Introduction to Elementary Particles

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Lecture 3 9 Oct 9, 2002 3 Special Relativity Before going any further, we quickly review the evidence and kinematics of special relativity. 3.1 A Short Review of Classical Mechanics All classical mechanics is based on F = m a m d v / dt m d 2 x / dt 2 , where F denotes the vector sum of all forces acting on the object of mass m . The result is an acceleration a ( t ) of the object in the direction of the net force. The goal of classical mechanics is to deduce/predict the full trajectory of the object over time x ( t ), i.e. predict where it will be at any time t in the future based on full knowledge of the forces at play and initial conditions (position x 0 and velocity v 0 at a starting time t =0). Vice versa, given a detailed measurement of the trajectory x ( t ) of an object, we want to deduce the details of the force(s) that played a role. In general, the force F may depend on location x and velocity v of the object, as well as depend on in- trinsic properties like orientation, charge, spin, color charge, mass, and so on: F = F ( x , v = d x / dt ,…, t ). Only in the simplest cases for F can we solve the equations of motion analytically, in all other cases computer solutions must be found. A few simple cases are listed Table 3 below. Table 3. Some common types of forces and resulting analytical trajectories Force type Equations of Motion Shape Constant; e.g. near- Earth gravity F = constant a = constant 0 0 00 0 0 22 0 0 11 () t tt d dt d t t dt d dt t dt d dt ≡⇒ = →= −→ = + ≡⇒ = + = →+ = −→=++ ∫∫ v v x x v aa v a v v v v x vv v ax vax xx x va a Parabolic trajectory Proportional; e.g. spring force F = – kx 2 max 2 max 0 0 0 2 max 0 0 2 0( ) s i n ( ) , with: , where and are determined by and : if 0; else: tan , and tan 1 dx mk x x t x t dt k m x xv x x v x ωφ ω φ +=⇒ = + = == += + Harmonic motion Inverse square e.g. Coulomb or Gravity 2 / kr =− Fr ± Central force problem: (Keppler equations) Elliptical or hyper- bolic trajectories The conservation of momentum and energy is a consequence of Newton’s Laws: 1. if 0 then constant Momentum conservation net i net i d dt ==⇒ = = p FF F p
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Lecture 3 10 Oct 9, 2002 2. 22 2 2 11 1 1 () 21 2 1 RHS of 1 (ork) ( ) 2 Fm a d m d m d m d m v v K K dt = ≡⋅= ⋅= = −≡−= ∫∫ xx v x v v x v v Fx x vv ±²³²´ Wd (I.15) The possibility for integration of the LHS of F = m a depends on the nature of the force F : many “well-behaved” forces can be written as a K gradient of a potential : F ( x ) = – dU / d x . Only for this type of “conservative” force the integral can be done, and as result total mechanical energy is conserved: x 1 x 2 F d x = U ( x 2 ) + U ( x 1 ) −∆ U = K U + K = 0. Thus, with total energy defined as E U + K we have (for conservative forces): E = 0, i.e. Energy conservation. For non-conservative forces the integral can- not be done: that means that such forces cannot be written as a gradient of a potential energy function. An example of a non-conservative force is simple friction: the total work done by friction depends on the detailed trajectory taken moving from position x 1
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Lecture 03 - Lecture 3 9 Oct 9, 2002 3 3.1 Special...

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