Lecture 13

[Music] [Music] Hamilton formalism okay so we need some more energy for for this one so the way we write hamiltonian for particles we can write hamiltonian for Fields okay so momentum density like we have momentum particles so what is this DL by DX dot partial partial L partial x dot so instead of that we have partial L not partial okay it's a it's a functional derivative now L is a function so we get partial L partial Delta L Delta Pi dot now I is a component it could be X component y component G component I get partial L partial 5 dot okay this is now L is integral l d4x so when I go inside then it's going to be partial when I I'm so this is total l d Cube not D sorry DQ so total L I have to do the functional derivative because it's is function of any F but when I go inside this legent density is function of five then I do partial D derivative okay now that is momentum I defined this is a definition Hamilton density is pi I this is pi I Pi I dot minus L is very similar to what we do for particles Pi i x i dot minus L okay and we Define total hamiltonian is integral over space okay so now uh equation of motion so like we have x dot is derivative of H know with pi dot so we have f i dot equal to partial h no there's no derivative this is old slide actually Delta H Delta pii and we have Pi dot is uh Delta H Delta pii okay so I think I have an example so we have this example for this I'm going to write down the hamiltoni this leg leg is function of 5 dot but hon is function of five and Pi okay so this dot is not there so Delta L Delta 5 so this this this dot is there this dot is there sorry this dot is there Delta l f a DOT like Delta L Delta XI dot is there no for for and that is electron derivative which is 5 dot this one squ I take the derivative I'll get Five Dot okay so Pi is nothing but Five Dot which is like momentum is x dot very similar the formalism is in fact motivated from particles so hamiltonian will be Pi 5 I do minus L now Pi is nothing but Pi dot so U this guy will give us um okay so what will I get Five Dot into Five Dot which is 5 dot Square minus this one minus half 5 dot square that is this one okay then this minus minus becom plus so plus gr 5^ squ plus V5 okay now Five Dot is nothing but Pi so this is my hamiltonan density it should be function of Pi and five and gra F but not Five Dot okay don't write five dot for hamiltonian the way for particles we don't write X do hamiltonian we write only momentum okay so equation of motion so F dot is Delta this is not there Delta H Delta Pi I and I = 1 for this is 1D so uh this is going to give us P do equal to Delta H by Delta Pi will be Pi okay now this is not do here so Pi dot is Delta H by Delta 5 with a minus sign there a minus sign here okay now um so so we get Delta H by Delta 5 so this is a functional derivative okay so functional derivative I again apply the logic of functional derivative is function of the D five as well as gra five will come okay both the terms will come so I get partial HED partial 5 minus this Divergence of partial HED partial gr F okay this is this will come and uh uh this one is this hamiltonian so function of derivative five will give us will come from here V5 so that is V5 and partial Ed partial gry will give us gry here and Divergence so Di of a grad is Lain okay that so this is pi dot okay Pi Dot and Pi dot is well Pi is p dot so this is nothing but Pi double dot or this it is square okay right so that I'm just using this this relation so this equation okay so uh field with the potential the equation oil equation is this okay and there are potential there are in fact we'll encounter uh Fields with potential okay you might have seen the free energy you know for for for phe transition is 5^ s minus 54 so these are the potential okay and U so this is the example I think I will do the other example later because I think is already close to saturation so this one is a quite important one so you can write down legin for short equation not short non relativistic Notting and I will do it in the next class in fact I think we need to spend some time on it so we'll do it in the next class [Music] [Music] [Music]