- #1
Blakely42
- 6
- 2
- Homework Statement
- Consider a mass m confined to the x axis and subject to a force F = kx where k > 0. Write down and sketch the potential energy U(x) and describe the possible motions of the mass. (Distinguish between the cases that E > 0 and E < 0.
- Relevant Equations
- E = T + U
L = T - U
F = -Grad(U)
H(p,x) = T + U
U(x) = - ∫Fdx = - (1/2)kx^2. T = (1/2)m(x')^2. E = (1/2)[m(x')^2 - kx^2]. We could write out the Lagrangian here, but the chapter this comes from (Taylor's Classical Mechanics 13.6) indicates we should probably write the Hamiltonian, H = T + U.
As far as I can tell, this doesn't tell me a single thing about what happens when E < 0 or E > 0. Even chapter 13 only deals with phase diagrams of x vs p. How am I supposed to translate this idea to an x vs U(x) diagram? I don't think there's enough information to do that here.Update: My professor answered my email, saying to think of it like a central force problem, but central force problems have energy from their rotational momentum; we use that energy in calculating U_eff(x). I don't see how we can do that here. It's not just an issue of getting the correct answer. I'm trying to understand how the equations yielded by the Hamiltonian actually tell me the behavior. i.e. how do the equations translate to English?
As far as I can tell, this doesn't tell me a single thing about what happens when E < 0 or E > 0. Even chapter 13 only deals with phase diagrams of x vs p. How am I supposed to translate this idea to an x vs U(x) diagram? I don't think there's enough information to do that here.Update: My professor answered my email, saying to think of it like a central force problem, but central force problems have energy from their rotational momentum; we use that energy in calculating U_eff(x). I don't see how we can do that here. It's not just an issue of getting the correct answer. I'm trying to understand how the equations yielded by the Hamiltonian actually tell me the behavior. i.e. how do the equations translate to English?
Last edited:
