Actually, the inverse-cubic force law produces trajectories that can be expressed analytically in polar coordinates. The 6 types of trajectories are named as Cote's spirals. Unlike Keplerian force where the type of orbit only depends on total mechanic energy, the trajectory type of an inverse cubic law force also depends on specific angular momentum. The 6 types of trajectories are r = r₀ sech (aθ) Poinsot's Spiral with maximum radius r = r₀ csch (aθ) Poinsot's Spiral with straight asymptote r = r₀ exp (aθ) Equiangular/Logarithmic Spiral r = r₀/(aθ) Reciprocal/Hyperbolic Spiral r = r₀ sec (aθ) Epispiral r = r₀ Perfect Circle
It's funny because in astrophysics we really only care about bound orbits and to a lesser extend stability. Whether an orbit is closed or not is entirely academic and not reflective of any real system since perturbations prevent anything approaching the perfection of a closed orbit. However, it seems to me that the stability of closed orbits may be somewhat profound in allowing for a greater possibility of bound systems. I'm sure there is some deep discussion of this somewhere in the literature, but I'm not sure where.
Absolutely love the Jupiter suite. It somehow fits the video type and animation style exactly while also matching what it is about. Couldn’t have chosen better. Also the rest of the video is also genius. It’s pretty much one of the most perfected educational video I have ever encountered.
I spend most of my time frolicking in electromagnetic fields, but I still enjoy Classical Mechanics. Non-linear oscillators, chaos and all that are still a big *WOW!* for me.
If you want to understand more about stable and "nearly circular" orbits and the like, I would recommend reading a book about Lagrangian and Hamiltonian dynamics (it's basically classical mechanics part 2) it is a profoundly beautiful introduction to variational methods, leading to fascinating insights such as liouville's theorem and Noether's theorem.
12:05 This is the same as the prediction for orbits in general relativity. The 1/r³ correction to the Newtonian potential causes orbital precession (in fact, I'm willing to bet that you got that gif from the wiki page on the general relativistic two-body problem lol). I wonder if the prediction of general relativity's effective potential could be expressed as a modification to the power law using one term, rather than a sum of the Newtonian term and the 1/r³ correction. Perhaps as one of the fractional exponents you tried. Or it might just be that the behavior of stable orbits for a fractional exponent mirrors the behavior when you have two different integer potentials together.
The derivation of Bertrand's thm is mostly just a statement about endbehavior and the second derivative at local minima of energy, so in principle, having a polynomial energy function shouldn't change much, except for maybe allowing circular orbits at many distances which each might or might not be stable based on local conditions.
Was there not more to show with the local maxima for greater negative exponents? Or does it just uninterestingly float away without much interaction beyond the set radius?
Well, the Newton's orbits are also unstable because if you put two particles with slightly different speeds, then one will go in ellipse, and another - on hyperbolla or parabola, and it's instability.
What if force isn't applied based on a power law? what about if instead of an inverse square law, why had an inverse catenary/cosh law? what about others?
Wait heres a question i see that they are going thru the origin in higher dimensions... or are they? Could they move past/thru the origin just along another dimension of movement?
Not if the origin itself, or what is in the origin, is also in higher dimensions. I assume a star in higher dimensions is also a higher dimensional object, or at least that would make sense.
If the force depends only on the radius, then any orbits happen the plan spanned by the initial velocity and position vectors. This is why it is fine to model everything in 2d, since it really only occurs in a aplane
Really unfortunate that star systems can't form in 4D, assuming that it is the inverse cube law rather than inverse square. And even with inverse square, double rotations screw everything up. Really sad that if the universe were 4D, there wouldn't be any cool 4D creatures alive to enjoy it. Somewhat ironically, 3D is the only dimension that can experience the beauty of 4D. (and I guess 2D as well, but it would be harder for them to grasp because they have less dimensional analogies to work with)
Great video! I just loved to study orbital mechanics in college. I remember solving a problem where the gravitational potential is perturbed by a weak 1/r force, which resulted in elliptical precession of the orbits and even some intersting closed polar curves.
I tried simulating motion with force laws other than power laws and inverse power laws, and found that there seem to be a few patterns that they follow. It seems the shapes tend to follow the theme of flower shaped orbits when the force decreases slowly, spirals when the force decreases too fast, a type of star shape that resembles the outline of a star shape when the force is attractive at some distances and repulsive at others, and another type of star shape, that involves going in near straight lines in between hitting the edges of the star shape when the force increases quickly with distance. I heard that when GR is taken into account that in 2 spatial dimensions there’s no spacetime curvature outside a mass, which would imply that when GR is taken into account there might be no stable orbits in 2 spatial dimensions.
It's true that in 4D you can have 2 independent planes of rotation, but both of them would be subject to the same effective potential analysis, so you still wouldn't get bounded orbits...probably. This might be something that CodeParade is better suited for.
I don't really know anything about string theiry, but the extra dimensions being compactified seems to imply that they aren't "available", so to say, for gravity to spread into, so the argument about which power law it should follow doesn't exactly apply in that case. In any case: unlike with light, gravity isn't made of "stuff" that is actually spreading out, so the whole argument is only an informed intuition anyway.
Holst's the planets was a great puck of music for the video
Actually, the inverse-cubic force law produces trajectories that can be expressed analytically in polar coordinates. The 6 types of trajectories are named as Cote's spirals. Unlike Keplerian force where the type of orbit only depends on total mechanic energy, the trajectory type of an inverse cubic law force also depends on specific angular momentum.
The 6 types of trajectories are
r = r₀ sech (aθ) Poinsot's Spiral with maximum radius
r = r₀ csch (aθ) Poinsot's Spiral with straight asymptote
r = r₀ exp (aθ) Equiangular/Logarithmic Spiral
r = r₀/(aθ) Reciprocal/Hyperbolic Spiral
r = r₀ sec (aθ) Epispiral
r = r₀ Perfect Circle
It's funny because in astrophysics we really only care about bound orbits and to a lesser extend stability. Whether an orbit is closed or not is entirely academic and not reflective of any real system since perturbations prevent anything approaching the perfection of a closed orbit. However, it seems to me that the stability of closed orbits may be somewhat profound in allowing for a greater possibility of bound systems. I'm sure there is some deep discussion of this somewhere in the literature, but I'm not sure where.
I think its on page 823
Absolutely love the Jupiter suite. It somehow fits the video type and animation style exactly while also matching what it is about. Couldn’t have chosen better.
Also the rest of the video is also genius. It’s pretty much one of the most perfected educational video I have ever encountered.
Also this deserves at least a few 100K views as far as I’m concerned.
Thanks! Honestly, I feel like I'm still learning how to do it.
I spend most of my time frolicking in electromagnetic fields, but I still enjoy Classical Mechanics.
Non-linear oscillators, chaos and all that are still a big *WOW!* for me.
Such a great interesting and beautiful video! You really deserve a lot of views on this one.
If you want to understand more about stable and "nearly circular" orbits and the like, I would recommend reading a book about Lagrangian and Hamiltonian dynamics (it's basically classical mechanics part 2) it is a profoundly beautiful introduction to variational methods, leading to fascinating insights such as liouville's theorem and Noether's theorem.
This was some good fun!
Those higher dimensional orbits sure look a lot like electron orbitals! And since electrons don’t interact with the nucleus they can be stable too
Thank you for this music ❤
Oooh this is a good one.
I read the theorem from Goldstein classical mechanics now I understand clearly..❤
Awesone video, mate!!! Great to include Holst music. +1 subscriber❤
Very cool, nice!
3:43 I would call those "completely different". They remain nearly identical *orbits,* just phase shifted.
That pentagon reminds me of the shape at the poles of jupiter.
i believe you're thinking about the hexagonal cloud formation at the north pole of saturn
12:05 This is the same as the prediction for orbits in general relativity. The 1/r³ correction to the Newtonian potential causes orbital precession (in fact, I'm willing to bet that you got that gif from the wiki page on the general relativistic two-body problem lol).
I wonder if the prediction of general relativity's effective potential could be expressed as a modification to the power law using one term, rather than a sum of the Newtonian term and the 1/r³ correction. Perhaps as one of the fractional exponents you tried.
Or it might just be that the behavior of stable orbits for a fractional exponent mirrors the behavior when you have two different integer potentials together.
Absolutely fascinating! Anything interesting happen with functions of distance that are not just powers? What about polynomials?
The derivation of Bertrand's thm is mostly just a statement about endbehavior and the second derivative at local minima of energy, so in principle, having a polynomial energy function shouldn't change much, except for maybe allowing circular orbits at many distances which each might or might not be stable based on local conditions.
Just Taylor series your function to reduce it to a sum of polynomials then truncate at some point.
@@IIAOPSWNot all functions are analytic at the origin!
Was there not more to show with the local maxima for greater negative exponents?
Or does it just uninterestingly float away without much interaction beyond the set radius?
Not fully sure what you mean. If you mean what happens if the planet drifts away from the star, it just flies off in a straight line.
Yeah, that's what I thought. But without seeing anything, I felt like there was a chance of a repulsive force like negative gravity.
Well, the Newton's orbits are also unstable because if you put two particles with slightly different speeds, then one will go in ellipse, and another - on hyperbolla or parabola, and it's instability.
Wow great video
Heh, good music choice.
What if n was an imaginary number?
super neat!!
Amazing
What if force isn't applied based on a power law? what about if instead of an inverse square law, why had an inverse catenary/cosh law? what about others?
Bertrand's theorem applies to ALL radial functions. For boundedness, you'd have to integrate them and compute the effective potential.
Wait heres a question i see that they are going thru the origin in higher dimensions... or are they? Could they move past/thru the origin just along another dimension of movement?
Not if the origin itself, or what is in the origin, is also in higher dimensions. I assume a star in higher dimensions is also a higher dimensional object, or at least that would make sense.
If the force depends only on the radius, then any orbits happen the plan spanned by the initial velocity and position vectors. This is why it is fine to model everything in 2d, since it really only occurs in a aplane
Gravity doesn’t follow the inverse square exactly though
I wonder what function could imitate the precession we see in mercury? Not that I’m a MOND believer myself.
Trivially, the inverse square law in a Schwarzschild metric would do the trick. 😉
Bertrand....I think the same guy who came up with Bertrand's Paradox.
Bertrand Russell?
@@authenticallysuperficial9874 No, not Russell. There is a Wikipedia entry for Bertrand paradox (probability).
Really unfortunate that star systems can't form in 4D, assuming that it is the inverse cube law rather than inverse square. And even with inverse square, double rotations screw everything up. Really sad that if the universe were 4D, there wouldn't be any cool 4D creatures alive to enjoy it. Somewhat ironically, 3D is the only dimension that can experience the beauty of 4D. (and I guess 2D as well, but it would be harder for them to grasp because they have less dimensional analogies to work with)
Stable but bound
Sounds like the majority of young people's mental health.
Its not called stable, but it doesn't decay so much to fall apart either.
yeah but gravity is a pressure gradiant this formulars are to easy understand this but in realtiy the planbet expands with g
if you want to build stuff you use this but if you want to understand stuff the planet accelerates to the speed of light iwht in under a day
if you want to build stuff you use relavivity but if you want to understand stuff you use reals
but if you want to do time jump or stargate then you better assume the planet expands with g
finally, a mathematical explanation for the stability of orbits
Or lack thereof ;)
Great video! I just loved to study orbital mechanics in college. I remember solving a problem where the gravitational potential is perturbed by a weak 1/r force, which resulted in elliptical precession of the orbits and even some intersting closed polar curves.
Hey, nice video. What do you use to integrate and graph the orbits?
Fascinating! I guess all those "what about this case?"s just go to show the importance of understanding the exact claim of the theorem
the music is a wonderful touch
I tried simulating motion with force laws other than power laws and inverse power laws, and found that there seem to be a few patterns that they follow. It seems the shapes tend to follow the theme of flower shaped orbits when the force decreases slowly, spirals when the force decreases too fast, a type of star shape that resembles the outline of a star shape when the force is attractive at some distances and repulsive at others, and another type of star shape, that involves going in near straight lines in between hitting the edges of the star shape when the force increases quickly with distance. I heard that when GR is taken into account that in 2 spatial dimensions there’s no spacetime curvature outside a mass, which would imply that when GR is taken into account there might be no stable orbits in 2 spatial dimensions.
Great video!
What about non-flat orbits in 4D? I hear there are some weird rotations in 4D where points trace non-planar curves.
It's true that in 4D you can have 2 independent planes of rotation, but both of them would be subject to the same effective potential analysis, so you still wouldn't get bounded orbits...probably. This might be something that CodeParade is better suited for.
🤍
amazing, now I'm curious how they make gravity work in string theory with higher dimensions
I don't really know anything about string theiry, but the extra dimensions being compactified seems to imply that they aren't "available", so to say, for gravity to spread into, so the argument about which power law it should follow doesn't exactly apply in that case.
In any case: unlike with light, gravity isn't made of "stuff" that is actually spreading out, so the whole argument is only an informed intuition anyway.