# Questions tagged [coordinate-systems]

A set of numbers used to quantify location in space.

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### Conservation and potential with non-cartesian forces

I understand how to determine if a force is conservative from $$\nabla\times \mathbf{F}=0 \implies \mathbf{F}\text{ is conservative}$$ When $F$ is in cartesian coordinates. ...
1 vote
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### Number of Degrees of Freedom of a Rigid Body System - Proof

Let us define the number of degrees of freedom of a material system as the number of scalar parameters needed to know the position of each particle of the system with respect to any inertial frame of ...
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• 61
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### How do we assume the direction of $u_{\theta}$ and $u_{r}$ in polar coordinate systems?

Is there a way to correctly predict the direction of the unit radial vector and the unit transverse vector in problems like the one below or is it just better to take a guess and solve the problem ...
1 vote
61 views

### Cart Pole kinetic energy

As explained in [1], the kinetic energy of a Cart Pole is: $$\frac{1}{2} (M+m)\dot x^2 + \frac{1}{2} m L^2 \dot \theta^2 - m L cos(\theta) \dot \theta \dot x$$ Where $m$ is the mass at the tip of ...
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1 vote
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### Would a gyroscope have solved the longitude problem?

So I was thinking about the longitude problem, which was the problem of determining the longitude at sea. It caused great problems in sea navigation. See: https://en.wikipedia.org/wiki/Longitude#...
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• 105
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### Different AdS metrics in global coordinates

In David Tong's lecture notes I came across the folloing AdS metric in global coordinates ds_3^2 = \left( \frac{dr^2}{\frac{r^2}{R^2} + 1 } - \left(\frac{r^2}{R^2} + 1 \right)dt^2 + ...
• 109
535 views

### Basis Vectors as Partial Derivatives Issues

I have been introduced a number of times to people defining vectors as derivatives of a curve, with basis vectors as partial derivatives, but I have several issues with this that make this formalism ...
• 592
1 vote
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### Can the metric tensor be treated as a linear transformation?

In general relativity, the metric tensor $g$ is a covariant, second rank, symmetric tensor that can be written down as a 4x4 matrix. The metric tensor generalizes the notion of distance between points ...
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### Equations of motion only have a solution for very specific initial conditions

An exercise made me consider the following Lagrangian $$L = \dot{x}_1^2+\dot{x}_2^2+2 \dot{x}_1 \dot{x}_2 + x_1^2+x_2^2.\tag{1}$$ If I didn't make a mistake the equations of motion should be given by: ...
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### Is the proper time invariant while going from stationery frame to freely falling frame in Genral Relativity?

While reading GTR, I found the following calculation: Consider an arbitrary gravitational field and let us take $x^{\mu}$ as the stationery/lab frame and $\xi^{\mu}$ as the freely falling frame, where ...
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### How to find canonical transformation to achieve desired Hamiltonian?

I am trying to find a way to transformation that will turn a Hamiltonian from one form into another form: $$(1)\;\;\;H=p^2+e^x\rightarrow\bar{H}=p'^2.$$ I don't know of any systematic ways to do this. ...
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