What is Connection: Definition and 517 Discussions
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Cartan in the first half of the 20th century as part of, and one of the principal motivations for, his method of moving frames. The connection form generally depends on a choice of a coordinate frame, and so is not a tensorial object. Various generalizations and reinterpretations of the connection form were formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object. On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold, rather than on an abstract principal bundle over it. Hence, despite their lack of tensoriality, connection forms continue to be used because of the relative ease of performing calculations with them. In physics, connection forms are also used broadly in the context of gauge theory, through the gauge covariant derivative.
A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because under a change of basis, the connection form transforms in a manner that involves the exterior derivative of the transition functions, in much the same way as the Christoffel symbols for the Levi-Civita connection. The main tensorial invariant of a connection form is its curvature form. In the presence of a solder form identifying the vector bundle with the tangent bundle, there is an additional invariant: the torsion form. In many cases, connection forms are considered on vector bundles with additional structure: that of a fiber bundle with a structure group.
My physics book explains that this is because of the conservation of charge and the fact that charge doesn't accumulate at a certain point.
Can someone please explain this in detail, especially the part where it says that charge can't accumulate.
Another question which i think is similar is...
I'd like to preface by saying that I'm not excellent at physics, so I"m still trying to figure this out. I know that when you have an internet connection you have multiple "channels" open to deliver data. The same with audio wires. I was listening to a podcast and they were talking about the new...
Given that heat capacity is a ratio of change of energy over change of temperature, while entropy is a change of energy over absolute temperature...
I was wondering if there is any basis for the idea that energy will tend to flow from media having low heat capacity to media having high heat...
hello!
Does anybody know the physical justification of the de broglie relation p=\hbar k? or (i guess equivalently) for p=-i\hbar ∂/∂x ?
i came across a more general "law" in laser physics, where momentum is seen as the gradient of the phase of a scalar light field (used in eg the paraxial...
Hi all,
I am trying to find the Christoffel connections of this metric:
ds2= -(1+2∅)dt2 +(1-2∅)[dx2+dy2+dz2]
where ∅ is a general function of x,y,z,t.
I tried to solve this through the least action principle, but some of my results(t-related terms) were different from the answer with a minus...
Homework Statement
The two polynomial eqns have the same coefficients, if switched order:
a_0 x_n+ a_1 x_n-1 + a_2 x_n-2 + … + a_n-2 x_2 + a_n-1 x + a_n = 0 …….(1)
a_n x_n+ a_n-1 x_n-1 + a_n-2 x_n-2 + … + a_2 x_2 + a_1 x + a_0 = 0 …….(2)
what is the connection between the roots of...
Hey all, making my way through Landau and Lifgarbagez classical theory of fields and i had a specific question on the Einstein equations. Following the palatini approach, we assume that the connection and metric are independent variables and are not related a priori. In the footnote, they say...
Homework Statement
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
##\sum _{n=1}\left( -1\right) ^{n}\dfrac {n} {n^{2}+1}## the sum goes to infinity.
Homework Equations
Theorem for absolute convergence.
Test for divergence
The Attempt...
A popular remote desktop connection issue at present shows up in my head making me wonder whether or not what I am thinking is correct at all. I am a Windows internals novice so please be prepared with my ignorance.
For security reason I suppose later versions of Windows (post-XP) have been...
I am looking for the best way to prevent corrosion between two different materials while maintaining electrical contact between them. Specifically, I am wondering about carbon fiber panels placed on an aluminum base and exposed to seawater. Carbon (graphite at least) has a much higher galvanic...
I'm trying to receive data from a electronic compass in simulink. The compass sends data using the NMEA protocol through USB using a serial connection.
The serial output format is: 4800 baud, 8 data bits, 1 stop bit, no parity
and uses the following sentences...
For the following two-dimensional metric
ds^2 = a^2(d\theta^2 + \sin^2{\theta}d\phi^2)
using the Euler-Lagrange equations reveal the following equations of motion
\ddot{\phi} + 2\frac{\cos{\theta}}{\sin{\theta}}\dot{\theta}\dot{\phi} = 0
\ddot{\theta} -...
My son pulled the wires off of the charging plug for his motorcycle (mini-bike).
I can use a Fluke to determine what side is positive comming from the charger, and trace the positive side from the battery to the charging socket.
My question is when charging do I connect the positive side of...
Hi all,
I've been fiddling around with this problem for a while. I intuitively understand that the parallel propagator is the path integral of the connection. I would like to be able to show the converse (connection is derivative of parallel propagator) mathematically, and I am having a...
I want to measure the temperature of a metal stage where I am holding a sample. What I have done is to clamp both wires from a thermocouple against the stage using a bolt and washer. Supposedly this is not the best practice but I don't see why this would be a problem.
I have read that you are...
Hi there, I am reading some introduction of crystal and the book mention that we can use the x-ray to explore the structure of the crystal. I more or less understand the idea between that, it is kinda like scatter the incoming wave and observe the outgoing wave. But the mystery stated in the...
Yes, it is about OPERA experiment, but it has nothing to do with neutrinos.
Can someone try to explain how is it technically possible that faulty connection could be responsible for additional signal delay? I understand it can make the signal unreadable, but delayed by 60 ns?
Main reason I...
The connection \nabla is defined in terms of its action on tensor fields. For example, acting on a vector field Y with respect to another vector field X we get
\nabla_X Y = X^\mu ({Y^\alpha}_{,\mu} + Y^\nu {\Gamma^\alpha}_{\mu\nu})e_\alpha
= X^\mu {Y^\alpha}_{;\mu}e_\alpha
and we call...
I don't know if this is the right place to ask this, but I'll give it a try.
I have a PC with a rather noisy fan which I'd like to replace so I got a new one with lower RPM and of better quality. However, there is one little problem: The PC won't boot with the new fan. I contacted the...
How to show:
Tabc = \Gammaabc - \Gammaacb
is a Tensor of rank (1,2)
Attempted solution:
1. Using definition of Covariant Derivative:
DbTa= ∂aTa+\GammaabcTc (1)
DcTa= ∂cTa+\GammaacbTb (2)
I subtracted (2) from (1) but I...
I'm trying to make a wireless energy experiment by using LC circuits as resonators. But within this circuit you can either make the inductor and capacitor in series or in parallel. What do the different combinations do to the circuit? Is there one that is ideal for some circumstances?
Moving from a schematic diagram to real circuits.
I have a few questions:
Since the ammeter should be connected in series, why should the positive pole of a battery be connected to the ammeter's "positive" plug. Won't it be connected in parallel that way?
If you didn't get what I mean see the...
Hey there!
Having some difficulty with this question... is there any proportionality law between potential difference and velocity?
An electric charge accelerated from rest through a potential difference of 250 V reaches a speed of 9.4 x 10^6 m/s. What speed will this same charge reach...
Hey All,
I am familiar with some of the practices of bolted connections between steel plates, or steel beams etc. I am currently facing a design problem at work dealing with a composite metal panel being screwed to a hallow PVC beam. The PVC beam is screwed into a hallow aluminum beam...
Homework Statement
[See Attachment]
Given: Thickness, load, G
Find:γ and horizontal displacement.
Homework Equations
τ=γG
Angle change should be the shear strain.
The Attempt at a Solution
I tried plugging 800 into tau and the given value of G to get γ, but that isn't...
Hi,
I was wondering if someone could explain how the Noether current and four-current are connected.
For context, I'm going through a derivation to show that local phase symmetry requires the electromagnetic field. I'm at a stage where the I have a Noether current of a complex field (for...
How can I see that the generators of the Galilean group correspond to energy, momentum, etc.?
References which cover the Galilean group and algebra as well as their realization in phase space are appreciated, especially if they are not too sophisticated.
Thanks
kith
Hey, how can we connect two auto-bots(one line follower & other follows the path of the other in another plane) by wires? Is it possible two program the line follower such that it will send signal to another through wires, to start the motor of the other? how is it possible?
Living out in the country has it's advantages and disadvantages. The wireless connection is bad. Trying to get from the house to a steel garage. Using a wireless device hard-wired to pc in house but the connection won't go into the garage. Garage is about 35 to 40 ft from house and have the...
Hi all,
I'm reading Sean Carroll's Space Time and Geometry and haven't figure out how equation 4.64 is derived, where he is in the process of deriving Einstein's equation from Hilbert action.
Given there is a variation of the metric,
g_{\mu\nu} \rightarrow g_{\mu\nu} + \delta g_{\mu\nu},
The...
So do I has the craziest question evers! YAY ME.
Anyhow, I have an old piece of video capture hardware, but the problem is I am missing the power plug thingamabobber :(. SADFACE.
And by that, I mean...
Now, I have a 5-12V adjustable power supply (cost me like $150 and only used it like...
Hi all,
To my understanding, the Levi-Civita connection is the torsion-free connection on the tangent bundle preserving a given Riemannian metric.
Furthermore, given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics...
I am working on a project that uses a Faraday cup. Its consists of a metallic cylinder(pipe like) with an electrode at the bottom.The ions enter the cylinder and hit the electrode.The resulting current is carried away from the electrode to an amplifier by a wire .The electrode that i am using is...
The first thing I remember hearing about in QM was the time-independent 1-D schrodinger equation, Hψ = (\frac{-\hbar^2}{2m}\frac{d^2}{dx^2} + V)ψ(x) = Eψ(x) . This is an eigenvalue equation, the Hamiltonian operator H operating on the energy eigenstate ψ to produce the product of the energy...
Homework Statement
Show that \frac{\partial\overline{\mathcal{L}}_G}{{\partial}\mathfrak{g}^{ab}_{,c}}=\Gamma^c_{ab}-\frac{1}{2}\delta^c_a\Gamma^d_{bd}-\frac{1}{2}\delta^c_b\Gamma^d_{ad}.Homework Equations...
Homework Statement
What is the connection between simple harmonic motion and pendulum motion?
Homework Equations
Harmonic motion period=T=2piroot(m/k)
Pendulum motion period=T=2piroot(L/g)
The Attempt at a Solution
Conservation of momentum??
Thanks for any help!
Hello,
I'm trying to find a table of basic connection coefficients. My GREAT desire is to find a table that includes the metric, connection coeffs, Riemann tensor, Ricci tensor and Ricci scalar for NORDSTROM's 1913 scalar theory of gravity.
Does anyone know an online source of this info...
It's been a while since I've met a girl I'm actually very attracted to and feel some kind of connection with, on a personal level. This is largely due to me not being out much and even more due to me not making the effort to try connecting at some level with the person in question. Why? At some...
Curvature form with respect to principal connection
Hi all,
I have a question. Let us suppose that P is a principal bundle with G standard group, \omega a principal connection (as a split of tangent space in direct sum of vertical and horizontal vectors, at every point in a differential way)...
Hi, I'm reading this wikipedia article on the metric connection, and it says that the Levi-Civita connection is a metric connection without torsion. If the metric connection is defined so that the covariant derivative of the metric is 0, how can there be torsion? Doesn't this condition force the...
Hi guyz,
I need some help in choosing the shaft couplings to connect the motor to the gear and from gear to the wheel.
The motor shaft is 11 mm and the gear shaft is 24 mm. (Max torque 2 Nm, Max speed 70 rpm)
Output of the gear shaft is 32 mm and the wheel shaft is 30 mm. (Max torque...
I was reading "Reinventing Gravity - A Physicist Goes Beyond Einstein" by Mr. John Moffat, btw it's a really good book for understanding introductary level of classical and theoretical physics, anyway a passage there caught my eye. I'm not finished with book yet but the author expressed/chose...
As in the case of fear how does thought stimlate adrenal glands? Not so much thought but the instant recognition of a threat causes the release of adrenaline. But there must be an electrical event which is recognition. EEG's show this. But there must be specialized receptors for the...
We have a short film project in Physics about the role of Physics in our course. I am a pharmacy student and me and my 3 teammates tried to write a plot summary but our storyline is not that good.
Can you provide me with any ideas we can do for our project? Thanks!
Bosoms (energy'ish) are 1 spin integers, Fermions (mass'ish) are 1/2 spin integers.
KE=1/2MV^2
E=1MV^2
I've only been trying to wrap my puny mind around spin for longer than 10 years, these spins creates magnetic poles right? In all 3 dimensions?
We all are familiar with the kind of differential geometry where some affine connection always exists to relate various tangent spaces distributed over the manifold, and from this connection two fundamental tensors, namely the Cartan's torsion and the Riemann-Christoffel curvature, arise.
Is it...
Homework Statement
A balanced delta load having 18ohm resistance and 24 capacitive reactance in series in each phase is supplied through lines each having 1 ohm resistance and 2 ohm inductive reactance. If the line to line voltage at the sending end is 250 V, find the phase current and line...