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PHYS195 NotesChapter14

# PHYS195 NotesChapter14 - Chapter 14 Dynamics 14.1...

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Chapter 14 Dynamics 14.1 Relativistic Action As stated in Section 4.4, all of dynamics is derived from the principle of least action. Thus it is our chore to find a suitable action to produce the dynamics of objects moving rapidly relative to us. It will be advantageous if that action possess the maximum amount of symmetry. This will produce the largest number of conserved quantities which in turn will simplify the analysis. In other words, in addition to having the usual symmetries of space and time translation, it would be nice to have the action be symmetric under Lorentz transformations. Remember that the classical actions are not symmetric under Galilean transformations but are invariant instead, see Section 5.4.4. This will expand the set of conserved quantities available for the solution of dynamical problems. In the following sections, we will be more careful in our handling of the notation and remember that there are three spatial directions, i. e. the position is x . Where it is unimportant for the interpretation, we will suppress the vector designation. 14.1.1 The Action for a Free Particle In order to discover the action for rapidly moving particles, we should look at simple situations. For the free particle, we know what the natural trajectory in space time is – a straight line. We want this to be the least action. In addition, if we want the set of Lorentz transformations to be a symmetry for for this action, we should construct it from form invariants of the Lorentz Transformations, see Section ?? , and Section 12.5. For timelike trajectories, the form invariant that characterizes the trajectory is the proper time. The action for the free particle should be dependent on the proper time. The 327

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328 CHAPTER 14. DYNAMICS simplest possibility is linearity. Since action has the dimensions of an energy times a time, we have to multiply the proper time by something with the dimensions of an energy. Fortunately, the relevant dimensionful parameters are available. One is the mass of the particle. In fact, when you think about it this is the definition of mass. Well, actually only that mass that is called the inertial mass. We will expand on this idea in Chapter 15 and below in Section 14.3. Also, since in the case of the free particle, the straight trajectory is the longest worldline between two events, see Section 12.7, the action should be proportional to the negative of the proper time. In this way, the greatest proper time will correspond to the least action. The unique combination that we have been led to is S ( x 0 , t 0 , x f , t f ; trajectory ) = - mc 2 τ ( x 0 ,t 0 ,x f ,t f ; trajectory ) (14.1) = - mc 2 x f ,t f trajectory,x 0 ,t 0 Δ τ (14.2) Place Figure Here This form is inappropriate for the interpretation of an action since it is not time sliced. To transform from segment slicing which is what we have in Equation 14.2 to time slicing, we use the fact that we can relate proper time intervals to coordinate time intervals as Δ τ = ( Δ t ) 2 - ( Δ x ) 2 c 2 . If we now factor out the ( Δ t ) 2 and realize that the velocity in space time is the
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