Classical Mech problems help

[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.

 

[quicknote]

Sure, I can provide some hints and guidance for solving these problems. First, it's important to remember that in classical mechanics, the momentum p is defined as mv, but in special relativity, it is defined as p = γmv, where γ is the Lorentz factor given by γ = 1/√(1-(v/c)^2). This is important to keep in mind when approaching problems involving relativistic dynamics.

For #3, you will need to use the definition of momentum p = ∂L/∂ẋ to express the relativistic momentum in terms of velocity. To do this, you will need to take the partial derivatives of the Lagrangian L with respect to ẋ and substitute in the expression for p given above. This will give you the relativistic momentum in terms of velocity.

For #4, you will need to use the Lagrangian from #3 to find the Hamiltonian. Remember that the Hamiltonian H is defined as H = ṗẋ - L, where ṗ is the time derivative of momentum. You will need to use the definition of momentum from #3 and the Lagrangian to express the Hamiltonian in terms of p and x only.

For #5, you will need to use Hamilton's equations of motion, which state that ṗ = -∂H/∂x and ẋ = ∂H/∂p. You can use the Hamiltonian from #4 to express these equations in terms of p and x, and then solve for ẋ in terms of p. This will give you the velocity of a relativistic particle in terms of its momentum.

I hope these hints are helpful in guiding you towards solving these problems. Remember to always keep in mind the differences between classical mechanics and special relativity, and to use the appropriate definitions and equations for each. Good luck!