Velocity phase space and energy diagrams

[phys2]

Homework Statement

Ok I have attached the pdf file and I have a problem with velocity phase spaces (Question 3a). Honestly, the lecture notes were not very helpful and looking online and in textbooks, they talked about solving Lagrange's equations but nothing to deal with the problem of Q3. These are the questions I have;

(a) So I understand that phase spaces are what you get when you solve Lagrange's equations; ie they will be sketches of paths?

For example; if I took the harmonic oscillator (with no driving forces or friction), solving using either lagrange or plain Newtonian way; i would get x(t) = Acos(wt) + Bsin(wt). What I planned to do was take dx/dt and then plot v (on y axis) and t on (x-axis). Would that give me the velocity phase space of the harmonic oscillator?

(b) The next problem would be dealing with energy diagrams. Suppose I have the energy diagram of the harmonic oscillator with V = 0.5kx² and I drew that on the y-axis and position on the x-axis. I would get a parabola originating at x = 0. Then what do I do?

Sorry for all the questions; preparing for a test soon! Thanks in advance

Homework Equations

The Attempt at a Solution

 

[Nate810]

For (a) - Yes, plotting the velocity and time on the x and y axis respectively would give you the velocity phase space of the harmonic oscillator. For (b) - The energy diagram is a representation of the total energy of the system at any given time. To plot the energy diagram, you need to first calculate the total energy of the system. This can be done by summing up the kinetic and potential energies of the system. You can then plot the total energy of the system on the y-axis and the position of the system on the x-axis - this will give you the energy diagram.