I've wondered about this before, and some of the answers here are really helpful. I think something that hasn't been mentioned that I've often thought must have some relevance is the fact that in the coordinates of an outside observer, falling objects never actually fall into a black hole...
Wrong isn't the word I would use, strictly speaking it's actually right, I just think it's badly phrased because it's better to use Newton's 1st law to define which frames are accelerating, not using the concept of acceleration to define Newton's 1st law.
The textbooks assume an inertial frame...
Newton's 1st law is actually more of a definition of which frames are not accelerating. It's true that Newton's laws have to be modified when you move into an accelerating reference frame, but what is an accelerating reference frame? Accelerating with respect to what? The best way that I've...
I think the very first line is wrong. The acceleration is not equal to mv^2/r, that formula applies to circular motion. If you like you can imagine the motion is instantaneously circular when the acceleration is perpendicular to the velocity, which is the argument you seem to be making, but...
It can be for any field, but you apply it to a specific field... I think... I'm confusing myself with the language now.
Here is how I would go about justifying this derivation, maybe this helps:
Ok so you have some Lagrangian e.g. the Klein-Gordon Lagrangian, and say you are given a...
I'm going through these notes at the moment and thinking about the same sort of stuff, so I'm not an expert or anything but I'll see if I can help. Explicit x dependence just means that an x appears somewhere in the expression for the Lagrangian.
You seem to be confused by the fact that the...
I'm not completely sure what you mean here, but I'm going to take it to mean this: If the acceleration and velocity are perpendicular, does the acceleration have to equal v^2/r as it does in circular motion?
I think the answer is yes, sort of, but you have to first ask what r means if the...
Because your feet are in contact with the floor and there is a force of friction on your feet from the floor exactly equal and opposite to the force on you from the crate, because friction acts in exactly the way it needs to to prevent your feet from moving.
There is a limit to how strong...
Are you sure that's what it's saying? I read it as saying that whenever the equations of motion are unchanged by a transformation (what the book defines as an invariance transformation) then the Lagrangian must change by a total time derivative and a conserved quantity therefore exists.
This...
Yes those notes seem to refer to cases where the Lagrangian is invariant under some transformation, but I think Noether's theorem can be used more generally than that.
Vanhees, do you think that the quotes I posted from the book are wrong?
Ok thank you both for your help, so a symmetry in the equations of motion is not sufficient for Noether's theorem to apply then. I'm happy to accept what both of you are saying on this, since you seem to agree and it explains the example in my original post. It does seem to contradict my book...
Hmm but this seems to contradict the book I have. It says that the Lagrangian can change as long as it changes by the total time derivative of some function, in which case a conserved quantity can still be found. I can also see that if the Lagrangian changes in such a way then the equations of...
No that would only work while the solar panels were colder than the hot sphere. I'm not sure how solar panels work so I can't tell you exactly why it breaks down when the solar panels heat up, but once everything is at the same temperature (which will eventually happen), you can no longer get...
No one has ever said that it is impossible to remove heat from something without doing work, just that you need something colder for that to happen, then it happens by itself! Just put a hot thing next to a cold thing and watch the heat leave the hot thing, you can even get some useful work out...
The solar panel is at a lower temperature than the sun, that's why it works. The solar panel is the second reservoir. If the solar panel was at the same temperature as the sun it wouldn't work. In any example you come up with you'll find there is always some object in there which has a lower...