How to Solve Special Relativity Dynamics Problems in Classical Mechanics?

[belleamie]

Does anyone know how to solves these, hints anything? I was able to do #1,2 and got lost at 3 and without 3 I couldn't do 4 and then i couldn't do 5 :(
PLease help

#3 in special relativity the dynamics are different. In particular, the momentum p is no longer mv. Failure to remeber this causes a lot of trouble. THe lagrangian of one dimensional speical relativity is
L(x, x(dot))= -mc^2 (sqroot of (1-(xdot/c)^2)-V(x)
where c is the speed of light. Use the definition of hte momentum p= partial derviatives of (L/xdot) to express the relativistic momentum in terms of velocity.

#4 use the lagrangian of #3 to find the hamiltonian of relativistic particle. Remember that H must be expressed as a function of p and x only. This hamiltonian is the energy of a relativistic particle.

#5 Use the hamiltonian of problem 4 and hamilton's equations of motion to express the velocity of a relativistic particle in terms of its momentum.

 

[HallsofIvy]

For #3, the best hint is to tell you to do exactly what the problem says: You are given L(x,xdot) so you can calculate L(x,xdot)/xdot. Now find the partial derivatives of that.
#4: I presume you know how to form the Hamiltonian of a classical particle from the Lagrangian. As the problem says, be sure you use p instead of xdot. In classical mechanics, p is just mass times xdot but in relativity you will need the formula you get in #3. Write the Hamiltonian in terms of x and xdot, solve the formula you got in #3 for xdot as a function of p, and substitute.

#5: Put the Hamiltonian you got in #4 into Hamilton's equations and solve for the velocity.