9 equations, similar in spirit to Newton's equations. Because of this similarity, eld theories are again referred to as \classical". Eventually, the ideas of special relativity, quantum mechanics and eld theories are combined into quantum eld theory. Here even the concept of particle gets subsumed into the concept of a eld. This is currently the best framework we have to describe the world around us. But we're getting ahead of ourselves. Let's rstly return to our Newtonian world.... 10 2. Forces In this section, we describe a number of di erent forces that arise in Newtonian me- chanics. Throughout, we will restrict attention to the motion of a single particle. (We'll look at what happens when we have more than one particle in Section 5). We start by describing the key idea of energy conservation, followed by a description of some common and important forces. 2.1 Potentials in One Dimension Let's start by considering a particle moving on a line, so its position is determined by a single functionx(t). For now, suppose that the force on the particle depends only on the position, not the velocity: F =F (x). We de ne the potentialV (x) (also called the potential energy) by the equation dV F (x) = (2.1) dx The potential is only de ned up to an additive constant. We can always invert (2.1) by integrating both sides. The integration constant is now determined by the choice of lower limit of the integral, Z x 0 0 V (x) = dx F (x ) x 0 0 Here x is just a dummy variable. (Do not confuse the prime with di erentiation In this course we will only take derivatives of position x with respect to time and always denote them with a dot over the variable). With this de nition, we can write the equation of motion as dV mx  = (2.2) dx For any force in one-dimension which depends only on the position, there exists a conserved quantity called the energy, 1 2 E = mx _ +V (x) 2 _ The fact that this is conserved means thatE = 0 for any trajectory of the particle which 1 2 obeys the equation of motion. While V (x) is called the potential energy, T = mx _ is 2 called the kinetic energy. Motion satisfying (2.2) is called conservative. 11 It is not hard to prove that E is conserved. We need only di erentiate to get   dV dV _ E =mx _x  + x _ =x _ mx  + = 0 dx dx where the last equality holds courtesy of the equation of motion (2.2). In any dynamical system, conserved quantities of this kind are very precious. We will spend some time in this course shing them out of the equations and showing how they help us simplify various problems. An Example: A Uniform Gravitational Field In a uniform gravitational eld, a particle is subjected to a constant force, F =mg 2 where g 9:8 ms is the acceleration due to gravity near the surface of the Earth. The minus sign arises because the force is downwards while we have chosen to measure position in an upwards direction which we call z. The potential energy is V =mgz Notice that we have chosen to have V = 0 at z = 0. There is nothing that forces us to do this; we could easily add an extra constant to the potential to shift the zero to some other height. The equation of motion for uniform acceleration is z  =g Which can be trivially integrated to give the velocity at time t, z _ =ugt (2.3) where u is the initial velocity at time t = 0. (Note that z is measured in the upwards direction, so the particle is moving up if z _ 0 and down if z _ 0). Integrating once more gives the position 1 2 z =z +ut gt (2.4) 0 2 where z is the initial height at time t = 0. Many high schools teach that (2.3) and 0 (2.4) the so-called \suvat" equations are key equations of mechanics. They are not. They are merely the integration of Newton's second law for constant acceleration. Do not learn them; learn how to derive them. 12 Another Simple Example: The Harmonic Oscillator The harmonic oscillator is, by far, the most important dynamical system in all of theoretical physics. The good news is that it's very easy. (In fact, the reason that it's so important is precisely because it's easy). The potential energy of the harmonic oscillator is de ned to be 1 2 V (x) = kx 2 The harmonic oscillator is a good model for, among other things, a particle attached to the end of a spring. The force resulting from the energy V is given by F =kx which, in the context of the spring, is called Hooke's law. The equation of motion is mx  =kx which has the general solution r k x(t) =A cos(t) +B sin(t) with = m HereA andB are two integration constants and is called the angular frequency. We see that all trajectories are qualitatively the same: they just bounce backwards and forwards around the origin. The coecients A and B determine the amplitude of the oscillations, together with the phase at which you start the cycle. The time taken to complete a full cycle is called the period 2 T = (2.5) The period is independent of the amplitude. (Note that, annoyingly, the kinetic energy is also often denoted by T as well. Do not confuse this with the period. It should hopefully be clear from the context). If we want to determine the integration constantsA andB for a given trajectory, we need some initial conditions. For example, if we're given the position and velocity at time t = 0, then it's simple to check that A =x(0) and B =x _(0). 2.1.1 Moving in a Potential Let's go back to the general case of a potential V (x) in one dimension. Although the equation of motion is a second order di erential equation, the existence of a conserved energy magically allows us to turn this into a rst order di erential equation, r 1 dx 2 2 E = mx _ +V (x) ) = (EV (x)) 2 dt m 13 This gives us our rst hint of the importance of conserved quantities in helping solve a problem. Of course, to go from a second order equation to a rst order equation, we must have chosen an integration constant. In this case, that is the energyE itself. Given a rst order equation, we can always write down a formal solution for the dynamics simply by integrating, Z x 0 dx tt = q (2.6) 0 2 0 x 0 (EV (x )) m 0 As before,x is a dummy variable. If we can do the integral, we've solved the problem. If we can't do the integral, you sometimes hear that the problem has been \reduced to quadrature". This rather old-fashioned phrase really means \I can't do the integral". But, it is often the case that having a solution in this form allows some of its properties to become manifest. And, if nothing else, one can always just evaluate the integral numerically (i.e. on your laptop) if need be. Getting a Feel for the Solutions Given the potential energy V (x), it is often very simple to gure out the qualitative nature of any trajectory simply by looking at the form ofV (x). This allows us to answer some questions with very little work. For example, we may want to know whether the particle is trapped within some region of space or can escape to in nity. Let's illustrate this with an example. Consider the cubic potential 3 V (x) =m(x 3x) (2.7) If we were to substitute this into the general form (2.6), we'd get a fearsome looking 1 integral which hasn't been solved since Victorian times . Even without solving the integral, we can make progress. The potential is plotted in Figure 2. Let's start with the particle sitting stationary at some position x . This 0 means that the energy is E =V (x ) 0 and this must remain constant during the subsequent motion. What happens next depends only on x . We can identify the following possibilities 0 1 Ok, I'm exaggerating. The resulting integral is known as an elliptic integral. Although it can't be expressed in terms of elementary functions, it has lots of nice properties and has been studied to death. 100 years ago, this kind of thing was standard fare in mathematics. These days, we usually have more interesting things to teach. Nonetheless, the study of these integrals later resulted in beautiful connections to geometry through the theory of elliptic functions and elliptic curves. 14

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