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1D, 2D and 3D


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#1 Nathan2222_old   Members   -  Reputation: -400

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Posted 08 February 2014 - 04:34 PM


I tried joining physicsforums.com to ask but kept getting an invisible random error.

I was just wondering what's the difference between 1D, 2D and 3D physics.
Does v = u + at apply when calculating the velocity of a 3d car.
Thanks.

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#2 ApochPiQ   Moderators   -  Reputation: 14291

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Posted 08 February 2014 - 04:47 PM

The basic form of velocity = start_velocity + (acceleration) * (time) is valid in any number of spatial dimensions. The only thing that changes is the number of components in the relevant vectors.

#3 Nathan2222_old   Members   -  Reputation: -400

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Posted 08 February 2014 - 04:56 PM

The basic form of velocity = start_velocity + (acceleration) * (time) is valid in any number of spatial dimensions. The only thing that changes is the number of components in the relevant vectors.


is this true for other physics formula/equations

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#4 Lactose!   GDNet+   -  Reputation: 2475

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Posted 08 February 2014 - 05:02 PM

All physics stuff I can think of that works on a vector works regardless of how many spatial dimensions you have, 1D, 2D or 3D doesn't really matter.

As ApochPiQ said, it just changes the vector from being e.g. direction.x to direction.x, direction.y, direction.z.



#5 Paradigm Shifter   Crossbones+   -  Reputation: 5131

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Posted 08 February 2014 - 05:07 PM

Cross product is 3D specific, that's maths not physics though. There is an analogy for any dimension, the wedge product.


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#6 Nathan2222_old   Members   -  Reputation: -400

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Posted 08 February 2014 - 05:10 PM

What of distance etc. because i saw a formula that is dx/something + dy/something. I can't remember what it was divided by

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#7 Bacterius   Crossbones+   -  Reputation: 8162

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Posted 08 February 2014 - 05:14 PM

Physics generalizes pretty well to higher dimensions, so the vector functions like v = u + at, F = ma, and so on apply in any number of dimensions. However some things are different due to the changes in geometry. For instance, there is no rotation in the usual sense in 1D, the cross product operation is not closed in neither one nor two dimensions (but you can still define something which works similarly from a physics perspective). While physics in general is typically first defined in three dimensions because, as far as we can tell, this is the space in which we exist,  along with a temporal dimension, there is no reason we can't come up with generalizations in lower or higher dimensions where all or most of the known laws of physics still apply in some sense, even if we can't visualize or test them directly. In fact, generalizations often guide physicists to find simplifications or improvements to existing theories, so it's actually beneficial to seek them.

 

So, to answer your question, yes, people have tried very hard to make sure that all of the physics equations are applicable in as many different situations as possible, and that includes different number of dimensions. However, you may need to make some changes to some of the code between 2D and 3D to account for fundamental changes in how the various operations which physics builds on are defined (and whether they apply at all in a meaningful way, e.g. rotation in one-dimensional space). Overall you can probably think of 2D as being 3D constrained to two dimensions.

 

However v = u + at probably doesn't "apply" that well in any dimension to find the velocity of a car, since a car generally is not accelerating in a constant direction at a constant rate. You probably want dv = a * dt where the acceleration of the car is integrated over time to obtain its change in velocity (and you can then integrate that to obtain its change in position).

 

--

 

More formally, modern physics does not assume that space is three-dimensional, or even euclidean. The formulas apply to any type of space-time geometry which meets specific mathematical requirements (which depend on the theory being considered, e.g. classical newtonian physics, general relativity, etc.). If you've learnt polymorphism and/or interfaces and/or contracts, this is the same idea: the Physics class uses a Space interface, such as euclidean R^3, which defines what distances, points, and vectors are, as well as a Time interface which describes the flow of time, and applies the (very generic) laws of physics to these. How these interfaces are implemented doesn't matter as long as they behave as expected by the Physics class. Thus you can create different "kinds" of physics which look different, but which ultimately all behave the same on some level. Apologies if I've used poor terminology, I'm not a physicist, just trying to convey that physics can be generalized. (and, no, those aren't *real* classes to be coded up, they are an analogy that I thought might be easier to understand to someone with a programming background)


The slowsort algorithm is a perfect illustration of the multiply and surrender paradigm, which is perhaps the single most important paradigm in the development of reluctant algorithms. The basic multiply and surrender strategy consists in replacing the problem at hand by two or more subproblems, each slightly simpler than the original, and continue multiplying subproblems and subsubproblems recursively in this fashion as long as possible. At some point the subproblems will all become so simple that their solution can no longer be postponed, and we will have to surrender. Experience shows that, in most cases, by the time this point is reached the total work will be substantially higher than what could have been wasted by a more direct approach.

 

- Pessimal Algorithms and Simplexity Analysis


#8 Lactose!   GDNet+   -  Reputation: 2475

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Posted 08 February 2014 - 05:14 PM

Euclidean distance -- Add the squared difference in each dimension. Square root of the sum.

Doesn't matter how many dimensions.


Edited by CoreLactose, 08 February 2014 - 07:48 PM.


#9 Álvaro   Crossbones+   -  Reputation: 11886

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Posted 08 February 2014 - 05:16 PM

What of distance etc. because i saw a formula that is dx/something + dy/something. I can't remember what it was divided by


Look, if you can't remember the formula, we won't be able to tell you whether it works in any number of dimensions. To first order, everything works the same. Things having to do with rotations might look a bit different, because the group of rotations in 3D is significantly harder to handle than the group of rotations in 2D (and there is only one "rotation" in 1D).

Edited by Álvaro, 08 February 2014 - 05:17 PM.


#10 Paradigm Shifter   Crossbones+   -  Reputation: 5131

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Posted 08 February 2014 - 05:19 PM

Distance depends on the topology though. The distance between 2 points on the surface of the Earth is not the Euclidean distance unless you build a tunnel. Same kind of thing applies in Manhattan.


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#11 Nathan2222_old   Members   -  Reputation: -400

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Posted 08 February 2014 - 05:19 PM

Physics generalizes pretty well to higher dimensions, so the vector functions like v = u + at, F = ma, and so on apply in any number of dimensions. However some things are different due to the changes in geometry. For instance, there is no rotation in the usual sense in 1D, the cross product operation is not closed in neither one nor two dimensions (but you can still define something which works similarly from a physics perspective). While physics in general is typically first defined in three dimensions because, as far as we can tell, this is the space in which we exist,  along with a spatial dimension, there is no reason we can't come up with generalizations in lower or higher dimensions where all or most of the known laws of physics still apply in some sense, even if we can't visualize or test them directly. In fact, generalizations often guide physicists to find simplifications or improvements to existing theories, so it's actually beneficial to seek them.
 
So, to answer your question, yes, people have tried very hard to make sure that all of the physics equations are applicable in as many different situations as possible, and that includes different number of dimensions. However, you may need to make some changes to some of the code between 2D and 3D to account for fundamental changes in how the various operations which physics builds on are defined (and whether they apply at all in a meaningful way, e.g. rotation in one-dimensional space). Overall you can probably think of 2D as being 3D constrained to two dimensions.
 
However v = u + at probably doesn't "apply" that well in any dimension to find the velocity of a car, since a car generally is not accelerating in a constant direction at a constant rate. You probably want dv = a * dt where the acceleration of the car is integrated over time to obtain its change in velocity (and you can then integrate that to obtain its change in position).
 
--
 
More formally, modern physics does not assume that space is three-dimensional, or even euclidean. The formulas apply to any type of space-time geometry which meets specific mathematical requirements (which depend on the theory being considered, e.g. classical newtonian physics, general relativity, etc.). If you've learnt polymorphism and/or interfaces and/or contracts, this is the same idea: the Physics class uses a Space interface, such as euclidean R^3, which defines what distances, points, and vectors are, as well as a Time interface which describes the flow of time, and applies the (very generic) laws of physics to these. How these interfaces are implemented doesn't matter as long as they behave as expected by the Physics class. Thus you can create different "kinds" of physics which look different, but which ultimately all behave the same on some level. Apologies if I've used poor terminology, I'm not a physicist, just trying to convey that physics can be generalized. (and, no, those aren't *real* classes to be coded up, they are an analogy that I thought might be easier to understand to someone with a programming background)

You consider it poor and i consider it new.
Thanks for the explanation and i still learning (physics and programming)

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#12 Nathan2222_old   Members   -  Reputation: -400

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Posted 08 February 2014 - 05:27 PM

It was dx/dt + dy/dt + dz/dt

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#13 Paradigm Shifter   Crossbones+   -  Reputation: 5131

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Posted 08 February 2014 - 05:35 PM

That would generalise to dv/dt where v = (x, y, z) [maybe more or less components of v depending on dimension]


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#14 Nathan2222_old   Members   -  Reputation: -400

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Posted 08 February 2014 - 05:41 PM

So i could calculate the momentum of an object in any dimension using p = m x v ?

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#15 Nathan2222_old   Members   -  Reputation: -400

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Posted 08 February 2014 - 06:00 PM

Does dimensional plane matter except when rotating or finding the position an object

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#16 frob   Moderators   -  Reputation: 18874

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Posted 08 February 2014 - 06:03 PM

Re-iterating the points from Bacterius, the math is still valid in different dimensions.

In college, my linear algebra teacher often gave us simple physics problems but in higher and lower dimensions. (He also had some fun things to help disprove the physics myths of 4+ spatial dimensions, such as "imagine light traveling in 4D space in this direction; what velocity would the light be when viewed on a 3D projection?") Math functions like cross product and dot product, also called the vector product and the inner product, respectively, still work exactly as expected mathematically in different spatial dimensions. It may take some extra work to calculate a 5D cross product, but if you have 4 5D vectors you can still use it to calculate something perpendicular to them.

We live in a world of 3 spatial dimensions. It may take some work to figure out what an operation means in a world with a different number of spatial dimensions, but once you understand it, the math still holds. Right now I expect you could easily explain perpendicular vectors in 2D space and 3D space. But in 4D, can you explain what it means to be perpendicular to a hyperplane? Or in 1D, can you explain what it means to be perpendicular to a point? (Hint, I can't for 1D.)

Going on with your 1D example. With only one spatial dimension you have a scalar universe. I can still imagine acceleration and position in a 1D universe, but concepts like angles don't work as well as they require two or more dimensions. Many operations that use n-1 dimensional components -- such as the cross product -- won't work directly because n-1 is zero. This is probably a perfectly valid thing since in a scalar universe perpendicular would be undefined. With only one component nothing can be orthogonal.


The math still works, you just need to make sure your formulas are the proper versions for that specific dimensionality. And to do that, you need to really understand the math, not just regurgitate formulas fed to you by your search engine.
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#17 Nathan2222_old   Members   -  Reputation: -400

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Posted 08 February 2014 - 06:18 PM

Re-iterating the points from Bacterius, the math is still valid in different dimensions.
In college, my linear algebra teacher often gave us simple physics problems but in higher and lower dimensions. (He also had some fun things to help disprove the physics myths of 4+ spatial dimensions, such as "imagine light traveling in 4D space in this direction; what velocity would the light be when viewed on a 3D projection?") Math functions like cross product and dot product, also called the vector product and the inner product, respectively, still work exactly as expected mathematically in different spatial dimensions. It may take some extra work to calculate a 5D cross product, but if you have 4 5D vectors you can still use it to calculate something perpendicular to them.
We live in a world of 3 spatial dimensions. It may take some work to figure out what an operation means in a world with a different number of spatial dimensions, but once you understand it, the math still holds. Right now I expect you could easily explain perpendicular vectors in 2D space and 3D space. But in 4D, can you explain what it means to be perpendicular to a hyperplane? Or in 1D, can you explain what it means to be perpendicular to a point? (Hint, I can't for 1D.)
Going on with your 1D example. With only one spatial dimension you have a scalar universe. I can still imagine acceleration and position in a 1D universe, but concepts like angles don't work as well as they require two or more dimensions. Many operations that use n-1 dimensional components -- such as the cross product -- won't work directly because n-1 is zero. This is probably a perfectly valid thing since in a scalar universe perpendicular would be undefined. With only one component nothing can be orthogonal.
The math still works, you just need to make sure your formulas are the proper versions for that specific dimensionality. And to do that, you need to really understand the math, not just regurgitate formulas fed to you by your search engine.

I'll have to understand the basics that is taught online and by textbooks first.
I just finished reading an article about dimensions greater than 3d or 1d and it basically said there'll be no earth/universe because there'll be no orbit if the universe was 4d or more.

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#18 frob   Moderators   -  Reputation: 18874

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Posted 08 February 2014 - 07:04 PM

I'll have to understand the basics that is taught online and by textbooks first.
I just finished reading an article about dimensions greater than 3d or 1d and it basically said there'll be no earth/universe because there'll be no orbit if the universe was 4d or more.

Yes, it is very easy to use currently-stated universal constants to show that our universe has 3 spatial dimensions, not 4 or more as some people claim. The speed of light (mentioned in my example) is one of the easiest ones to use for that. If we lived in a universe with 4 spatial dimensions our 3D physics systems based on 3D constants would not work; the speed of light would not be constant from a 3D reference if it can also angle through 4D. Tautology is great.

For your source of learning, the basics when taught online probably will not be enough unless you are using college level online courses and textbooks. The mathematics of 3D space is called linear algebra. It builds on many earlier levels of mathematics, such as trigonometry and basic algebra. Many linear algebra courses will assume that you also have a good grasp of calculus, others assume some number theory or set theory, but you might be fine without them. It is often taught in the 3rd or 4th year of computer science students, perhaps 2nd year for math and physics majors.

If you discover the material is too advanced go back and get the prerequisite material.
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#19 Lactose!   GDNet+   -  Reputation: 2475

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Posted 08 February 2014 - 07:36 PM

Distance depends on the topology though. The distance between 2 points on the surface of the Earth is not the Euclidean distance unless you build a tunnel. Same kind of thing applies in Manhattan.

True, I should have specified I meant Euclidean distance.



#20 Álvaro   Crossbones+   -  Reputation: 11886

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Posted 08 February 2014 - 07:45 PM

Distance depends on the topology though. The distance between 2 points on the surface of the Earth is not the Euclidean distance unless you build a tunnel. Same kind of thing applies in Manhattan.


We are talking about the difference between 1D, 2D and 3D (see title). Introducing more general frameworks like Differential Geometry will only confuse things at this stage, so let's assume we are talking about R^1, R^2 and R^3 with the Euclidean metric.

[Oh, and saying that it "depends on the topology" is not very precise anyway. It depends on the manifold we are on and what metric we are using. The metric determines a topology, but not all topologies correspond to metrics, and there could be more than one metric that results in the same topology. For instance, an open disk of radius 1 has the same topology as the surface of a half sphere, which is the same as the topology of R^2 with the Euclidean metric; but the distances between points are quite different (bounded or not bounded, angles of triangles adding to Pi or something else...).]

Edited by Álvaro, 08 February 2014 - 07:55 PM.





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