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    Invariance of scalar products on Lie algebras

    Hi folks, If I have a Lie algebra \mathfrak{g} with an invariant (under the adjoint action ad of the Lie algebra) scalar product, what are the conditions that this scalar product is also invariant under the adjoint action Ad of the group? For instance, the Killing form is invariant under...
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    Chern Simons form: calculation

    Hey folks, I'm trying to understand a couple of calculations in Chern-Simons theory and I'm stuck. I want to prove the following in four spacetime dimensions: Let G be a Lie Group with generators T_{a}. Further let A=T_{a} A^{a}_{\mu} dx^{\mu} be a connection and F=dA+A \wedge A its...
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    Phi 3 scalar field theory

    You don't have to guess here. I will just cover the phi cubed case. Are you allowed to use Feynman's rules? If yes, just draw the 2+3 external lines and two vertices with three outgoing lines each and draw all topologically distinct diagrams. After that, you can write down the terms very easily...
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    Needing help with projectile trajectories

    You need to set up the differential equation of motion in two dimensions. In a simple model there are only two forces involved: gravity, which is constant and air resistance, which is proportional to velocity squared. There is no analytical solution, therefore you have to integrate the equations...
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    Fluid model question

    I have a specific question about fluid analysis that is bugging me. I am no expert in this field but very interested in it. Lets's say we want to model a flow where air expands out of an air reservoir across a valve into a cylinder and applies a pressure force on an airtight piston, which...
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    Introducing LaTeX Math Typesetting

    \lim_{\substack{x\rightarrow 0\\}} f(x,y)
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    Find the distance/height, impact ,etc of the tsunami?

    Perhaps you are looking for something like this:
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    Drag Equation Derivation:

    All I wanted to show is that there is a way to come up with the drag formula, nothing more. Perhaps someone knows the true origin, if I didn't capture it. This formula works amazingly well for what it's intended, and it's used quite often for simple models on a level where students don't know...
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    Drag Equation Derivation:

    Okay, I'll do the derivation in terms of energy because it's easier than with momentum and doesn't even involve calculus. Let's start with the kinetic energy of the fluid in front of the body: E=\frac{1}{2}m v^2 The mass is given by the area times an arbitrary length times the density...
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    Drag Equation Derivation:

    Actually, there is a derivation for the drag formula. You have to write down the momentum contained in a fluid element and differentiate with respect to time, after which you integrate over dV. This will give you the formula without the drag coefficient, which is plugged into the formula...
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    Rail Gun physics, need assistance

    Recoil does not depend on the mechanism used to accelerate the projectile. Whenever you accelerate something, there is a force in the direction of acceleration. This is the force the gun exterts on the projectile. There is an equal and opposite force acting on the gun, which accelerates the...
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    Fighter aircrafts have the jet engines buried inside the fuselage

    The moment of inertia along the longitudinal axis is also kept small by placing the engine inside the fuselage, increasing maneuverability. Another benefit would be lower air resistance.
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    How much money is sufficient for you?

    Foreigners are usually treated good. There are some folks who are prejudiced when it comes to foreigners but I believe they are the minority. Italian, Portuguese and Spanish people have established a good presence in Switzerland. After all, 21.6% of the Swiss inhabitants are foreigners. I've...
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    How much money is sufficient for you?

    Well, I must say I can't complain about life over here. For young people, finding a good job without degree can be difficult. Also, living costs are said to be very high. You can have a hard time finding a nice apartment with 4 rooms for less than 1200$ a month. Decent restaurants are also...
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    How much money is sufficient for you?

    Yes, it's most likely Lake Geneva. It's a very nice place to stay. And since this thread is about money, I have to say that it's also pretty expensive, but this applies to almost every place in Switzerland to be honest :smile:
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    How much money is sufficient for you?

    Might I ask what area she is going to? And is she visiting Switzerland as an exchange student? Just curious :rolleyes:
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    Center of Gravity/Torque

    I thought that the center of gravity is where the sum of all turning moments is equal on both sides. If you saw the object apart at the center of gravity, one piece could be considerably heavier than the other. Imagine a long section of wood with a solid steel sphere at one end. The center of...
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    2d motion problem

    If you solve y^2=16x for y, you get y=4*Sqrt[x] and y=-4*Sqrt[x]. Plot the two curves and decide for yourself which one looks like a canyon. I think you should be able to solve it then.
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    Location-dependent acceleration

    robphy, of course you're right, I misunderstood you. Back to my question: What does the imaginary part of the solution represent?
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    Location-dependent acceleration

    Ok. Thank you for the hint. Everything has worked out concerning the DE. One more question has come up: When I set the initial values so that a0>a1, which means that acceleration decreases with the distance travelled, I get complex values for s[t] and its derivatives. What does the...
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    Location-dependent acceleration

    I think s1 must be able to remain positive even when a1-a0<0. That would mean that the acceleration decreases with the distance and nothing else. I don't see why a1-a0<0 leads to s1<0, could you explain that? In fact, as I said in the initial post, the problem I'm trying to solve is for...
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    Location-dependent acceleration

    It is not given that a1-a0=0. Where does he state that?
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    Location-dependent acceleration

    That was exactly what I was looking for, thank you. I think that I wasn't able to solve this because I'm fairly new to calculus.
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    Location-dependent acceleration

    First of all, thanks for replying. My problem lies in setting up the DE. Normally, t is the only independent variable. However, in this situation, s is also independent in a(s) and therefore s(t) will contain the function a(s). I'm stuck at this point. Could you please elaborate on...
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    Location-dependent acceleration

    I'm having a really hard time figuring this one out. At first glance it seems very trivial, but I think it is not. This is not a homework assignment but rather a calculation I need to do for personal reasons. The problem is as follows: An object is accelerated along a distance of s. We have...
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    Difficult question !

    Ok, I've got a VERY difficult one for you guys! All units are metric. This is projectile motion WITH air resistance! The muzzle of a cannon is placed at the point of origin in a 3d grid (P(0;0;0)). It fires a spherical aluminium projectile with a radius of 3cm at a velocity of 150 m/s. A...
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    Simulating ballistics

    Hello everyone. I'm new to this forum and very impressed so far. I'm making a program which simulates the flight of an object. In order to do that, I have to compute the coefficient of drag. This is the first problem. From what I know, it should be possible to determine this number (at least...