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Unformatted text preview: Ph-507. Homework 1 (due Mon, January 24).PROBLEM 1-1 (1 pts).Starting with the Newtons laws, prove additivity of mass.PROBLEM 1-2 (3 pts).By minimizing the functionalS, derive the diferential equation forz(t):S=TZdt"d2zdt22+dzdt4+z4#.PROBLEM 1-3(5 pts).Special theory of relativity is based on the postulate thatspeed of lightcis independentof the choice of(inertial)reference frame (RF). Consider two observers, one moving with aninfnitesimalvelocitywith respect to the other. In order to satisfy the above postulate, the Galilean transformation of the coordinatesbetween the two RFs:r=r+t,(1)should be combined with the following time correction:t=t+rc2.(2)The two equations, (1)-(2), are the linearized (in) versions of the relativistic Lorentz transformations.a) Find the transformation for the velocity of a moving particle (vv) associated with this change of the RF(neglect the subdominant termsO2). Show that speed of lightcis indeed independent of RF.b) Use the physical equivalence of the two RFs, tofnd the general form of LagrangianL(v)for a free relativisticparticle (i.e.v/cis not small). (Hint: use the fact that the change of action due to the transition to a new RF is:S=RL(r, v, t)dtRL(r, v, t)dt=RhL(r, v, t)dtdtL(r, v, t)idt, weredt/dtis the full derivative takenalong the particle trajectoryr(t)).c) Determine the unknown parameter in your Lagrangian by taking the limitvcand comparingL(v)with theknown classical result....
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