is there a nice way to think of flows in category theory? like you have a category Diff which has objects manifolds and morphisms smooth maps, and isomorphisms are diffeomorphisms. A flow is a ... particular? family of these diffeomorphisms?
yep, but I'm trying ot understand what lagrangian mechanics is saying about the actual path
the actual path will be a combination of the path that (a) keeps energy constant, and something else, because (a) isn't sufficient to define the motion
so if I assume (a)
what's left
and it looks like the bit that's left is int (p dx) right
so that's why intuitively I feel like there's two principles at play (a) conservation of energy and (b) some principel that stops the zig zag back and forth
force isn't a concept in lagrangian mech
lagrangian is an alternative way of looking at things
but essentially from a NEwtonian point of view you're correct
and the second hidden step is that the least action is then minimzing somethin gelse subject to constant energy
My goal is to be able to fully see what the lagrangian is doing when least action is occuring and know intuitively why it leads to newton's laws essentially
and if I express L = pv - H
then I can see the H will disappear from the minimization
so that's the energy conservation bit
and the int pvdt is the other bit
where int pvdt = int pdx
so now I just need to understand why int pdx stops the zig zagging if you will
and it seems like it's because the universe is trying to do two things, the universe is selecting a path that conserves energy, and also a path that minizes int pdx
now intuitively I would have thought that once the universe chooses a path that conserves energy, it will then try to do the rest in the shortest time
so it would be nice if I could show that int pdx is actually a geodesic subject to energy conservation
I'm hoping someone can show this, or explain how it differs from what a geodesic would be (subject to a constraint of energy conservation)
I guess "shortest time" idea doesn't stop the zig zagging
because we're fixing time at 1s in my earlier example
and that one possible energy give syou a max height h
so there is some height h
if we assume no zig zagging
so it seems solved
but when you add zig zagging
that means you can increase your energy
to something greater
and ordinarily that higher energy will mean you get back to pos 0 quicker than 1s, but the zig zagging slows you down so you get there in 1s after all
but that means your max height will exceed h
the path that minimizes zig zagging though will be the original h we calculated
which is what the physics selects
so that's why I'm pretty sure in the 1D case with a potential, if you look at all energy conservingv paths, then pick the one that is shortest distance, you get the right outcome
but there are more possible energy conserving paths, such as throwing the ball up twice as fast, have it go up aand down and up and down and only then return to 0
yeah I did say to ignore that trivial case before, but thinking further I agree with you it probably can't be ignored
@Kenshin i think it is consistent with Maupertuis's principle. its just that we should not specify time for Maupertiuss's principle to hold
just specify xi=0, xf=0 and some value of E
depending on the specified E, the path where the ball doesn't move may or may not be an actual pat
@Kenshin i also think, if u want intuition for stationary action principle, u should work with Hamilton's principle. Maupertuis's principle seems outdated
wiki says this principle only gives u the shape of the path. it doesn't give x(t)
but I don't have any intuition for Hamilton's principle either. I take it as as axiom of physics @Kenshin
@RyderRude, to recap so far we have L = pq - H. Lagrangian minimizes int Ldt, which thus minimizes int (pq) - int(H)dt
and if we then assume this will select a path of conserved energy
this means Hamilton's principle is a combo of (a) selecting least energy path and (b) selecting the path that also satisifies Maurapas principle
ok, now we then have int p dx
but let's actually write this instead as
int (pv dt)
now I was hoping to be general, but let's temporarily set p = mv
then we have
int (mv^2 dt)
int (2*KE dt)
now the factor of 2 won't impact the minimization
so we have int (KE dt) that we want to minimze
so essentially this means the Maritipaus principle is trying to minimze the average kinetic energy over the path, subject to conservation of total energy
this is progress I think
the problem I have now is that this only seems to apply where p = mv
@SillyGoose I don't think so, because it's not particularly specific to groups: This follows mostly from a general (equally nameless) theorem about covering spaces and group action, where for a properly discontinuous action of some $H$ on simply connected $X$ we have that $\pi_1(X/H) = H$.
@Kenshin so in general, u have the stationary action principle. And u also know independently that energy is conserved. so u only want to stationarise over paths along which energy is conserved, right?
but this does not allow u to disregard the $\int Hdt$ term from the action
I don't know independently. Conservation of energy is a consequence of least action as well. But I'm conditioning on it, to see what else least action is doing
"but this does not allow u to disregard the $\int Hdt$ term from the action" sure it does
H is a constant
so int H dt = H int dt = H(tf - ti) = const
no path variation will change since tf and ti are fixed
@qwerty You cannot state what the "flow map" is inside of the category of smooth finite-dimensional manifolds as the space of vector fields on some manifold $X$ is not a smooth finite-dimensional manifold itself. What you would want to say is that there is a smooth map $V(X)\to \mathrm{Hom}_\text{groups}(\mathbb{R},\mathrm{Diff}(X))$ that associates to a vector field its flow.
Say if I split the lagrangian into $L(m , \lambda) + L_{CT}$ where $L_{CT}$ is the counterterm, and I use a different renormalisation scheme, wont I have to use a different mass and coupling constant in the final calculations as $m$ and $\lambda$ will only be the experimentally realised values for the on-shell scheme?
E.g. say if I calculated a correlation function and the result was $f(m,\lambda)$, wouldn't I have to input a different $m$ and $\lambda$ if I used a different renormalisation scheme?
i think it is not a co incidence that all known physical diff eqns come from an action principle. Don't u think an action principle is a good way to do physics? @Kenshin
but "what's happening" is ill defined. U only prefer Newton's laws as "what's happening" cuz u can intuit it. but that doesn't mean Newton's laws are what's happening
what's happening is observations
anything that gives the same observations counts as "what's really happening"
@Kenshin I think you're not even wrong - the level at which you are making these statements is so far removed from what the actual research is about it just doesn't apply
So say if I split my Lagrangian in two different ways e.g. $L_{OS} + L^{CT}_{OS}$ and $L _{MS} + L^{CT}_{MS}$ where $OS$ refers to on-shell and MS refers to the MS renormalisation scheme. And say if I calculated a correlation function in the first scheme $f^{OS}(m_{OS},\lambda_{OS})$ and a correlator in the second scheme $f^{MS}(m_{MS},\lambda_{MS})$. Will I have $$f^{OS}(m_{OS},\lambda_{OS}) = f^{MS}(m_{MS},\lambda_{MS})$$ where I have to independently work out what $m_{MS}$ and $\lambda_{MS}$ are?
like, thinking that people are "just guessing an action" is such a staggeringly poor description of how we actually construct theories it doesn't even rise to the level of being an intelligible criticism - there are worlds of nuance between being able to derive an action and "just guessing it"
And if you understood the slightest bit about the conflict between GR and QM you would understand that no one hopes to resolve this conflict by just finding a new action
Depends on what you're doing and what you mean by "observed mass", but yes, there will be some relation between the renormalized mass and the "physical mass", i.e. the pole of the propagator. It can be arbitrary - Chiral Anomaly discusses a neat toy example here
@ACuriousMind to get solutions to Einstein field equation, does one have to fix topology of spacetime apriori, or are there formulations in which that is not required
Can one perform the sommerfeld expansion to integrals of the type: $\int H(\epsilon)d\epsilon \frac{d f(\epsilon)}{d \epsilon}$, where $f(\epsilon)$ is the fermi dirac function? Essentially is it possible to perform the sommerfeld expansion to integrals that contain a quadratic power of the F.D function?
In our condensed matter theory
we were considering the relevant integrals for transport theory and
When one speaks about the density of states, usually the definition is as: nr. of states per unit energy, which in itself is understandable. The problem I have is how this notion makes sense when we ask: "What is the system considered here"?
