Dynamics and Relativity

Transcription

1 Dynamics and Relativity Stepen Siklos Lent term 2011 Hand-outs and examples seets, wic I will give out in lectures, are available from my web site Lecture notes, wic I will not give out, are also available on my web site: tey will appear after eac lecture.

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3 Capter 0 Preliminaries 0.1 Course contents Section 0: Preliminaries 0.1 Course plan 0.2 Some vector calculus. Section 1: Newtonian dynamics: basic concepts 1.1 Newton s Laws 1.2 Dimensional Analysis Section 2: Forces 2.1 Potentials 2.2 Conserved quantities (including angular momentum) 2.3 Friction 2.4 Te Lorenz force 2.5 Gravitational force 2.6 Escape velocity Section 3: Orbits 3.1 Motion in a plane 3.2 Orbits in a central force: te u-θ equation 3.3 Closed orbits 3.4 Stability of circular orbits 3.5 Orbits in an inverse square force 3.6 Ruterford scattering 3.7 Kepler s Laws 3.8 Orbits under more general forces 3

4 4 CHAPTER 0. PRELIMINARIES Section 4: Rotating Frames 4.1 Angular velocity 4.2 Rotating axes 4.3 Coriolis force 4.4 Centrifugal force Section 5: Systems of particles 5.1 Equations of motion 5.2 Variable mass problems 5.3 Two-body problem 6.4 Moments of inertia 5.5 Inertia tensor 5.6 Motion of a rigid body Section 6: Special Relativity 6.1 Concepts and laws 6.2 Space-time diagrams 6.3 Lorentz transformations 6.4 Lengt contraction and time dilation 6.5 Velocity composition 6.6 Proper time and te Minkowski metric 6.7 Four-vectors 6.8 Dynamical examples 6.9 *Doppler effect (if time allows)*

5 0.2. THE COURSE Te course It migt seem odd to combine two seeming very disparate teories Newtonian dynamics and special relativity in one course. However, tere are many ideas common to te two teories and tinking about te fundamental ideas beind Newtonian dynamics can ligt up one s understanding of relativity (and vice versa). It is very striking tat te problems tat Newton was wrestling wit in te seventeent century are exactly tose occupying Einstein 220 years later. Many students will ave seen quite a lot of Newtonian dynamics at scool and in te past ave complained tat tere is little new for tem in te course. Te same students also complained tat tey understood te material better at scool. Of course, wat tis means is tat tey could do te questions at scool witout really understanding wat tey were doing and, believing tat tey didn t need to work at te course, ten found tat tey couldn t catc up. To try to prevent tis appening, I ave avoided setting questions on examples seets tat look like souped up A-level or STEP questions. Many oter students will ave seen very little Newtonian dynamics at scool and will find tis course quite ard to start wit. I advise tese students to stick at it. Very soon, tey sould get to grips wit wat is after all mainly matematics. For te benefit of bot sets of students, I ave provided printed lecture notes. In tese notes, I ave put in more explanation and matematical steps tan is possible (or desirable) in large lectures, and I ave tried also to include some deeper discussions and ideas beyond te course for tose wo really ave covered some of te material. I will not distribute tese notes in lectures: tey can be seen on my web site (wic is presumably wat you are now looking at). 0.3 Some vector calculus Muc of tis course is formulated in terms of vectors. No doubt te following lemma will also be proved in te vector calculus course, but it is important enoug to be proved twice. 1 Lemma Let u(t) be a time-dependent vector 2 wit components u i (i = 1, 2, 3) wit respect to a set of fixed Cartesian axes. Ten ( = 1, 2, ) 3. Note Tis just says tat vectors can be differentiated component by component, in Cartesians. Of course, we assume tat te axes are not time-dependent (moving) and in Capter 4 we deal wit a case wen tis lemma does not old: rotating Cartesian axes. Similarly, if te axes are not Cartesian, te lemma may not old. 3 Proof We can t make progress wit tis witout knowing wat differentiation of a vector means. 1 Make te most of tis: it is probably te last lemma-proof-corollary in te course. 2 To be more precise, we want u to be a differentiable function R R 3. 3 For example, in sperical polar coordinates and axes, te position vector of te particle is (r, 0, 0), i.e. rbr, (NOT of course (r, θ, pi)), but te velocity is (ṙ, r θ, r sin θ φ).

6 6 CHAPTER 0. PRELIMINARIES Taking te usual definition in terms of limits, we ave lim u(t + ) u(t) 0 u 1(t + ) 0 0 u 2 (t + ) u 3 (t + ) u 1(t) u 2 (t) u 3 (t) u 1(t + ) u 1 (t) u 2 (t + ) u 2 (t) u 3 (t + ) u 3 (t) u 1 (t + ) u 1 (t) u 2 (t + ) u 2 (t) 0 u 3 (t + ) u 3 (t) wic is te required result. An alternative approac migt ave been to write (definition of differentiation) (same, in Cartesian components) (usual rules for adding vectors) (usual rule for multiplying vectors by scalars) (definition of differentiation) u = u i e i (e i e j = δ ij, summation convention applies) and ten differentiate to obtain te result immediately: = d(u ie i ) = i e i + de i u i = i e i (since te cartesian axis vectors e i are fixed). If te axes are not fixed, tis result would not old. For example, in plane polar coordinates, r = re r, and dr = dr e r + r de r = dr e r + r dθ e θ. (1) Te second term can be obtained by converting to cartesian axes so tat e r = (cos θ, sin θ). Ten differentiating wit respect to t, using te cain rule, gives de r = dθ de r dθ = θ( cos θ, sin θ) = θe θ. Te last term in (1) represents te compenent of velocity (if t is time) tangent to te circle r = constant, as will be discussed in section 4. Adopting tis alternative approac assumes tat we know ow to differentiate te sum of vectors and te proct of a scalar (u i ) and a vector (e i ). We would ave to start wit a couple of lemmas: d (u + v) = + dv and d(λv) = λ dv + dλ v

7 0.3. SOME VECTOR CALCULUS 7 were u and v are arbitrary t-dependent vectors and λ is and arbitrary t-dependent scalar. Tese are bot easily proved from te definition of a derivative, applied to vectors. 4 For example: d(λv) λ(t + )v(t + ) λ(t)v(t) 0 ( ) λ(t + ) v(t + ) v(t) 0 =λ dv + dλ v ( ) v(t) λ(t + ) λ(t) + lim 0 Corollary Te derivatives of scalar and vector procts obey te Leibnitz (proct) rule. Proof For scalar procts, we ave d(u v) = d(u iv j δ ij ) = d(u iv i ) (using suffix notation and summation convention) For vector procts, = i v dv i i + u i = v + u dv d(u v) (using Leibnitz for eac term in te summation) = v + u dv. (using te above Lemma) Tis can be proved in te same way as for te scalar proct, except te vector proct as ɛ ijk rater tan δ ij ; tis does not affect te proof, because te compenents of ɛ ijk are constants, being 0 or ±1. 4 But ten we migt ask wat exactly we mean by a vector tat is a function of a parameter. It is like a set of Babuska dolls: everytime you go a layer deeper, anoter even deeper layer appears.