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3:43 AM
Is the formula for area of the hexagonal lattice in the wiki article wrong?

https://en.wikipedia.org/wiki/Bravais_lattice
Sorry I miscalculated
 
 
1 hour later…
123
5:08 AM
Hi , Good Morning
 
I just looked up the meaning of the word foray, and I was surprised to see its usage versus time graph.
I mean, how come this 19th century word, which almost went extinct in the 20th century, rise back up, higher than ever, in the early 21st century?
 
5:37 AM
@SirCumference I protest! Not even a single L&L and Multon :P
 
vzn
lol its simple, foray has gone on a new foray :P
 
It's for array();
 
vzn
and who else on 1st glance saw a coronavirus curve there? ... cabin fever, stir crazy o_O
 
_Bool Hand.raise = true;
if (!anyone_listening)
    bye();
else
    say.hi(&username, *Azmuth);
 
 
9 hours later…
2:28 PM
When we do regular QM and deal with the spin matrices which are the spin-1/2 representation of $\mathfrak{su}(2)$, are we dealing with left or right handed Weyl spinors? Does it make a difference?
Perhaps it depends whether we are technically talking about a spin-1/2 particle or its anti-particle (even though they don't appear in non-rel qm)
 
Anti-Particle comes only in QFT.
 
Sure, but Dirac spinors are the direct sum of left and right handed Weyl representations of $\mathfrak{su}(2)$ and we are (at least to my understanding) strictly using Weyl spinors in the treatment of spin in qm
Also RQM allows anti-particles, not just QFT afaik
 
Not sure about $SU(2)$ thingy, I skipped all those :P
 
Always a dangerous thing to do :P
 
hehehe XD :)
@ACuriousMind
 
3:04 PM
@Charlie There are no right- or -left handed spinors in non-relativistic QM
 
oh
 
The handedness is a peculiarity of the representation theory of $\mathfrak{so}(1,3)$, which is in finite dimensions equivalent to that of $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$. It's the two copies that lead to the handedness - a chiral particle transforms in the trivial rep of one of these $\mathfrak{su}(2)$s but in a non-trival rep of the other. In non-relativistic QM the symmetry algebra is $\mathfrak{su}(2)$ to begin with and there is no notion of handedness
 
ohh I see, ok tyvm
 
the non-relativistic $\mathfrak{su}(2)$ sits diagonally inside the relativistic $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$, which is why we say that things transforming in the $(s_1,s_2)$ rep of $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$ have a spin of $s_1 + s_2$ - the $\mathfrak{su}(2)$ rep labeled by $s_1 + s_2$ is what they transform under in the non-rel limit
so both a left-handed $(1/2,0)$ and a right-handed $(0,1/2)$ get mapped to a simple spin-1/2 particle in the non-rel scenario
 
Oh that's why in qed we use $A_\mu$, a four-vector in the $(1/2,1/2)$ rep for spin-1 photons
never crossed my mind
 
3:43 PM
Uh quick question on the term "unitarily equivalent", in the context of "all $4\times 4$ representations of the Dirac algebra are unitarily equivalent", a phrase used in P&S, this means there exists an invertible intertwiner between them that is unitary wrt. an inner product on the vector space?
I asked in the math chat and got laughed at for using the word intertwiner :(
 
4:09 PM
@Charlie yes
 
tyty
 
123
4:29 PM
Yo
 
5:14 PM
Oct 22 at 19:06, by Sir Cumference
being a physicist means ignoring the angry mathematicians :P
 
@JohnRennie :P
 
123
5:46 PM
Hi all.
 
hi
 
123
@Charlie Symon CM is very good book i am reading
 
oh nice
 
123
Does LM work for only conservative energy
HM i think work on symmetry
 
both Lagrangian and Hamiltonian mechanics do poorly when trying to describe dissipative (="with friction") systems.
some dissipative systems can be described in a Lagrangian framework, but not all, see physics.stackexchange.com/q/147341/50583
 
123
5:58 PM
@ACuriousMind Hi, Okay.. Now i am interested to learn LM and HM.
For HM it is necessary to understand quaternions?
 
not at all, why would you think so?
 
123
We don't need to learn quarternion for HM?
 
I don't see why you would need to. Just because Hamilton also did stuff with quaternions doesn't mean they must feature in everything named after him :P
 
123
Because hamilton work on quaternions for 17years of his life. I thought it uses it
 
No, Hamiltonian mechanics has nothing to do with quaternions, he was just a bit obsessed with them :P
 
123
6:00 PM
@ACuriousMind :P my fault
what is special about HM over LM?
I think law of conservation of energy is that right?
 
I'm not sure what you mean by this
You can describe a system in which energy is conserved with both Lagrangian and Hamiltonian mechanics. Energy is conserved regardless of which formalism you use
 
123
I learn somewhere , it says HM is special kind of LM.
 
No, that's not true. The two formalisms are different formalisms, not subsets of each other, and the Legendre transformation can translate between them.
(alas, the precise nature of that translation can get a bit complicated once you get to constraints and/or gauge symmetries, but don't worry about that for now)
 
123
If the system maintain its motion. Is this the criteria to be system is following conservation of energy?
 
Intertwiner is the math word, they laugh at their own conventions
 
6:08 PM
@123 depending on the situation it can simplify the problem more than LM or Newtonian formalism
also it is theoretically very useful in general
 
123
Also functional analysis is necessary to learn LM or HM?
 
there's also a fourth formalism
In classical mechanics, Routh's procedure or Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions. As with the rest of analytical mechanics, Routhian mechanics is completely equivalent to Newtonian mechanics, all other formulations of classical mechanics, and introduces no new physics. It offers an alternative way to solve mechanical problems. == Definitions == The Routhian, like the Hamiltonian, can be obtained from a...
but neither me nor my professor have ever seen anyone use this in practice
 
It you take quaternions seriously, as a number system, they are basically irrelevant to physics, you can pretend you're using them when using a certain group in qm but you're using the group that just happens to behave similarly
 
@123 just some basics of functional analysis
 
123
Yes i have seen this in wikipedia. But non of book use this formulism
@bolbteppa I see.. good information it helped me what to learn.
 
6:14 PM
Any 'functional analysis' one needs is most likely explained in a QM book as if it's linear algebra
 
123
@bolbteppa Hmm.. Ookay.
Is this the criteria of conservation of energy if motion does not produce heat?
Once i know this criteria i can digest this concept easily. That's why i asked question also to find close deal with physical quantity.
 
if you're looking to understand conservation of energy, other formalisms aren't really gonna help
 
I'm not sure what you want to know - "conservation of energy" means simply that the expression of energy you've written down for whatever system you're considering is constant
 
the hamiltonian is more of a generalization of energy, but it's not gonna be more enlightening on conservation
 
Conservation of energy is a law, it doesn't depend on whether heat is involved @123
 
6:21 PM
if you're currently in your first classical mechanics course, stick to the newtonian formalism for now
 
With the L/H formalisms, you have access to Noether's theorem where we can see that conservation of energy corresponds to time translation symmetry. It's not about heat, although heat is the most common way to dissipate energy away, it's not the only one (another is e.g. radiation)
 
and learn LM after that
@ACuriousMind that's true, i somehow forgot about noether's theorem
is it necessary to use a non newtonian formalism for that tho?
 
123
@ACuriousMind but there should be some criteria
 
(i'll admit my courses spent very little time on noether's theorem)
 
@123 I don't know what you mean by "criteria". Criteria for what?
 
6:24 PM
@123 for conservation of energy?
conservation of total energy is always held in an isolated system
 
123
@SirCumference Yes
 
mechanical energy is only conserved when we don't have e.g. heat, electricity, etc.
 
What kind of criteria do you have in mind?
 
jesus i'm sleepy
 
123
At which we can say this system follow conservation of energy. Like Motion of the system is repeating as it is etc..
 
6:26 PM
All closed systems follow the conservation of energy
 
which energy are you talking about?
 
123
@SirCumference :P :-) don't sleep
 
@Charlie well but what's the definition of a closed system other than "it has conservation of energy" :P
 
From classical mechanics to qft, energy is a conserved quantity
 
it's easy for students to confuse mechanical and total energy until the difference is explicitly pointed out
 
6:27 PM
I feel 123 is looking for something about a system that could tell you it conserves energy other than "it conserves energy", but really the only other thing all conserving systems have in common is time translation invariance via Noether's theorem, as I already said
 
@ACuriousMind Well I'm a chemistry student so I feel compelled to answer "if it has a lid on it"
 
123
@SirCumference Yes you are right. But it can be explained easily.
 
@SirCumference Brownian Mechanics too!
 
@123 conservation of total energy holds if you can move your entire system to a new position without seeing a change in the behavior of the system
 
Wait that's translational invariance, not time translation invariance
if you're talking about Noether's theorem
 
6:28 PM
conservation of mechanical energy holds if total energy is conserved, and we only deal with kinetic and potential energy
 
@SirCumference ...that would be momentum conservation, not energy conservation :P
 
@Charlie sigh did i really say that
 
nuke the chat logs!
 
you have 1 minute until it is enshrined for eternity :P
 
oh well I tried :P
 
123
6:29 PM
@ACuriousMind Time translation invariance is specific to the position. I don't know much about that. Like similar to equipotential surface. Pls correct me.
 
argh it's not enough time
 
What does "specific to the position" mean
 
whatever, ignore that @123
 
C'mon tell me Noether's Theorem is half of classical mechanics
I'll believe you
 
123
As i given example of equipotential surface. If we move along equipotential surface we find symmetry
 
6:31 PM
Symmetry of what?
 
symmetry? along the surface? kiaaa?
 
@Azmuth Brownian mechanics? you mean brownian motion?
 
123
At which we find energy number is same everywhere at that surface.
 
Brownian dynamics (BD) can be used to describe the motion of molecules for example in molecular simulations or in reality. It is a simplified version of Langevin dynamics and corresponds to the limit where no average acceleration takes place. This approximation can also be described as 'overdamped' Langevin dynamics, or as Langevin dynamics without inertia. In Langevin dynamics, the equation of motion is M X ¨ = − ∇ U ( X...
 
You'd find the potential is the same everywhere at that surface, not necessarily the energy
 
6:32 PM
@SirCumference *dyanamics... my bad
 
whatever that would even mean
 
123
@Azmuth :P you are using words of hindi. :O
 
or, I guess the potential energy, nvm
 
@Charlie didn't get it
 
123
@Charlie Oooo... I see
 
6:33 PM
@Azmuth ah, well it's kind of a simplification rather than an equivalent reformulation
 
@SirCumference whatever... Crazy physicists even use Kepler''s laws instead of Newton
 
that is true
 
lmao!
 
The bottom line is that conservation of energy always applies, if you really want to search for a "reason" for this you need Noether's theorem
 
noether's theorem basically tells you that every type of conservation corresponds to a way you can change the system without affecting its behavior
 
6:36 PM
Well there is some intuition on the forms of KE and PE. I told him already ...
 
in loose terms
 
Oct 24 at 4:12, by Azmuth
$$ \int -F \, dx = -m \int v dv $$
 
@Azmuth every time a differential is used, a mathematician dies inside
 
@SirCumference LMAO, I feel hurt!
 
123
I got dizziness...
 
6:38 PM
well i'm not gonna blame you, i've yet to find one physics book that doesn't use differentials in a derivation
 
@SirCumference Well, that's difficult, but I'll write one that uses limits :P
 
@Azmuth You really don't need to repost the entire thing you wrote, everyone can go back to read the transcript if they want to read it
 
gotcha
 
4 messages moved to Trash
 
@123 I feel that
 
6:39 PM
@123 At any rate, it's important to remember the difference between total and mechanical energy
 
123
How to check whether system following conservation of energy or not? few criteria you told me. Is it necessary to put in equation to find conservation?
 
they are the same thing if only KE and PE are present, but they're different if other forms of energy are present
 
@123 Yes. Energy is an abstract quantity.
 
@123 It is always true.
 
@123 are you in an introductory class mech class?
because then you don't need to worry about that
 
6:40 PM
@SirCumference I guess he is high school (just took calculus classes)
 
@Azmuth It is not. There are plenty of systems that don't conserve energy.
 
123
Physicist fight first time in my life :P
 
@ACuriousMind (in classical mech)
 
@Azmuth even globally it doesn't hold in GR
 
@123 There are better fights on internet...
 
6:40 PM
@Azmuth So you think classical mechanics does not involve friction?
 
123
But in your discussion i learnt a lot. Pls carry on
 
@ACuriousMind well total energy is still conserved there :P
 
@ACuriousMind I'm talking in purest classical terms. (also, friction also does works against the system, everytime)
 
just not mechanical
 
Energy is only conserved if you have an isolated system
 
6:41 PM
@SirCumference sure, but classical mechanics does not consider heat
 
123
@JMac Don't confuse me :O
 
so from the viewpoint of classical mechanics a system with friction is a dissipative system
 
@ACuriousMind depends what you call "classical mechanics"
if you just mean "not GR or QM", then it can include heat
 
@SirCumference Newtonian/Lagrangian/Hamiltonian mechanics, isn't that what we're talking about here the whole time? :P
 
i'm sure there were healthy models of heat before QM came along
 
6:43 PM
Friction isn't a hurdle, Total energy always remains conserved. It just gets converted to heat and stuff like that
@SirCumference Let's bring string theory too
AdS/CfT
 
@Azmuth has anyone ever used string theory to model heat?
 
123
My question is that. There must be something at which we can tell about the system is conservative without putting everything in equations.
 
@SirCumference Ask ACM, I'm for sure not the guy you are searching around
 
@SirCumference sure, classical statistical mechanics works fine to some degree, it's just not what I picture under "classical mechanics" when we're talking about using Lagrangian mechanics to describe motion.
 
I don't know what they are, but they are fun
 
6:44 PM
@ACuriousMind i guess it's more of an application of classical mechanics
 
@123 Why "must" there be something? If you think it is a requirement that all physics can be done without math, you are sorely mistaken.
 
@Azmuth You still need to be careful when you say that. Energy is conserved universally; but locally energy can be lost or taken from the surroundings.
 
Raise your hand if you hate Classical Mechanics Homework!
 
i consider physics classical until we modify some fundamental axioms, e.g. assume time doesn't flow the same in all reference frames, or assume particles can be in superposed states
 
@JMac Oh yes, I should mention, globally
 
6:45 PM
@SirCumference yes it's classical, but it's not "classical mechanics" in the narrow sense. Whatever, we're not actually disagreeing about anything except naming :P
 
@SirCumference You are speaking the language of gods!
Local Time (exists)
 
123
@ACuriousMind No i did not say this. But if any one parameter (distance, speed, acceleration , angle etc..) change it effect other. On the basis of this if system does not follow conservation of energy there must be some clear effect on motion at which we can say system is conservative or not.
 
again, I don't know how you arrived at "must" there
the only universal characteristic of a system that does not conserve energy is that the number we call "energy" is not constant :P
 
@SirCumference I think you might be thinking of "classical physics" vs "classical mechanics"; which is honestly a difference I haven't thought of until now.
 
@123 please be clear whether you are talking about total or mechanical energy
@JMac i mean all of classical physics is based on classical mechanics, i'm not sure if the difference is too significant
 
123
6:48 PM
@ACuriousMind sorry exclude "must" my words.
 
classical mechanics in its basic form is actually very logical, rather than just intended to be an approximation. it just makes some wrong assumptions
 
123
@SirCumference mechanical energy first. Because total energy also contain other types of energy.
 
If I may throw an unrelated question in here, in P&S the relationship $\Lambda_{1/2}^{-1}\gamma^\mu \Lambda_{1/2}={\Lambda^\mu}_\nu\gamma^\nu$ is seemingly derived from the infinitesimal Lorentz and $SU(2)$ transformations. But this relationship is also one of Wightmann's axioms? Why is it necessarily to axiomatize something that can be derived mathematically?
 
123
mechanical energy depend on velocity and position.
 
@SirCumference ah, "beautiful but wrong", the most common fate of theories :P
 
6:49 PM
Total energy in the universe is constant and is equal to = $m^2 c^4 + p^2c^2$
 
In which $\Lambda_{1/2}$ are the spinor reps of $\mathfrak{so}(1,3)$
 
@123 OK, knowing whether mechanical energy will be conserved is actually tricky to answer
 
@Azmuth ...and what do you think $m$ is in there?
 
kind of depends how much of an approximation you want to make
 
ah wait crap that's not the axiom, ignore me!
 
6:50 PM
@ACuriousMind mass of universe
:P
@Charlie Do you understand Lie Algebra rigorously?
 
@SirCumference The difference I'm talking about is that "classical mechanics" seems to actually usually refer to the classical theories on the motions of macroscopic objects; whereas "classical physics" is usually broader and includes things like classical electromagnetism and thermodynamics and stuff.
 
@Azmuth "mass" is a surprisingly tricky thing to define for something that's not a point mass in GR.
 
@ACuriousMind welp once you tweak the assumptions you can turn classical mech into a more accurate theory :P
 
@Azmuth Idk what you would define as rigorously :P
 
If you want to talk about the universe's total energy, you should rather be integrating some stress-energy tensors.
 
6:51 PM
this doesn't relate to how they work, more why it was necessary to axiomatize it
 
@ACuriousMind Umm.. I see problem here.
 
123
@SirCumference Why it is tricky. If mechanical energy conserved does it mean system maintain its motion?
 
@JMac that's a fair point, though classical thermodynamics is arguably still classical mechanics, just at a different scale
 
@ACuriousMind I heard that $E = mc^2$ can be derieved from Field Equations, how?
 
actually open a GR book instead of just claiming to have read one and find out!
 
6:52 PM
or don't open a gr book
it hurts
 
they're usually heavy
 
probably better for one's sanity, yes :P
 
No GR book hurts (except MTW)
 
@Slereah eric poisson's gravity
this book is trash
 
@SirCumference TBH I'm mostly basing this off the differences in wikipedia articles, and it's kinda making sense to me as a distinction en.wikipedia.org/wiki/Classical_physics en.wikipedia.org/wiki/Classical_mechanics
 
6:53 PM
@JMac fwiw this is what I was getting at, thanks for putting it into words
 
123
We all open GR book
 
@JMac i dunno, it's prob just a personal philosophy of mine
 
123
:
:P
 
@SirCumference still got 4/5
 
@Azmuth so did Townsend QM
and that's one of the worst books i've ever read
if not the worst
 
6:55 PM
@SirCumference Lmao, that's the best!!! LOLOLLOLOLOL
 
i don't think those reviewers are real humans
 
123
@SirCumference Why it is tricky. If mechanical energy conserved does it mean system maintain its motion?
 
I never picked Townsend's again!
Sakurai is good... atleast for reference..
 
@123 it depends on the situation and approximation. e.g. obviously there's no such thing as a perfectly elastic collision
 
@ACuriousMind I honestly never thought of the terms that closely but when I looked it up it made a lot of sense what difference you were trying to get at.
 
6:57 PM
now if you just have an object in the presence of a conservative force, then sure, mechanical energy is conserved
but in the presence of nonconservative forces, e.g. friction, it's not conserved
 
123
@SirCumference Hmm.. that's good answer it solve my very big problem. if we choose ideal situation and object maintain its motion. We can definitely say conservation of energy is in placed?
 
@123 depends on whether the force is conservative
i.e. it has a scalar potential
only in that case is potential energy defined for it
 
123
How can we say force is conservative or not?
 
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. Conservative vector fields have the property that the line integral is path independent; the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected. Conservative...
it'll take some multivariable calculus knowledge
 
123
Force doesn't change with time
 
6:59 PM
but in a 1D case you can just check if it's integrable over distance
e.g. conservation of mechanical energy isn't held in a changing electromagnetic field, since some energy is transferred between the particles and the field itself
 
123
Ookay i solved it in mathematical method book
 
but in a static electromagnetic field, it's conserved
gtg
 
@SirCumference @Slereah This is my favourite GR book - I'd pay even if someone is able to read first 5 pages of it :P archive.org/details/einsteintheoryof032414mbp/page/n11/mode/2up
lmao!
 
Oh yes, that one
It's cute
 
@Slereah lmao!
I found it first!
Best GR Book ever!
 
7:05 PM
MWT does have a few GR poems
 
Never read that one....
I wish Carrol to add some poems too :P
Oh okay... Good Night...
I've some homework left... :-)
Have fun :-D
 
123
GooD Night guys. See you
 

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