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193 votes

Why does nature favour the Laplacian?

Nature appears to be rotationally symmetric, favoring no particular direction. The Laplacian is the only translationally-invariant second-order differential operator obeying this property. Your "...
knzhou's user avatar
  • 102k
184 votes

How can time dilation be symmetric?

The answer to this is that our twins, $A$ and $B$, are not measuring the same thing on their clocks. Since they are not measuring the same thing there is no paradox in the fact that each twin thinks ...
117 votes

Why isn't the Euler-Lagrange equation trivial?

Ah, what a tricky mistake you've made there. The problem is that you've simply confused some notions in multivariable calculus. Don't feel bad though-- this is generally very poorly explained. Both ...
user1379857's user avatar
  • 11.5k
90 votes

What is really curved, spacetime, or simply the coordinate lines?

Congratulations! You stumbled upon an important question of differential geometry: How can I know whether the curvature is caused by my choice of coordinates or the space I live in? As has been ...
Lukas Juhrich's user avatar
72 votes

Partial derivative notation in thermodynamics

That's because in thermodynamics we sometimes use the same letter to represent different functions. For example, one can write the volume of a system as $V=f_1(P,T)$ (a function of the pressure and ...
1__'s user avatar
  • 1,584
62 votes

Rotate an object about the time axis

This a great question, and leads to some interesting ideas. Firstly, the notion of a "rotation axis" is restricted to three dimensions. In more than three dimensions the rotation axis ...
mike stone's user avatar
  • 53.7k
52 votes

Why is this vector field curl-free?

The vector $\hat \varphi$ is not defined at the origin, because the coordinate transformation $$(x,y) \mapsto (r,\varphi) = \left(\sqrt{x^2 + y^2}, \arctan(y/x)\right)$$ is singular there. Hence your ...
Robin Ekman's user avatar
  • 14.8k
44 votes

What are holonomic and non-holonomic constraints?

If you have a mechanical system with $N$ particles, you'd technically need $n = 3N$ coordinates to describe it completely. But often it is possible to express one coordinate in terms of others: for ...
DK2AX's user avatar
  • 4,789
43 votes

Does it make sense to take an infinitesimal volume of shape other than a cube?

Infinitesimal volume elements do not have to be cubes. Some familiar examples come from typical solids of revolution problems from calculus 1/2. Typically one discusses using either the "disk/...
BioPhysicist's user avatar
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40 votes

Given two vectors (with no common point), is a dot product between them possible?

Vector by definition only has magnitude and direction. Origin is not part of the vector. In particular, you can unambiguously define a vector between two points $A$ and $B$ by finding the distance ...
Ruslan's user avatar
  • 29k
40 votes

Does Newtonian mechanics work in polar coordinates?

Your teacher is incorrect. $\vec F = m \vec a$ is valid in any inertial (non-accelerating) coordinate system. You must account for the fact that the unit vectors for position in some coordinate ...
John Darby's user avatar
  • 9,381
40 votes

Why are angles so weird?

Your concerns are valid, and for that reason, there is also an alternative way to view angular velocities, angular momentums, etc.: They are not vectors, but bivectors. A bivector is essentially an ...
Vercassivelaunos's user avatar
34 votes

In the theory of special relativity speed is relative so who decides which observer’s time moves slower?

A core idea of special relativity is there is no right frame of reference. It doesn't matter which of the two observers you use as your point of reference, the math will work out either way. Yes, they'...
TheEnvironmentalist's user avatar
32 votes

Why/When can we separate spacetime into space and time?

The notion that a spacetime can be decomposed into a spacelike and timelike part is generally called a spacelike foliation, where your spacetime manifold can be decomposed into purely spatial ...
Slereah's user avatar
  • 16.4k
31 votes

Why is clock synchronisation such a big deal in physics?

All of special relativity is based on the assumption that any observer can set up a coordinate system and then label spacetime events with their coordinates in that system. Then we can use the Lorentz ...
John Rennie's user avatar
30 votes

Derive vector gradient in spherical coordinates from first principles

You asked for a proof from "first principles". So let's do it. I'll highlight the most common sources of errors and I'll show an alternative proof later that doesn't require any knowledge of tensor ...
Wood's user avatar
  • 1,954
30 votes

Is Fermat's principle only an approximation?

In general relativity, it's not entirely clear what "least time" means, since you have to ask "whose time are you talking about"? Are you talking about the time as measured by the emitter? The ...
Michael Seifert's user avatar
30 votes

Does light really travel more slowly near a massive body?

The simple way to show that the speed derived from the Schwarzschild coordinates has no absolute meaning is to derive an expression for the speed measured by a different observer and show that they ...
John Rennie's user avatar
30 votes

Why does nature favour the Laplacian?

This is a question that hunted me for years, so I'll share with you my view about the Laplace equation, which is the most elemental equation you can write with the laplacian. If you force the ...
fredwhileshavin's user avatar
30 votes

Massless Kerr black hole

It's simply flat space in Boyer-Lindquist coordinates. By writing $\begin{cases} x=\sqrt{r^2+a^2}\sin\theta\cos\phi\\ y=\sqrt{r^2+a^2}\sin\theta\sin\phi\\ z=r\cos\theta \end{cases}$ you'll get good ol'...
Mauro Giliberti's user avatar
29 votes

What is the cause of the constancy of the speed of light in vacuum?

The invariance of the speed of light follows from the principle of relativity. This says there is no experiment that can distinguish between inertial reference frames: physical laws are the same in ...
Aiden's user avatar
  • 1,691
27 votes

Is it strange that there are two directions which are perpendicular to both field and current, yet the Lorentz force only points along one of them?

The universe is not preferentially selecting one direction over another. The fact that it appears that this is happening is an artifact of how we represent the magnetic field. It is well-known that ...
Brian Bi's user avatar
  • 6,551
27 votes

Why can vector components not be resolved by Laws of Vector Addition?

Indeed, any vector can be resolved in terms of two components (in $n$-dimensional space in terms of $n$ components). For this being possible the components should be linearly independent, i.e. in your ...
Roger V.'s user avatar
  • 59.2k
27 votes

Does Newtonian mechanics work in polar coordinates?

Actually, Newtonian mechanics can be made to work over any Riemanian manifold of any dimension. It's actually a specialisation of Lagrangian mechanics. This is called Geometric mechanics as is ...
Mozibur Ullah's user avatar
26 votes

How do Rindler coordinates fit into special relativity?

This can only be answered by pointing out the difference between special and general relativity. There is the historically motivated definition, which is still in widespread use in popular and semi-...
Batiatus's user avatar
  • 604
26 votes

Lagrange's equation is form invariant under EVERY coordinate transformation. Hamilton's equations are not under EVERY phase space transformation. Why?

You should really think about the variables we use as being like coordinates on some manifold, the configuration space (roughly the same as the phase space, I won't be careful about the distinction). ...
Richard Myers's user avatar
26 votes

On mathematical level, what exactly is time in Newtonian mechanics?

Here is one way to address your question. "Time is defined so that motion looks simple." - Misner, Thorne, and Wheeler in Gravitation, p.23. Continue through to p. 26 where they say "...
robphy's user avatar
  • 11.9k
25 votes

Why is this vector field curl-free?

There already are very good answers so I would just like to give some physical intuition why this vector field is curl-free even though it has non zero circulation. We can make an analogy of the curl ...
Diracology's user avatar
  • 17.7k
25 votes

Does Newtonian mechanics work in polar coordinates?

Your teacher is definitely incorrect. In fact, the whole point $\vec{F}=m\vec{a}$ is written as a vector equation is to emphasize that the equation does not depend on the coordinate system you choose ...
Smiley Lens's user avatar
25 votes

Why is Noether's theorem not guaranteed by calculus?

Since the problem here appear to be coordinates, let's just stop using coordinates, and for simplicity consider the theory of a single scalar field on space(time) $M$: Our field is a function $\phi : ...
ACuriousMind's user avatar
  • 126k

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