Question

Hamiltonian

Answer 1

@Michele_Laino

Answer 2

Ok so my question is, when you get to the end of the problems and need to find the equation of motions we use \[\huge \dot{q_i} = \frac{ \partial H}{ \partial P_i }~~~~P_i = - \frac{ \partial H }{ \partial q_i }\] right

Answer 3

So when you find these is that the final solution or do we still have to take the time derivative of these and add them up to get the final equation of motion?

Answer 4

you have to take the derivative of H with respect to q_i and p_i respectively, and then you have to integrate them with respect to time

Answer 5

so you will get the functions q_i and p_i

Answer 6

Oh right because that should P_i dot there right, which then gives you P_i and q_i

Answer 7

So once you integrate it, do you just add the two functions or leave them be?

Answer 8

yes! The hamiltonian equations are:
\[\large \begin{gathered}
\dot q = \frac{{\partial H}}{{\partial p}} \hfill \\
\dot p = - \frac{{\partial H}}{{\partial q}} \hfill \\
\end{gathered} \]

Answer 9

Oh wait so we integrate it as \[\int\limits \dot{P_i} dt\]

Answer 10

yes!

Answer 11

similarly for \[{\dot q}\]

Answer 12

Oh I kept thinking we integrate it as \[\int\limits \dot{p_i} di\] haha

Answer 13

:)

Answer 14

So once the integrate is done, that will be our equation of motion right, mhm interesting, it's hard to tell sometimes without units xD.

Answer 15

yes! please keep in mind that the functions q and p are important in statistical physics

Answer 16

Mhm ok weird in my notes I have an example that is integrated \[\int\limits \dot{P_i} di \]

Answer 17

I think it is a typo error, since i is only an index which runs over the degrees of freedom of our mechanical system

Answer 18

So it shows \[\dot{P_y} = - \frac{ \partial H }{ \partial y } = - mg \implies P_y = - mgy+C\]

Answer 19

please wait...

Answer 20

No problem

Answer 21

\[\large {p_y} = -\int {mg} \;dt\]

Answer 22

Yes that would be -mgt+C no?

Answer 23

yes!

Answer 24

Mhm wonder why there's a y.

Answer 25

since you have found H

Answer 26

\[\large H = mgy + f(x,t)\]

Answer 27

since, in general H= H(x,y,t)

Answer 28

Oooh ok, thanks :)

Answer 29

thanks! :)