# Is Position A Derivative?

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I was just wondering if position could be a derivative of something itself, for example existence over time?. If not then can it be broken down any further in any way? What does position mean?

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Position, in mathematical terms, is $\overrightarrow{r}(t)$, at time t. It is usually a parametrized function.

It corresponds to the displacement from the origin to your point. It itself is not a derivative, but it has two frequently used derivatives, the velocity and acceleration expressions. From these things, you can compute all sorts of interesting tangential and normal components, and the magnitude of the velocity derivative is the speed of the particle at time t. But it is off-topic somewhat.

Edited by A Tripolation
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I was just wondering if position could be a derivative of something itself, for example existence over time?. If not then can it be broken down any further in any way? What does position mean?

Note that in your question you use both the words "position" and "time".

Your question may be more profound than you realize. In physics there are a few quantities that are taken as primitive. Time and position (aka space) are two of them. There are others, electric charge being one.

There is currently no definition for either time or space that are any more fundamental than the operational definitions "Time is what clocks measure" and "Distance is what rulers measure". There are any number of crackpots with alternate viiews on the internet, but they cannot produce a viable rigorous theory. There are also researchers doing legitimate research that may someday present a different picture, but no such theory currently is available.

General relativity, and also special relativity, unify the notions of space and time in a single entity, spacetime. Spacetime is both space and time and in a sense neither. Both space and time are observer dependent and only local concepts in general relativity. They are mixed together by curvature of the spacetime manifold. There is no global concept of either space or time. What there is is a metric that can be used to determine the "length' of the "world line" of an object's existence (what is called a timelike curve), and the units of that length are units of time -- what is called proper time. That same metric applied to a spacelike curve yields length in conventional units of length.

All of this starts with the basic idea of the cartesian plane with a 3-D grid established with "rulers" and a time axis determined by a "clock". There is no more fundamental definition of what constitutes a clock or a ruler. The rest of the edifice is built using some relatively sophisticated mathematics -- Riemannian geometry.

Any deep insight that produces a viable (as opposed to crackpot) more fundamental theory of space and time will be a truly major advance in physics, comparable to Einstein's invention of relativity.

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Take the anti-derivative of distance with respect to something. The derivative of that with respect to the same something, will be distance. At the moment I can't think of any that might be useful though.

There are a few things that you could consider distance to be. For example if you use the Plank wavelength, then the wavelength of something (which is a distance) will tell you about the momentum of a particle.

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Mini off-topic remark:

There are a few things that you could consider distance to be. For example if you use the Plank wavelength, then the wavelength of something (which is a distance) will tell you about the momentum of a particle.

There is no such thing as the "Planck wavelength" (it's Planck with a "ck", btw.). What you probably meant to say was "Planck length". The correct object would have been the "Planck constant". The Planck constant is a natural constant and conceptually something entirely different than those dreaded Planck units which are essentially just a convention for units (and sound cool because they appear in the context of quantum gravity).

Edited by timo
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Take the anti-derivative of distance with respect to something. The derivative of that with respect to the same something, will be distance. At the moment I can't think of any that might be useful though.

There are a few things that you could consider distance to be. For example if you use the Plank wavelength, then the wavelength of something (which is a distance) will tell you about the momentum of a particle.

When you throw in quantum mechanics that procedure may not work.

Not every function is a derivative. A derivative cannot have a "jump" discontinuity -- derivatives have the intermediate value property.

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I was just wondering if position could be a derivative of something itself, for example existence over time?. If not then can it be broken down any further in any way? What does position mean?

Position only has meaning in reference to something.

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There is no such thing as the "Planck wavelength" (it's Planck with a "ck", btw.).

Oops! I meant the de Broglie wavelength.

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So as I now understand it; position is a fundamental attribute/component that only has meaning when applied to an object in space-time and cannot be broken down any further (at least in our current understanding of physics).

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• 1 month later...

Wonder some more Charlie , if you ever find your solutions they will be worthy of high accolades , possibly .

Edited by hal_2011
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So as I now understand it; position is a fundamental attribute/component that only has meaning when applied to an object in space-time and cannot be broken down any further (at least in our current understanding of physics).

Position in spacetime is a 4 coordinates. So I suppose it can be broken, at least in four.

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I would ask whether Existence can be mathmatically described .

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• 4 weeks later...

One could imagine a scalar function $\phi(\mathbf{x})$ which gives a perturbed position (i.e. displacement) as $\nabla \phi$. Could be used as a math trick to reduce a system of equations to a single PDE under certain circumstances. Don't think $\phi$ would have any physical interpretation, though..

Edited by baxtrom
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well, you can take the millionth integral of position if you really want, but it won't mean much.

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