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Is the volume in general relativity independent or dependent on the coordinates?

The answer to this question will become more meaningful once we define the commonly used 3+1 decomposition of spacetime. Under the assumption that the spacetime manifold $\mathcal{M}$ is hyperbolic, ...
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Contradicting Changes in a Lagrangian under Transformation

In one case you've made an infinitesimal transformation, in the other a finite one. They're only equal to first order. Or to put it another way, your first equation $(*)$ is not exact: to make it ...
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Contradicting Changes in a Lagrangian under Transformation

TL;DR: The infinitesimal symmetry transformation needs to be properly integrated into a corresponding finite symmetry transformation in order to preserve the invariance of $L$ to all order in the ...
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Time and speed of light in Relativity

from your comment to Eric's fine answer, I think you do not really understand how the second is defined. Please refer to the relevant Wiki page. In particular.... When the atom is irradiated with ...
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Finding a new hamiltonian from a given canonical transformation

Since you know the new $q(Q)$, it’s easier to use another generating function $F(p,Q)=pq(Q)$ (Legendre transform of your function) which gives: $$ q = \frac{\partial F}{\partial p}(=q) \\ P = \frac{\...
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Time and speed of light in Relativity

Don't think of time as a "thing" that "moves" or "runs". Think of it as just another direction in which we can move. People in different places in spacetime will have ...
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Degrees of Freedom for an Asymmetric top

Let's work from the first principles. Assume that the top has $N$ particles. Choose the any one of them, say the one at the point at which the top is fixed. Let's call it $P$. That point can be ...
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Is the volume in general relativity independent or dependent on the coordinates?

On any orientable pseudo-Riemannian manifold you have naturally defined volume n-form. You can integrate this to get a coordinate independent volume. However, you are asking about spatial volume, not ...
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4 votes

Is the volume in general relativity independent or dependent on the coordinates?

The actual volume should be independent from the way we measure it. Untrue. As an example, one can mention the metré des archives, since that is easier to understand than dealing with volumes. The ...
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Are curvilinear coordinates inertial?

Indeed, it is valid to consider that polar coordinates are non-inertial. You should be aware that the term “reference frame” does not have one unique meaning. Different authors may use it to mean ...
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Are curvilinear coordinates inertial?

Reference frames can be inertial or non inertial. Coodinate systems are not reference frames unless the frame is somehow being tied to the coordinate system. Does the book explain how the frame is ...
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Lagrangian Dynamics of an inverted Spherical Cart Pendulum

You are modelling this using a simple pendulum. $\phi$ describes the angular displacement of the point about the $z$-axis. This only makes sense if $\theta\neq0$. As $\theta\to0$, using conservation ...
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Lagrangian Dynamics of an inverted Spherical Cart Pendulum

Imagine the pendulum swinging along $y=0$, with $x$ changing. As the pendulum swings directly below the cart, $\phi$ instantly changes from 0 to $\pi$, so you would expect $\ddot{\phi}$ to be infinite ...
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How to calculate Proper Distance as an arc length in Schwarzschild metric?

For a fixed $r$ and $t$, $$ds^2 = r^2(d\theta^2 +\sin^2\theta\ d\phi^2)\ ,$$ which is the same arc length as in Euclidean space. Then if $\phi = f(\theta)$ then $$s =r \int \left(1 + \sin^2\theta\ \...
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What is the process of finding a good canonical transformation to describe a system? How do I choose the correct generating function?

Problem-specific Solution I stumbled upon the exact same question while studying the same material (Goldstein), and after a while I have it figured out. Since we're trying to get the expression $f(P)$ ...
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Does existence of an analytic solution to an equation of motion given by Newton's second law depend on coordinates?

No! If there is a "analytic" solution for some equation of motion, at all, it may be displayed in any coordinate system, specially in a time-dependent one, with orientation of acting forces.
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Does existence of an analytic solution to an equation of motion given by Newton's second law depend on coordinates?

I'd say no, since once you have the solution in one coordinate system, you can always use coordinate transformation equations to then find the solution in the other coordinate system. For example, if ...
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Does existence of an analytic solution to an equation of motion given by Newton's second law depend on coordinates?

I don't think you're using "analytic" in the strict mathematical sense. You're probably thinking of "closed-form solutions in terms of elementary functions". Anyway, existence of ...
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Soldering Spinors in cylindrical coordinate

It seems that the answer is yes and we can write the soldering in deferent coordinate system.
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Confusion about the action variable definition

The circle integrals/action variables $I_k$ are e.g. used in the construction of the angle variables, cf. e.g. this related Phys.SE post.
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Conservation and potential with non-cartesian forces

The curl of any vector $\vec{F}$ in spherical polar coordinates is: $$ \nabla \times \vec{F} = \frac{1}{r\sin \theta}\left(\frac{\partial (F_\phi \sin \theta) }{\partial \theta} - ...
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Number of Degrees of Freedom of a Rigid Body System - Proof

Definition. The dimension of the configuration space is called the number of degrees of freedom. Thus, if we find the dimension of the configuration space of a rigid body, we can deduce its degrees of ...
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Contravariant Vector Component Transformation from Polar to Cartesian

It's because basis are defined with satisfying normalization condition. Basis are covariant, because they are partial derivatives(Consider them as gradient). $$\hat{r} = \frac{\partial}{\partial r} = ...
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Change of Metric Under Coordinate Transformation

I think that you are trying to compute the Lie derivative of the metric. If so, there should be no factor of 1/2. Under an infinitesimal shift $x^\mu\to x^\mu +\eta^\mu$ we have $g \to g+ {\mathcal ...
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How do we assume the direction of $u_{\theta}$ and $u_{r}$ in polar coordinate systems?

As a rule, $\hat{u}_X$ is always pointing in the direction along which $X$ grows. It works when $X$ is a linear parameter ($x$, $z$, $r$...) as well as when it's an angular parameter ($\theta$, $\phi$....
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How do we assume the direction of $u_{\theta}$ and $u_{r}$ in polar coordinate systems?

The unit radial vector is always away from the origin. The unit tangential vector is always anti-clockwise around the origin such that $\hat{r}\times\hat{\theta}$ is out of the diagram. Not knowing ...
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Cart Pole kinetic energy

Let us start with an example first. Consider a pendulum with constantly accelerated support (instead of the support being a degree of freedom of the system). The position of said pendulum can be ...
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Why can't we use integral of $x$, $y$ and $z$ in calculating moment of inertia

The moment of inertia is defined for a specific rotation axis. The radius $r$ is the distance from this axis, not from the origin point. For example, for the moment of inertia around the $z$-axis the ...
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Is this hamiltonian of the form of some well-known physical system?

One likely candidate for what they "want you to say" is a Kepler potential (i.e. produced by an inverse distance-squared force) viewed in a rotating frame that scales the rotation rate by a ...
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Why is it necessary that different observers agree on the value of the spacetime interval $ds^2$?

space intervals in Newtonian mechanics In Newtonian mechanics different observers can disagree on the position of events. As an example, let's say I am $100$ m to the left of you. An event, $A$, ...
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Why is it necessary that different observers agree on the value of the spacetime interval $ds^2$?

FIRST POSTULATE OF SPECIAL RELATIVITY The laws of physics are the same and can be stated in their simplest form in all inertial frames of reference. SECOND POSTULATE OF SPECIAL RELATIVITY The speed of ...
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How would the following image look like, if we didn't use $ct$ for time?

What will happen depends on your units. If you are using SI units, and use $t$ instead of $ct$, then you will effectively stretch the graph by an enormous factor such that the light cone will be ...
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How would the following image look like, if we didn't use $ct$ for time?

That is the same as setting c equal to 1, so the scale should not change at all. In general, yes, changing the units of c or t would just visually expand or contract the horizontal axis, equivalent to ...
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Change of coordinates from an arbitrary frame to a locally inertial frame in General Relativity

I have looked at the three-dimensional analogue of the problem Michael treats. Their are 27 equations involved, of the form \begin{equation*} g_{ \rho\sigma,\lambda }=a~(b_{\sigma\lambda\rho}+b_{\rho\...
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Notion of Present

The notion of 'the present' presupposes a notion of simultaniety and Einstein pretty much deconstructed this notion in his paper on special relativity when he promoted the constancy of the speed of ...
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3 votes

String theory: Conformal invariance and Conformal Killing Vectors

There are two potential connections to be made here: the relation between metric moduli and global gauge choice, and the relation between conformal Killing vectors and the Virasoro algebra. Firstly, ...
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