Classical Dynamics Lecture Notes

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Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2004 and 2005 Classical Dynamics University of Cambridge Part II Mathematical Tripos Dr David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK 1 Recommended Books and Resources  L. Hand and J. Finch, Analytical Mechanics This very readable book covers everything in the course at the right level. It is similar to Goldstein's book in its approach but with clearer explanations, albeit at the expense of less content. There are also three classic texts on the subject  H. Goldstein, C. Poole and J. Safko, Classical Mechanics In previous editions it was known simply as \Goldstein" and has been the canonical choice for generations of students. Although somewhat verbose, it is considered the standard reference on the subject. Goldstein died and the current, third, edition found two extra authors.  L. Landau an E. Lifshitz, Mechanics This is a gorgeous, concise and elegant summary of the course in 150 content packed pages. Landau is one of the most important physicists of the 20th century and this is the rst volume in a series of ten, considered by him to be the \theoretical minimum" amount of knowledge required to embark on research in physics. In 30 years, only 43 people passed Landau's exam A little known fact: Landau originally co-authored this book with one of his students, Leonid Pyatigorsky. They subsequently had a falling out and the authorship was changed. There are rumours that Pyatigorsky got his own back by denouncing Landau to the Soviet authorities, resulting in his arrest.  V. I. Arnold, Mathematical Methods of Classical Mechanics Arnold presents a more modern mathematical approach to the topics of this course, making connections with the di erential geometry of manifolds and forms. It kicks o with \The Universe is an Ane Space" and proceeds from there...Contents 1. Newton's Laws of Motion 1 1.1 Introduction 1 1.2 Newtonian Mechanics: A Single Particle 2 1.2.1 Angular Momentum 3 1.2.2 Conservation Laws 4 1.2.3 Energy 4 1.2.4 Examples 5 1.3 Newtonian Mechanics: Many Particles 5 1.3.1 Momentum Revisited 6 1.3.2 Energy Revisited 8 1.3.3 An Example 9 2. The Lagrangian Formalism 10 2.1 The Principle of Least Action 10 2.2 Changing Coordinate Systems 13 2.2.1 Example: Rotating Coordinate Systems 14 2.2.2 Example: Hyperbolic Coordinates 16 2.3 Constraints and Generalised Coordinates 17 2.3.1 Holonomic Constraints 18 2.3.2 Non-Holonomic Constraints 20 2.3.3 Summary 21 2.3.4 Joseph-Louis Lagrange (1736-1813) 22 2.4 Noether's Theorem and Symmetries 23 2.4.1 Noether's Theorem 24 2.5 Applications 26 2.5.1 Bead on a Rotating Hoop 26 2.5.2 Double Pendulum 28 2.5.3 Spherical Pendulum 29 2.5.4 Two Body Problem 31 2.5.5 Restricted Three Body Problem 33 2.5.6 Purely Kinetic Lagrangians 36 2.5.7 Particles in Electromagnetic Fields 36 2.6 Small Oscillations and Stability 38 2.6.1 Example: The Double Pendulum 41 1 2.6.2 Example: The Linear Triatomic Molecule 42 3. The Motion of Rigid Bodies 45 3.1 Kinematics 46 3.1.1 Angular Velocity 47 3.1.2 Path Ordered Exponentials 49 3.2 The Inertia Tensor 50 3.2.1 Parallel Axis Theorem 52 3.2.2 Angular Momentum 53 3.3 Euler's Equations 53 3.3.1 Euler's Equations 54 3.4 Free Tops 55 3.4.1 The Symmetric Top 55 3.4.2 Example: The Earth's Wobble 57 3.4.3 The Asymmetric Top: Stability 57 3.4.4 The Asymmetric Top: Poinsot Construction 58 3.5 Euler's Angles 62 3.5.1 Leonhard Euler (1707-1783) 64 3.5.2 Angular Velocity 65 3.5.3 The Free Symmetric Top Revisited 65 3.6 The Heavy Symmetric Top 67 3.6.1 Letting the Top go 70 3.6.2 Uniform Precession 71 3.6.3 The Sleeping Top 72 3.6.4 The Precession of the Equinox 72 3.7 The Motion of Deformable Bodies 74 3.7.1 Kinematics 74 3.7.2 Dynamics 77 4. The Hamiltonian Formalism 80 4.1 Hamilton's Equations 80 4.1.1 The Legendre Transform 82 4.1.2 Hamilton's Equations 83 4.1.3 Examples 84 4.1.4 Some Conservation Laws 86 4.1.5 The Principle of Least Action 87 4.1.6 William Rowan Hamilton (1805-1865) 88 4.2 Liouville's Theorem 88 2 4.2.1 Liouville's Equation 90 4.2.2 Time Independent Distributions 91 4.2.3 Poincar e Recurrence Theorem 92 4.3 Poisson Brackets 93 4.3.1 An Example: Angular Momentum and Runge-Lenz 95 4.3.2 An Example: Magnetic Monopoles 96 4.3.3 An Example: The Motion of Vortices 98 4.4 Canonical Transformations 100 4.4.1 In nitesimal Canonical Transformations 102 4.4.2 Noether's Theorem Revisited 104 4.4.3 Generating Functions 104 4.5 Action-Angle Variables 105 4.5.1 The Simple Harmonic Oscillator 105 4.5.2 Integrable Systems 107 4.5.3 Action-Angle Variables for 1d Systems 108 4.5.4 Action-Angle Variables for the Kepler Problem 111 4.6 Adiabatic Invariants 113 4.6.1 Adiabatic Invariants and Liouville's Theorem 116 4.6.2 An Application: A Particle in a Magnetic Field 116 4.6.3 Hannay's Angle 118 4.7 The Hamilton-Jacobi Equation 121 4.7.1 Action and Angles from Hamilton-Jacobi 124 4.8 Quantum Mechanics 126 4.8.1 Hamilton, Jacobi, Schr odinger and Feynman 128 4.8.2 Nambu Brackets 131 3 Acknowledgements These notes rely heavily on the textbooks listed at the beginning and on notes from past courses given by others, in particular Anne Davis, Gary Gibbons, Robin Hud- son, Michael Peskin and Neil Turok. My thanks also to Michael Efroimsky and Matt Headrick for useful comments. I am supported by the Royal Society. 4 1. Newton's Laws of Motion \So few went to hear him, and fewer understood him, that oftimes he did, for want of hearers, read to the walls. He usually stayed about half an hour; when he had no auditors he commonly returned in a quarter of that time." Appraisal of a Cambridge lecturer in classical mechanics, circa 1690 1.1 Introduction The fundamental principles of classical mechanics were laid down by Galileo and New- th th ton in the 16 and 17 centuries. In 1686, Newton wrote the Principia where he gave us three laws of motion, one law of gravity and pretended he didn't know cal- culus. Probably the single greatest scienti c achievement in history, you might think this pretty much wraps it up for classical mechanics. And, in a sense, it does. Given a collection of particles, acted upon by a collection of forces, you have to draw a nice diagram, with the particles as points and the forces as arrows. The forces are then added up and Newton's famous \F = ma" is employed to gure out where the par- ticle's velocities are heading next. All you need is enough patience and a big enough computer and you're done. From a modern perspective this is a little unsatisfactory on several levels: it's messy and inelegant; it's hard to deal with problems that involve extended objects rather than point particles; it obscures certain features of dynamics so that concepts such as chaos theory took over 200 years to discover; and it's not at all clear what the relationship is between Newton's classical laws and quantum physics. The purpose of this course is to resolve these issues by presenting new perspectives on Newton's ideas. We shall describe the advances that took place during the 150 years after Newton when the laws of motion were reformulated using more powerful techniques and ideas developed by some of the giants of mathematical physics: people such as Euler, Lagrange, Hamilton and Jacobi. This will give us an immediate practical advantage, allowing us to solve certain complicated problems with relative ease (the strange motion of spinning tops is a good example). But, perhaps more importantly, it will provide an elegant viewpoint from which we'll see the profound basic principles which underlie Newton's familiar laws of motion. We shall prise open \F = ma" to reveal the structures and symmetries that lie beneath. 1 Moreover, the formalisms that we'll develop here are the basis for all of fundamental modern physics. Every theory of Nature, from electromagnetism and general relativity, to the standard model of particle physics and more speculative pursuits such as string theory, is best described in the language we shall develop in this course. The new formalisms that we'll see here also provide the bridge between the classical world and the quantum world. There are phenomena in Nature for which these formalisms are not particularly useful. Systems which are dissipative, for example, are not so well suited to these new techniques. But if you want to understand the dynamics of planets and stars and galaxies as they orbit and spin, or you want to understand what's happening at the LHC where protons are collided at unprecedented energies, or you want to know how electrons meld together in solids to form new states of matter, then the foundations that we'll lay in in this course are a must. 1.2 Newtonian Mechanics: A Single Particle In the rest of this section, we'll take a ying tour through the basic ideas of classical mechanics handed down to us by Newton. We'll start with a single particle. A particle is de ned to be an object of insigni cant size. e.g. an electron, a tennis ball or a planet. Obviously the validity of this statement depends on the context: to rst approximation, the earth can be treated as a particle when computing its orbit around the sun. But if you want to understand its spin, it must be treated as an extended object. The motion of a particle of massm at the position r is governed by Newton's Second Law F =ma or, more precisely, F(r; r _) = p _ (1.1) where F is the force which, in general, can depend on both the position r as well as _ _ _ the velocity r (for example, friction forces depend on r) and p =mr is the momentum. Both F and p are 3-vectors which we denote by the bold font. Equation (1.1) reduces to F =ma if m _ = 0. But if m =m(t) (e.g. in rocket science) then the form with p _ is correct. General theorems governing di erential equations guarantee that if we are given r and r _ at an initial timet =t , we can integrate equation (1.1) to determine r(t) for all 0 t (as long as F remains nite). This is the goal of classical dynamics. 2 Equation (1.1) is not quite correct as stated: we must add the caveat that it holds only in an inertial frame. This is de ned to be a frame in which a free particle with m _ = 0 travels in a straight line, r = r + vt (1.2) 0 Newtons's rst law is the statement that such frames exist. An inertial frame is not unique. In fact, there are an in nite number of inertial frames. Let S be an inertial frame. Then there are 10 linearly independent transformations 0 0 SS such thatS is also an inertial frame (i.e. if (1.2) holds inS, then it also holds 0 in S ). These are 0  3 Rotations: r =Or where O is a 3 3 orthogonal matrix. 0  3 Translations: r = r + c for a constant vector c. 0  3 Boosts: r = r + ut for a constant velocity u. 0  1 Time Translation: t =t +c for a constant real number c 0 If motion is uniform in S, it will also be uniform in S . These transformations make up the Galilean Group under which Newton's laws are invariant. They will be impor- tant in section 2.4 where we will see that these symmetries of space and time are the underlying reason for conservation laws. As a parenthetical remark, recall from special relativity that Einstein's laws of motion are invariant under Lorentz transformations which, together with translations, make up the Poincar e group. We can recover the Galilean group from the Poincar e group by taking the speed of light to in nity. 1.2.1 Angular Momentum We de ne the angular momentum L of a particle and the torque acting upon it as L = r p ;  = r F (1.3) Note that, unlike linear momentum p, both L and  depend on where we take the origin: we measure angular momentum with respect to a particular point. Let us cross both sides of equation (1.1) with r. Using the fact that r _ is parallel to p, we can write d (rp) = rp _ . Then we get a version of Newton's second law that holds for angular dt momentum: _  = L (1.4) 3 1.2.2 Conservation Laws From (1.1) and (1.4), two important conservation laws follow immediately.  If F = 0 then p is constant throughout the motion  If = 0 then L is constant throughout the motion Notice that  = 0 does not require F = 0, but only r F = 0. This means that F must be parallel to r. This is the de nition of a central force. An example is given by the gravitational force between the earth and the sun: the earth's angular momentum about the sun is constant. As written above in terms of forces and torques, these conservation laws appear trivial. In section 2.4, we'll see how they arise as a property of the symmetry of space as encoded in the Galilean group. 1.2.3 Energy Let's now recall the de nitions of energy. We rstly de ne the kinetic energy T as 1 T = m r _ r _ (1.5) 2 Suppose from now on that the mass is constant. We can compute the change of kinetic dT _ _ _ energy with time: = p r = F r. If the particle travels from position r at time t 1 1 dt to position r at time t then this change in kinetic energy is given by 2 2 Z Z Z t t r 2 2 2 dT _ T (t )T (t ) = dt = F rdt = Fdr (1.6) 2 1 dt t t r 1 1 1 where the nal expression involving the integral of the force over the path is called the work done by the force. So we see that the work done is equal to the change in kinetic energy. From now on we will mostly focus on a very special type of force known as a conservative force. Such a force depends only on position r rather than velocity r _ and is such that the work done is independent of the path taken. In particular, for a closed path, the work done vanishes. I Fdr = 0 , r F = 0 (1.7) 3 It is a deep property of at space R that this property implies we may write the force as F =rV (r) (1.8) for some potential V (r). Systems which admit a potential of this form include gravi- tational, electrostatic and interatomic forces. When we have a conservative force, we 4 necessarily have a conservation law for energy. To see this, return to equation (1.6) which now reads Z r 2 T (t )T (t ) = rVdr =V (t ) +V (t ) (1.9) 2 1 2 1 r 1 or, rearranging things, T (t ) +V (t ) =T (t ) +V (t )E (1.10) 1 1 2 2 So E = T +V is also a constant of motion. It is the energy. When the energy is considered to be a function of position r and momentum p it is referred to as the Hamiltonian H. In section 4 we will be seeing much more of the Hamiltonian. 1.2.4 Examples  Example 1: The Simple Harmonic Oscillator This is a one-dimensional system with a force proportional to the distance x to the 1 2 origin: F (x) =kx. This force arises from a potential V = kx . Since F 6= 0, 2 momentum is not conserved (the object oscillates backwards and forwards) and, since the system lives in only one dimension, angular momentum is not de ned. But energy 1 2 1 2 E = mx _ + kx is conserved. 2 2  Example 2: The Damped Simple Harmonic Oscillator We now include a friction term so thatF (x;x _) =kx x _. SinceF is not conservative, energy is not conserved. This system loses energy until it comes to rest.  Example 3: Particle Moving Under Gravity Consider a particle of mass m moving in 3 dimensions under the gravitational pull of 2 a much larger particle of mass M. The force is F =(GMm=r ) r which arises from the potential V =GMm=r. Again, the linear momentum p of the smaller particle is not conserved, but the force is both central and conservative, ensuring the particle's total energy E and the angular momentum L are conserved. 1.3 Newtonian Mechanics: Many Particles It's easy to generalise the above discussion to many particles: we simply add an index to everything in sight Let particle i have mass m and position r where i = 1;:::;N i i is the number of particles. Newton's law now reads _ F = p (1.11) i i 5 th where F is the force on thei particle. The subtlety is that forces can now be working i between particles. In general, we can decompose the force in the following way: X ext F = F + F (1.12) i ij i j6=i th th ext where F is the force acting on thei particle due to thej particle, while F is the ij i th external force on the i particle. We now sum over all N particles X X X ext F = F + F i ij i i i;j withj6=i i X X ext = (F + F ) + F (1.13) ij ji i ij i where, in the second line, we've re-written the sum to be over all pairs ij. At this stage we make use of Newton's third law of motion: every action has an equal and opposite reaction. Or, in other words, F =F . We see that the rst term vanishes ij ji and we are left simply with X ext F = F (1.14) i i P ext ext where we've de ned the total external force to be F = F . We now de ne the i i P total mass of the system M = m as well as the centre of mass R i i P m r i i i R = (1.15) M Then using (1.11), and summing over all particles, we arrive at the simple formula, ext  F =MR (1.16) which is identical to that of a single particle. This is an important formula. It tells that the centre of mass of a system of particles acts just as if all the mass were concentrated there. In other words, it doesn't matter if you throw a tennis ball or a very lively cat: the center of mass of each traces the same path. 1.3.1 Momentum Revisited P The total momentum is de ned to be P = p and, from the formulae above, it is i i ext _ simple to derive P = F . So we nd the conservation law of total linear momentum ext for a system of many particles: P is constant if F vanishes. 6 P Similarly, we de ne total angular momentum to be L = L . Now let's see what i i happens when we compute the time derivative. X _ L = r  p _ i i i X X ext = r  F + F (1.17) i ij i i j=6 i X X ext = r  F + r  F (1.18) i ji i i i;j withi6=j i P ext The last term in this expression is the de nition of total external torque:  = r i i ext F . But what are we going to do with the rst term on the right hand side? Ideally we i would like it to vanish Let's look at the circumstances under which this will happen. We can again rewrite it as a sum over pairs ij to get X (r r ) F (1.19) i j ij ij which will vanish if and only if the force F is parallel to the line joining to two particles ij (r r ). This is the strong form of Newton's third law. If this is true, then we have a i j statement about the conservation of total angular momentum, namely L is constant if ext  = 0. Most forces do indeed obey both forms of Newton's third law: 1 F =F and F is parallel to (rr ). For example, gravitational ij ji ij i j and electrostatic forces have this property. And the total momentum and angular momentum are both conserved in these systems. But some forces don't have these properties The most famous example is the Lorentz force on two moving particles with electric charge Q. This is given by, 2 F =Qv  B (1.20) ij i j th where v is the velocity of the i particle and B is the magnetic i j Figure 1: The th eld generated by the j particle. Consider two particles crossing magnetic eld for each other in a \T" as shown in the diagram. The force on particle two particles. 1 from particle 2 vanishes. Meanwhile, the force on particle 2 from particle 1 is non-zero, and in the direction F "  (1.21) 21 7 Does this mean that conservation of total linear and angular momentum is violated? Thankfully, no We need to realise that the electromagnetic eld itself carries angular momentum which restores the conservation law. Once we realise this, it becomes a rather cheap counterexample to Newton's third law, little di erent from an underwater swimmer who can appear to violate Newton's third law if we don't take into account the momentum of the water. 1.3.2 Energy Revisited P 1 2 The total kinetic energy of a system of many particles is T = m r _ . Let us i i 2 i decompose the position vector r as i r = R + r (1.22) i i where r is the distance from the centre of mass to the particle i. Then we can write i the total kinetic energy as X 2 1 2 1 _ _ T = MR + m r (1.23) i i 2 2 i Which shows us that the kinetic energy splits up into the kinetic energy of the centre of mass, together with an internal energy describing how the system is moving around its centre of mass. As for a single particle, we may calculate the change in the total kinetic energy, Z Z X X ext T (t )T (t ) = F dr + F dr (1.24) 2 1 i ij i i i i=6 j Like before, we need to consider conservative forces to get energy conservation. But now we need both ext  Conservative external forces: F =rV (r ;:::; r ) i i 1 N i  Conservative internal forces: F =rV (r ;:::; r ) ij i ij 1 N wherer =r . To get Newton's third law F =F together with the requirement i i ij ji that this is parallel to (rr ), we should take the internal potentials to satisfyV =V i j ij ji with V (r ;::: rN) =V (jr rj) (1.25) ij 1 ; ij i j th th so that V depends only on the distance between the i and j particles. We also ij insist on a restriction for the external forces, V (r ;:::; r ) =V (r ), so that the force i 1 N i i on particle i does not depend on the positions of the other particles. Then, following the steps we took in the single particle case, we can de ne the total potential energy P P V = V + V and we can show that H =T +V is conserved. i ij i ij 8 1.3.3 An Example Let us return to the case of gravitational attraction between two bodies but, unlike 1 2 1 2 _ _ in Section 1.2.4, now including both particles. We have T = m r + m r . The 1 2 1 2 2 2 potential isV =Gm m =jr rj. This system has total linear momentum and total 1 2 1 2 angular mometum conserved, as well as the total energy H =T +V . 9 2. The Lagrangian Formalism When I was in high school, my physics teacher called me down one day after class and said, \You look bored, I want to tell you something interesting". Then he told me something I have always found fascinating. Every time the subject comes up I work on it. Richard Feynman Feynman's teacher told him about the \Principle of Least Action", one of the most profound results in physics. 2.1 The Principle of Least Action Firstly, let's get our notation right. Part of the power of the Lagrangian formulation over the Newtonian approach is that it does away with vectors in favour of more general coordinates. We start by doing this trivially. Let's rewrite the positions of N particles A with coordinates r as x where A = 1;::: 3N. Then Newton's equations read i V p _ = (2.1) A A x A where p =m x _ . The number of degrees of freedom of the system is said to be 3N. A A These parameterise a 3N-dimensional space known as the con guration spaceC. Each point inC speci es a con guration of the system (i.e. the positions of all N particles). Time evolution gives rise to a curve in C. Figure 2: The path of particles in real space (on the left) and in con guration space (on the right). The Lagrangian A A De ne the Lagrangian to be a function of the positions x and the velocities x _ of all the particles, given by A A A A L(x ;x _ ) =T (x _ )V (x ) (2.2) 10 P 1 A 2 A where T = m (x _ ) is the kinetic energy, and V (x ) is the potential energy. A A 2 Note the minus sign between T and V To describe the principle of least action, we A consider all smooth paths x (t) in C with xed end points so that A A A A x (t ) =x and x (t ) =x (2.3) i f initial nal t Of all these possible paths, only one is the true path x taken by the system. Which one? To each path, let us final assign a number called the action S de ned as Z t f A A A Sx (t) = L(x (t);x _ (t)) dt (2.4) t i x initial The action is a functional (i.e. a function of the path which x is itself a function). The principle of least action is the fol- Figure 3: lowing result: Theorem (Principle of Least Action): The actual path taken by the system is an extremum of S. Proof: Consider varying a given path slightly, so A A A x (t)x (t) +x (t) (2.5) A A where we x the end points of the path by demanding x (t ) = x (t ) = 0. Then i f the change in the action is Z  t f S =  Ldt t i Z t f = Ldt t i Z   t f L L A A = x + x _ dt (2.6) A A x x _ t i At this point we integrate the second term by parts to get Z      t t f f L d L L A A S = x dt + x (2.7) A A A x dt x _ x _ t i t i A But the nal term vanishes since we have xed the end points of the path so x (t ) = i A x (t ) = 0. The requirement that the action is an extremum says that S = 0 for all f A changes in the path x (t). We see that this holds if and only if   L d L = 0 for each A = 1;::: 3N (2.8) A A x dt x _ 11 These are known as Lagrange's equations (or sometimes as the Euler-Lagrange equa- tions). To nish the proof, we need only show that Lagrange's equations are equivalent A A to Newton's. From the de nition of the Lagrangian (2.2), we haveL=x =V=x , A whileL=x _ =p . It's then easy to see that equations (2.8) are indeed equivalent to A (2.1).  Some remarks on this important result:  This is an example of a variational principle which you already met in the epony- mous \variational principles" course.  The principle of least action is a slight misnomer. The proof only requires that S = 0, and does not specify whether it is a maxima or minima of S. Since L = TV , we can always increase S by taking a very fast, wiggly path with T  0, so the true path is never a maximum. However, it may be either a minimum or a saddle point. So \Principle of stationary action" would be a more accurate, but less catchy, name. It is sometimes called \Hamilton's principle".  All the fundamental laws of physics can be written in terms of an action principle. This includes electromagnetism, general relativity, the standard model of particle physics, and attempts to go beyond the known laws of physics such as string theory. For example, (nearly) everything we know about the universe is captured in the Lagrangian  p 1   / L = g R F F + D (2.9)  2 where the terms carry the names of Einstein, Maxwell (or Yang and Mills) and Dirac respectively, and describe gravity, the forces of nature (electromagnetism and the nuclear forces) and the dynamics of particles like electrons and quarks. If you want to understand what the terms in this equation really mean, then hang around for Part III next year  There is a beautiful generalisation of the action principle to quantum mechan- ics due to Feynman in which the particle takes all paths with some probability determined by S. We will describe this in Section 4.8.  Back to classical mechanics, there are two very important reasons for working with Lagrange's equations rather than Newton's. The rst is that Lagrange's equations hold in any coordinate system, while Newton's are restricted to an inertial frame. The second is the ease with which we can deal with constraints in the Lagrangian system. We'll look at these two aspects in the next two subsections. 12 2.2 Changing Coordinate Systems We shall now show that Lagrange's equations hold in any coordinate system. In fact, this follows immediately from the action principle, which is a statement about paths and not about coordinates. But here we shall be a little more pedestrian in order to explain exactly what we mean by changing coordinates, and why it's useful. Let q =q (x ;:::;x ;t) (2.10) a a 1 3N where we've included the possibility of using a coordinate system which changes with time t. Then, by the chain rule, we can write dq q q a a a A q_ = = x _ + (2.11) a A dt x t In this equation, and for the rest of this course, we're using the \summation convention" in which repeated indices are summed over. Note also that we won't be too careful about whether indices are up or down - it won't matter for the purposes of this course. To be a good coordinate system, we should be able to invert the relationship so that A A A x =x (q ;t) which we can do as long as we have det(x =q )6= 0. Then we have, a a A A x x A x _ = q_ + (2.12) a q t a A A A Now we can examine L(x ;x _ ) when we substitute in x (q ;t). Using (2.12) we have a   A 2 A 2 A L L x L x x = + q_ + (2.13) b A A q x q x _ q q tq a a a b a while A L L x _ = (2.14) A q_ x _ q_ a a A A We now use the fact that we can \cancel the dots" and x _ =q_ =x =q which we a a A can prove by substituting the expression forx _ into the LHS. Taking the time derivative of (2.14) gives us       A 2 A 2 A d L d L x L x x = + q_ + (2.15) b A A dt q_ dt x _ q x _ q q q t a a a b a So combining (2.13) with (2.15) we nd       A d L L d L L x = (2.16) A A dt q_ q dt x _ x q a a a 13 Equation (2.16) is our nal result. We see that if Lagrange's equation is solved in the A x coordinate system (so that ::: on the RHS vanishes) then it is also solved in the q coordinate system. (Conversely, if it is satis ed in the q coordinate system, so the a a A LHS vanishes, then it is also satis ed in thex coordinate system as long as our choice A of coordinates is invertible: i.e det(x =q )6= 0). a So the form of Lagrange's equations holds in any coordinate system. This is in contrast to Newton's equations which are only valid in an inertial frame. Let's illustrate the power of this fact with a couple of simple examples 2.2.1 Example: Rotating Coordinate Systems Consider a free particle with Lagrangian given by 1 2 L = mr _ (2.17) 2 with r = (x;y;z). Now measure the motion of the particle with respect to a coordinate system which is rotating with angular velocity = (0; 0;) about the z axis. If 0 0 0 0 r = (x;y;z ) are the coordinates in the rotating system, we have the relationship 0 x = x cost +y sint 0 y = y costx sint 0 z = z (2.18) Then we can substitute these expressions into the Lagrangian to ndL in terms of the rotating coordinates, 0 0 2 0 0 2 2 0 0 2 1 1 L = m(x _ y ) + (y _ +x ) +z _ = m(r _ + r ) (2.19) 2 2 In this rotating frame, we can use Lagrange's equations to derive the equations of motion. Taking derivatives, we have L 0 0 = m(r _  ( r )) 0 r   d L 0 0 = m( r + r _ ) (2.20) 0 dt r _ so Lagrange's equation reads   d L L 0 0 0 =m( r + ( r ) + 2 r _ ) = 0 (2.21) 0 0 _ dt r r The second and third terms in this expression are the centrifugal and coriolis forces respectively. These are examples of the \ ctitious forces" that you were warned about in 14

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