# Tag Info

### 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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### 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.

### 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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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 $g \to g+ {\mathcal ... • 41.3k 1 vote ### How do we assume the direction of$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$.... • 809 1 vote ### How do we assume the direction of$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}$is out of the diagram. Not knowing ... 0 votes ### Cart Pole kinetic energy Let us start with an example first. Consider a pendulum with constantly accelerated support (instead of the support being a degree of freedom of the system). The position of said pendulum can be ... • 1,528 2 votes Accepted ### Why can't we use integral of$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$-axis the ... • 25.9k 5 votes Accepted ### Is this hamiltonian of the form of some well-known physical system? One likely candidate for what they "want you to say" is a Kepler potential (i.e. produced by an inverse distance-squared force) viewed in a rotating frame that scales the rotation rate by a ... • 1,277 0 votes ### Why is it necessary that different observers agree on the value of the spacetime interval$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,$A$, ... • 5,370 0 votes ### 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 ... • 35 5 votes Accepted ### 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 ... • 33.3k 1 vote ### 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\...