Chinny84

@Luthier415Hz True, but typically you specify IC because of the model (could be a Physical model). IC fixes the model, but the IC themselves have to mean something - an example, if we are are integrating a function with singularities then having a definite integral with that singularity between the bounds of integration is unlikely to be informative.

@Luthier415Hz in short yes - you only want the same number of ICs as free coefficients - if you have Second Order and only one IC then you would only get to $c_1 = -c_2$ and not define what value either one would be exactly. This may be ok for some work. (look at en.wikipedia.org/wiki/Overdetermined_system)

I think you are confusing a few things - but the solution until you specify the IC can only be written in terms of coefficients. Once you pick ICs you then fix what the coefficients are. If you have too many ICs e.g. y'' = k then you may run into a problem as you have too many constraints for too few coefficients so you may not satisfy all the ICs.

@student91 it is a different ODE solution he is using as an example.

yes fix parameters/coefficients based on the IC values - as the function can take any form. It is only once you fix the integration constant do you satisfy your IC (now not all IC can be satisfied by your function) but then you go back the drawing board then :)

the question of different coefficient values given the same IC comes down to uniqueness of your solution. Depends on what you mean, a limit cycle can have the same final form (shape of solution after some time period) regardless of initial conditions (subject to criterion for limit cycle behaviour)

You solve an ODE to give you a function of the form you see with $C1, C2$ these are integration constants - you will then need the initial conditions and or boundary conditions to specify the constants that yield the specified IC. So ODE => Solve ODE => Fix parameters

When solving integrals you have definite and indefinite? Basically with limits and without. Since I did an integration without limits I needed a constant (which you find by inserting in the starting information)

The C is an integration constant

It came from when I moved the half from the one side to the right.

You can relabel terms to make it clearer .. So substitute $y = v^2$ and see what you get

A real mathematician (which I am not) would disagree with my terminology - but you divide both sides by the right hand side leaving unity on the rhs

You are correct in terms of the last equation

A real mathematician (which I am not) would disagree with my terminology - but you divide both sides by the right hand side leaving unity on the rhs

Let's not worry about numerical values at the moment.

but essentially you are correct. Then you move the $dx$ over to the rhs

almost remember that you have $\frac{1}{2}\dfrac{d}{dx}v^2$

This leads to $v\frac{dv}{dx} = \frac{1}{2}\dfrac{d}{dx}v^2$

$\frac{d}{dx}v^2 = 2v\frac{dv}{dx}$ this is implicit differentiation. But we require only $v\frac{dv}{dx}$ so to achieve this we divide both sides by 2.

then you can define $y = v^2$ then you have a linear ode $\frac{1}{2}\frac{dy}{dx} = g - ky$ which you may be able to solve?

I am running out of ideas how to prove the solution. So essentially what I do is state $$\dfrac{d^2 x}{dt^2} = \frac{dv}{dt} = \frac{dx}{dt}\frac{dv}{dx} = v\frac{dv}{dx} = \frac{1}{2}\dfrac{d}{dx}v^2$$

Ok, so you understand the chain rule so from that you can integrate the ode.

NP. Are you happy with the chain rule $\dfrac{dv}{dt} = \dfrac{dx}{dt}\times \dfrac{dv}{dx}$

I am not sure how just knowledge of integration by parts will help you here. I will keep thinking. Do you have any idea how the problem should be solved? initial equations? - failing that I will give a short edit on solving the equation you start in the comment above.

This is why I think my solution does not satisfy your requirements. Have you worked with differential equations?

Ah ok. So I only dealt with equilbria for rotating plasmas in the two fluid limit. So which Department are you atracked to? I only know two personally warwick (place of study) and leuven (having given a talk and had my an examiner for my viva from there goosens)

One more question whilst working thought model I am assuming you only have variation in the z axis? For all variables? I.e $B_x(z,t)$ and similarly for pressure?

You should check a paper by Ken mcclements and chippy dealing with the this very problem where the toroidal rotation is much. Larger than and poloidal so can be neglected in expansion. So are you working on tokamak dynamics? As I was looking at rotational plasma with the grad shafranov equation?

Hi there. It actual fact I was working on nonlinear axis metric solutions for rotating plasmas. So what area of MHD you looking at?

I meant uniqueness instead of existence.

I would modify the question with the full model as outlined in your comment..this will get my brain churning (thumbing over my thesis ;) )

I assume you are trying to prove existence? Or the work done by the wave is constant? I have a background in nonlinear MHD and I haven't come across this formulation of conservation? Also do you have other initial conditions ?

@gage if you check his profile he has a non trivial account on MO so I assume he understands the rules? No?

Ah my beloved MHD waves. :).

@ArthurFischer I second the opinion of alex. The moderators here seem to take pride in what they do and do it well..I guess I am bias since I haven't been on the wrong end of the "law" ;)

@JasperLoy basically I only thought the rep score for answers were +10 or -2 but I had a score of +1? But not to worry I was looking for a quick answer :).

Good evening (London) everyone, Random question which I felt shy to ask on meta..but what does a +1 and a "removed" message on an answer I posted a while back mean?

@jasperloy I don't know why? I think I know why.

I get you on that point. I am a data scientist /risk developer now (so maybe not the best person in terms of advice for academia ;) ) . See ya.

That last msg was rather convoluted for me. But I feel the best way to resolve the issue you have is to focus on the mathematics and use the physics as an application. I was asked in my viva was a trained mathematician ;/..

It is an issue but then you have to look at Edward Witten and realise that you can be both. I was always a more capable maths guy than the rest of the physics phd students at the time.. But I would definitely say I can hold my own with most math undergrads in the more applied maths (and not so applied) but as for beyond that I would have to accept that I am more of physics than maths.

My quantum mechanics days were gone before I did my phd. Sadly. I was looking at MHD theory so a smoothed scale above the stat mech (kinetic theory). I was always intrigued about the transport equations as they have an important place in turbulence dynamics for tokamaks.

Awesome. So you used used minimisation of an eigenvalue problem? Of sorts.

I used WKB whilst studying plasma physics (my research area) .. But back then I would apply (try) all sorts of techniques.. Especially perturbation theory for nonlinear odes. So which area are you focused on? Or better still which length scale ;)?

@semicassical given your awesome answers I guessing you are theoretical physicist like I was ;)? If not then you could be a "mentalist" (here I am covering all bases) .. So what is your research field?

@nickolas used to be? I hold a phd in physics.

So what topics are people (in this chat room) covering at the moment maths wise? Self taught, course etc? I can kick off with probability theory in particular the mathematics underpinning stochastic calc

Hey @IceBoy and the rest of the room. Slowly my MSE world is increasing!!

I suggest that you put that in to another question "what is the geometrical inter. of the second order derivative and higher orders" as this is beyond the scope of this particular question. But please check this site for other occurrences of this question, and be specific as such a question can be closed down if certain criteria are not met.