Solving Variable Change: Difficulties Understanding ##V(x)## & ##y##

[Moolisa]

Summary: When $V (x) = \frac 1 2 mω^2x^2 + mgx$
$H=\frac p 2m +V(x)$
Difficulty understanding how these change on variables came about
$y = x+\frac mg mω^2 = x+\frac g ω^2$

Apologies if this is not the appropriate thread. I chose this one because even though it's physics, I'm having issues with what seems to be basic math principles

$V (x) = \frac 1 2 mω^2x^2 + mgx$

$y = x+\frac{mg} {mω^2} = x+\frac g ω^2$

So that

$H=\frac p {2m} +\frac 1 2 mω^2y^2 - \frac {mg^2} {2ω^2}$

I don't understand how the variables changed

By the way the solution looks like, I tried

$0= \frac 1 2 mω^2x^2 + mgx$
Rearranged so $x=-\frac {2mg^2} {mω^2}$

But that is obviously not how to do it

Any help is appreciated, If you can, please tell me what this technique/concept this is, so I can read up on it. I looked up changing variables online but can only find the one for integrations, which I don't think is what this is

[Moderator's note: Moved from a technical forum and thus no template.]

 

[tnich]

I don't understand how the variables changed

I think the point of the change of variables is to get rid of the $mgx$ term in the Hamiltonian by completing the square. Note that there is no linear $y$ term in the resulting Hamiltonian. As to how to do the change of variables, try this:
1) Solve $y = x+\frac{mg} {mω^2} = x+\frac g ω^2$ for $x$.
2) Note that adding a constant to $x$ does not change $p$.
3) Substitute the resulting expression into $H=\frac {p^2} {2m} +\frac 1 2 mω^2x^2 + mgx$
4) Collect terms.

 

[SammyS]

Summary: When $V (x) = \frac 1 2 mω^2x^2 + mgx$
$H=\frac p 2m +V(x)$
Difficulty understanding how these change on variables came about
$y = x+\frac mg mω^2 = x+\frac g ω^2$

Apologies if this is not the appropriate thread. I chose this one because even though it's physics, I'm having issues with what seems to be basic math principles.
...

Yes, the questions in this thread deal mainly with algebra. However, you do have some typos in your expressions. It looks like most of those result from errors in using LaTeX. Fixing some of these requires knowledge which falls outside the scope of basic algebra. Dimensional Analysis is useful in correcting these. The Second equation, the one for the Hamiltonian, is missing an exponent on $p$, the momentum. This and many other of your LaTeX coded expressions, fractions in particular, have what look to be errors due to lack of the LaTeX grouping symbols, " { " and " } " . Other expressions toward the bottom of the OP have similar fractions coded correctly .

Your LaTeX code for that initial Hamiltonian expression is, $H=\frac p 2m +V(x)$ .
It should be $H=\frac {p^2} {2m} +V(x)$ , which is rendered as: $H=\frac {p^2} {2m} +V(x)$ .

Similarly, $y = x+\frac mg mω^2 = x+\frac g ω^2$ should be $y = x+\frac {mg} {mω^2} = x+\frac g {ω^2}$

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