/sci/ - Science & Math

[Anonymous No.6758581]

Dear /sci/

What is the name for a system of ODEs of the form:<div class="math">A_{ \kappa \mu }(x) \, \ddot x^ \mu +B_{ \kappa \mu \nu }(x) \, \dot x^ \mu \dot x^ \nu =C_ \kappa(x)</div>where <span class="math">\kappa , \mu , \nu[/spoiler] run over <span class="math">1,2, \ldots ,N[/spoiler]; the summation convention was used, i.e.<div class="math">p_{ \mu \nu }q^ \nu \equiv \sum _{ \nu =1}^N p_{ \mu \nu }q^ \nu</div>etc.; and where <span class="math">A,B,C[/spoiler] are generally nonlinear functions of the <span class="math">x^ \mu[/spoiler], but independent of the first and second derivatives..?

tl;dr, what is the specific name for a system of autonomous second-order ODEs which are linear in the second derivatives, quadratic in the first derivatives, and arbitrarily nonlinear in the variables?
Or what are some theorems about such systems?

[Anonymous No.6758759]

Page 3 bump

Assuming that the Lagrangian <span class="math">L[/spoiler] is quadratic in the generalized velocities, this is also the general form of the Euler-Lagrange equations, since<div class="math">p^ \mu = \frac { \partial L}{ \partial \dot q^ \mu }</div> is then linear in the velocities, so that<div class="math">\frac { \mathrm d}{ \mathrm dt} \frac { \partial L}{ \partial \dot q^ \kappa }= \frac { \mathrm dp^ \kappa}{ \mathrm dt}= \frac { \partial p^ \kappa}{ \partial \dot q^ \mu } \ddot q^ \mu + \frac { \partial p^ \kappa}{ \partial q^ \mu } \dot q^ \mu</div>and since <span class="math">p^ \kappa[/spoiler] is linear in the velocities, it follows that <span class="math">\displaystyle \frac { \partial p^ \kappa}{ \partial \dot q^ \mu }=A_{ \kappa \mu }(q)[/spoiler] is independent of the velocities and <span class="math">\displaystyle \frac { \partial p^ \kappa}{ \partial q^ \mu } \dot q^ \mu[/spoiler] can be written as a quadratic form of the velocities <span class="math">B^1_{ \kappa \mu \nu }(q) \dot q^ \mu \dot q^ \nu[/spoiler].

Finally, <span class="math">\displaystyle \frac { \partial L}{ \partial q^ \kappa }[/spoiler] must be the sum of a quadratic form of the velocities <span class="math">B^2_{ \kappa \mu \nu }(q) \dot q^ \mu \dot q^ \nu[/spoiler] and a velocity independent term <span class="math">C_ \kappa (q)[/spoiler], and then we let <span class="math">B=B_1-B_2[/spoiler] and recover the form of the differential equation above.

Now since this is the general form of the EL equations, there must be some literature on them, no?
in particular, what is the stability of these and general qualitative behavior?

So a mathematical treatment of the physical equations, basically

[Anonymous No.6758791]

In deriving B, you assume that p is a(q)·q'+b(q) with b(q)=0, but whatever.

If you introduce v:=q', X:=(q,v), then you just have a non-linear ode system, quadratic in v. Don't know why that needs it's own term, but for C=0, you can look at
http://en.wikipedia.org/wiki/Solving_the_geodesic_equations#The_geodesic_equation

[Anonymous No.6758805]

>>6758581
>>6758759
>>6758791
What a refreshing change from all the science vs. religion threads the mods seem to prefer

[Anonymous No.6758846]

>>6758805
proceed with what, though?

Since I'm pushing a System-F agenda atm. (>>6758816) let me point out this funky paper I came across

http://bentnib.org/conservation-laws.pdf

[Anonymous No.6759031]

>>6758791
>In deriving B, you assume that p is a(q)·q'+b(q) with b(q)=0, but whatever.
>p is a(q)·q'+b(q)
Yes, I explicitly assumed that the Lagrangian was quadratic in the velocities, so the conjugate momenta must be linear in the velocities. I should have made it more clear, but by "quadratic" I specifically mean "quadratic form", so there are no linear velocity terms in the Lagrangian. This is always the case in classical mechanics. So then there cannot be a term depending only on q in the momenta
>with b(q)=0
Yes.

>Don't know why that needs it's own term, but for C=0
No, C must be there. Unless you know how to get rid of it? That would be nice. Mind you it's (extremely) nonlinear in the <span class="math">x^\mu[/spoiler].

>geodesic equation
Yeah, I noticed that it had a similar form. No surprise though—the geodesic equation follows from a least action principle

>>6758846
>proceed with what, though?
I really just want some stability theorems to do with this equation, or any sort of qualitative analysis.

>>6758805
Not so fast buddy

[Anonymous No.6759057]

>>6759031
>I should have made it more clear, but by "quadratic" I specifically mean "quadratic form", so there are no linear velocity terms in the Lagrangian. This is always the case in classical mechanics.
No
Consider
http://en.wikipedia.org/wiki/Lorentz_force#Lorentz_force_and_analytical_mechanics
or the Landau problem, or any expansion of non-polynomial dispersion relations in solid state physics.

>No, C must be there
You parsed that sentence differently, than I intended, for C=0 the geodesics are an example.
And "term" meant name, as in new name for that kind of equation.

>I really just want some stability theorems to do with this equation, or any sort of qualitative analysis.
There is certainly much literature.. I don't really know but since you want to talk about a broad class of systems, the answer will be broad. Do you know of stuff like KAM theory, or ergodicity stuff. If not, maybe that leads somewhere
http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem

[Anonymous No.6759085]

>>6759057
Shit, I meant "Classical" classical mechanics, like rigid bodies and stuff
>pic related

>"Don't know why that needs it's own term"
This was the phrase I was parsing. Still confused what you meant by that

>KAM theory
>ergodicity stuff
Thanks. Will read

[Anonymous No.6759182]

>Shit, I meant "Classical" classical mechanics, like rigid bodies and stuff
>pic related

It's still wrong, because not-linearity of the Lagrangian is a coordinate dependent statement
http://en.wikipedia.org/wiki/Canonical_transformation#Type_1_generating_function

>Still confused what you meant by that
doesn't matter