Position as a function of energy

[dsaun777]

I've seen position as a function of time in Newtonian physics and potential energy as a function of position, is there an inverse? Any instance where position is a function of energy eg KE, PE. Maybe this is more appropriate for quantum mechanics or modern physics.

 

[PeterDonis]

Any instance where position is a function of energy eg KE, PE.

I'm not sure what you mean by "position is a function of energy". Obviously in mathematical terms any invertible function can be inverted, and typical potential energy functions are invertible. But that doesn't tell you anything about the physics. You need to explain what you mean in terms of physics before we can answer your question.

 

[DrStupid]

and typical potential energy functions are invertible

Really? The gravitational potential energy is typical but the function of position is not invertible - not even in the one-dimensional case.

 

[dsaun777]

I'm not sure what you mean by "position is a function of energy". Obviously in mathematical terms any invertible function can be inverted, and typical potential energy functions are invertible. But that doesn't tell you anything about the physics. You need to explain what you mean in terms of physics before we can answer your question.

For instance, we can make accurate enough predictions for a particles trajectory but there comes a point where we have to measure the particle to define exactly where it is.

 

[PeterDonis]

The gravitational potential energy is typical but the function of position is not invertible - not even in the one-dimensional case.

Huh? $1/r$ is invertible (except for the edge case of $r = 0$, but that can easily be handled with limits).

 

[PeterDonis]

we can make accurate enough predictions for a particles trajectory but there comes a point where we have to measure the particle to define exactly where it is

What does this have to do with "position as a function of energy"? I still don't understand what you're trying to ask.

 

[DrStupid]

Huh? $1/r$ is invertible

$1/|r|$ is not

 

[PeterDonis]

$1/|r|$ is not

It is for the range $0 < r < \infty$, which is the relevant range.

 

[dsaun777]

What does this have to do with "position as a function of energy"? I still don't understand what you're trying to ask.

Not too sure how to clarify it further. But you sort of answered by saying that potential functions are invertible. Could you elaborate on how they are invertible?

 

[PeterDonis]

you sort of answered by saying that potential functions are invertible

Actually, that's not always true; @DrStupid correctly pointed out that in 3 dimensions, the standard gravitational potential function is not invertible (since it depends on the radius and there are multiple positions at any given radius).

In any case, as I said before, "invertible" is a mathematical property, not a physical property. If all you're interested in is what mathematical functions are invertible, that is a separate discussion that belongs in the math forum, not the physics forum. Here we assume your question is about physics.

 

[PeterDonis]

Not too sure how to clarify it further.

In that case, this thread is closed.