Motion of a mass m confined to the x-axis (Hamiltonian)

In summary: F is the force, d is the displacement, and U is the potential. If the potential is positive, then the displacement will be positive, and if the potential is negative, the displacement will be negative.
  • #1
jmm5872
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Consider a mass m confined to the x-axis and subject to a force Fx=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.


It is the part in parenthesis that confuses me. I can't picture what a negative value of energy would be.

I know the potential is U(x) = (1/2)kx2, and that Total energy is kinetic plus potential (E = T + U). I also assume that the potential is always positive. If this is true, then the only was for the energy to be negative is to have the kinetic be negative and larger than the potential.

Does this refer to the case when the mass is moving in the -x direction giving T = -(1/2)mv2?

It seems to me that the motion should be the same whether the energy is negative or positive since this is a classical mass and confined to the parabolic potential well. It seems like it should oscillate back and forth for any energy.

For some reason I don't think I am picturing this correctly.
 
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  • #2
You've written down the potential for the force F = -kx, but your force doesn't have the negative sign. The potential should be U(x) = -1/2 kx2. Now E<0 should make sense to you.
 
  • #3
The problem specifically states that k > 0, and F = kx, which gives U = 1/2 kx^2. I don't really understand what you are trying to say, or where the F = -kx that you wrote comes from.
 
  • #4
I'm saying your belief that F=kx implies U = 1/2 kx2 is wrong. Look up how to find the potential from a force or vice versa.
 
  • #5
Ah, okay, I understand now. I forgot that the potential is opposite the sign of force! Thank you!

F = -dU/dx
 

FAQ: Motion of a mass m confined to the x-axis (Hamiltonian)

What is the Hamiltonian in relation to the motion of a mass m confined to the x-axis?

The Hamiltonian is a mathematical function that describes the total energy of a system in classical mechanics. For the motion of a mass m confined to the x-axis, the Hamiltonian represents the total energy of the mass along the x-axis, taking into account its kinetic and potential energy.

How is the Hamiltonian related to the equations of motion for a mass m confined to the x-axis?

The Hamiltonian is related to the equations of motion through Hamilton's equations, which state that the time evolution of a system is determined by the partial derivatives of the Hamiltonian with respect to the position and momentum variables. In the case of a mass m confined to the x-axis, the equations of motion describe how the position and momentum of the mass change over time.

What are the advantages of using the Hamiltonian formalism for the motion of a mass m confined to the x-axis?

The Hamiltonian formalism has several advantages for studying the motion of a mass m confined to the x-axis. It allows for a more concise and elegant formulation of the equations of motion, and it provides a clearer understanding of the conservation of energy and momentum in the system. Additionally, the Hamiltonian formalism is more useful for systems with multiple degrees of freedom, making it a valuable tool for studying more complex systems.

Can the Hamiltonian be used to describe motion in other dimensions besides the x-axis?

Yes, the Hamiltonian can be used to describe motion in any number of dimensions. In the case of a mass m confined to the x-axis, the Hamiltonian only takes into account the motion along the x-axis. However, for systems with multiple dimensions, the Hamiltonian will take into account the motion along all of the dimensions.

How does the Hamiltonian change if there are external forces acting on the mass m confined to the x-axis?

If there are external forces acting on the mass m confined to the x-axis, the Hamiltonian will include terms that account for these forces. This will result in a change in the equations of motion, as the external forces will affect the position and momentum of the mass along the x-axis. However, the overall structure of the Hamiltonian will remain the same, with the addition of these external force terms.

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