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seen Jul 14 at 17:40

Mar
22
awarded  Notable Question
Mar
12
accepted Bogoliubov transformation with a slight twist
Mar
12
answered Bogoliubov transformation with a slight twist
Mar
11
asked Bogoliubov transformation with a slight twist
Jan
20
awarded  Quorum
Jan
13
accepted Missing terms in Hamiltonian after Legendre transformation of Lagrangian
Jan
9
comment Missing terms in Hamiltonian after Legendre transformation of Lagrangian
Another question if you will. I was advised to try the problem in the coulomb gauge (which is where I'm ultimately heading), and in this case where $\nabla \cdot \vec{A}=0$ and $\phi = 0$ I have a similar situation where my derived Hamiltonian is the same as the Lagrangian apart from sign flips and the fact I am 'missing' the term $\frac{1}{c}\vec{A}\cdot\frac{\partial \vec{P}}{\partial t}$. However in this gauge there is no $\phi$ field and thus no conjugate momenta which are equal to zero. In this situation can we still use similar arguments as you presented above? Thanks.
Jan
9
comment Missing terms in Hamiltonian after Legendre transformation of Lagrangian
Thank you once again @Qmechanic, Dirac_Bergmann analysis is certainly new to me, I don't suppose there are any texts you'd particularly recommend on the subject? My internet hunt hasn't returned anything that good so far.
Jan
8
awarded  Commentator
Jan
8
comment Missing terms in Hamiltonian after Legendre transformation of Lagrangian
Hmm yes, as is the case in my example here, the 'missing' terms are linear in time derivatives of quantities that are functions of spatial coordinates, so kind of like a velocity. I could believe my Hamiltonian is correct, but I am doubting myself due to these 'missing' terms, something I have not before encountered.
Jan
8
revised Missing terms in Hamiltonian after Legendre transformation of Lagrangian
edited body
Jan
8
asked Missing terms in Hamiltonian after Legendre transformation of Lagrangian
Jan
8
comment Is it physically realistic to have an electric field and polarisation density but no displacement field?
Ah you make a good point, yes as $\vec{D} = \epsilon \vec{E}$ if $\vec{D}$ is zero then the permittivity must be zero, which it isn't as I derived it earlier. Ok back to the start I suppose!
Jan
7
asked Is it physically realistic to have an electric field and polarisation density but no displacement field?
Dec
31
accepted Derivation of Lagrangian density for an infinite classical dielectric in interaction with the EM field
Dec
31
comment Derivation of Lagrangian density for an infinite classical dielectric in interaction with the EM field
This seems like just the kind of answer I was looking for. Thank you Qmechanic. Allow me to digest this over the next week and I may well ask for clarification on a few points if I become stuck. Many thanks.
Dec
22
revised Derivation of Lagrangian density for an infinite classical dielectric in interaction with the EM field
edited tags
Dec
20
comment Derivation of Lagrangian density for an infinite classical dielectric in interaction with the EM field
As a further comment, after he provides the equations he says "This is the Lagrangian density for an oscillating polarisation density $\mathbf{P}$ with a restoring force, as can be seen by comparing (3) with the Lagrangian for a moving charged particle"
Dec
20
comment Derivation of Lagrangian density for an infinite classical dielectric in interaction with the EM field
Thanks for the response. Firstly I'm not very familiar with this Lagrangian using electromagnetic tensors. The change from relativistic to non-relativistic, is this all quite straightforward? From that point onwards I'm with you but it is strange you're not getting those other terms. He doesn't seem to define either of these quantities, it's rather irritating. At a very rough guess $\beta$ is the wave vector inside the medium and $\omega_0$ is something to do with natural frequency of the displaced electrons perhaps? I'm not sure though
Dec
20
revised Derivation of Lagrangian density for an infinite classical dielectric in interaction with the EM field
edited tags