Physics 69 Hamiltonian Mechanics (2 of 18) The Oscillator - Example 1

In this video I will find the equations of a simple oscillator of a mass attached to a spring using the Hamiltonian equations.

Next video in this series can be seen at:

[KlausDieckmann]

He is the only professor whom I can follow without difficulties. My hope is to understand all the stuff.

[PatrickMulwa-uddw]

Thanks professor with you the concepts looks very easy

[finnmulder]

I have a question: Why must the m*d/dt( ∂H/ ∂p) + ∂H/ ∂x equal 0? Can that be understood from the equations above, and if so how? The rest is all more than clear. Thanks

[sidereal]

Excellent video series. Thank you for all the help!

[abhishikthraj]

a very clear explanation which i really like

[qeditz]

If we solve problem..with lagrangian and Hamilton result will be same in every cases both proof Newtonian Difference is that in lagrangian we take q dot...and in Hamilton we convert q dot into momenta..(p)...but I want to know more difference..plz help me wid that

[willie]

Dr. van Brezen Please note that I apologized for my complaints after you replied but it got buried in the replys. Thankyou

[mark.p]

Sir would it be possible to have some more lecturs on hamiltoniam mechanics and maybe one or two explaining the difference between lagrangian and hamiltonian ?

[TheUNdErMiNdEd]

Great video! Is it possible that you could show us how to derive the first order differential equations as opposed to second order?

[vih]

So why do Hamiltonian AND Lagrangian mechanics exist? I mean, is there any difference between them at the end (the output of using them is the same, isnt it)? It doesn't matter which one do you choose when solving a problem right? Or is it more useful if you solved a specific example with hemiltonian instead of lagrangian or vice versa, Because, if I am not mistaken, they both yield the differential equation x(t)... Thank you

[bhagswag]

What caused to to take those derivates after you wrote down the T + V equations?

As in, how did you know to do dH/dx, dH/dp, and d(dH/dp)/dt?

[eucsstamticc]

I have a question about the solution to the differential equation at the 4:00 mark. My understanding is that when you solve for the solution you find C1e^(at)cos(wt) + C2e^(at)sin(wt), where C1 and C2 would be your amplitudes and 'a' and 'b' are the values from the imaginary equation a + bi >> 0 +wi. My question is this: Why are we only using the cosine portion of the solution and not both cos and sin? Is it because the oscillation is only happening along the x-direction? My differential equation textbook(11th edition by Zill, 'a first course in DEs with modeling applications) shows both cos and sin in the solution for free undamped motion. I am a math & physics major so I am trying to figure out why a DE class would utilize the entire solution but Hamiltonian mechanics only uses part of it. Thanks in advance.

[jimpoppiti]

When you take the partial derivative dH/dp you get dx/dt (velocity). The mass cancels out. Then you take the derivative with respect to time. You should just get d2x/dt2. In the video you show this derivative as
m d2x/dt2. Where does the m come from?

[kevincarrasco]

Hi excellent video,
I just didn't understood the part where you rewrite the equation why is m below and where do we take the omega and the amplitude?



Could you please clarify this part a little bit


Thank you

[madridboy]

I don’t know how to solve the differential at last. Can you explain me.

[abeshudug]

I'm guessing this is only for ideal oscillators as there is no damping ratio involved?

[olgermannik]

What if you have many moving objects that have different momentums and positions? Do you take different deriatives of these?

[Dantheon]

But isn't the momentum a function of x? Don't you need to differentiate the first term of the hamiltonian as well?

[noreenbargah]

thank you so much for the video.
but i want to know what book or source you are lecturing ?

[homamhassn]

How can the energy (the Hamiltonian) changes with position
Does that violet the conversation of energy