How do we decide dimension of motion?

[Pushoam]

How do we decide dimension of motion?
Consider a particle moving along $\hat x$ direction.
This motion is known as one dimensional motion as only one coordinate i.e. x is changing with respect to time.
Consider a particle having circular motion in X-Y plane.
In Cartesian Coordinate system, both x and y are changing with respect to time and hence the motion is two dimensional.
But in Polar coordinate system(r,θ) ,only one coordinate i.e. θ is changing with respect to time and hence the motion is one dimensional.
So, does the dimension of motion depend on the coordinate system?
Or,
Do we always use Cartesian Coordinate system to determine the dimension of motion ?

 

[Chestermiller]

If the motion is in a straight line, it is one dimensional. If it is not in a straight line, but is occurring in a plane, it is two dimensional. If it is not in a straight line, and is not occurring in a plane, it is three dimensional.

 

[Pushoam]

Thank you for replying,

If the motion is in a straight line, it is one dimensional. If it is not in a straight line, but is occurring in a plane, it is two dimensional. If it is not in a straight line, and is not occurring in a plane, it is three dimensional.

According to this ,
no.of dimension of motion = no.of Cartesian coordinates (needed to describe the motion) changing with respect to time

Can you please give me some reference for it?

 

[Chestermiller]

Thank you for replying,

According to this ,
no.of dimension of motion = no.of Cartesian coordinates (needed to describe the motion) changing with respect to time

Can you please give me some reference for it?

In some contexts, 1D means that the problem can be described using 1 spatial independent variable, 2D means that the problem can be described using 2 spatial independent variables, and 3D means that the problem must be described using 3 spatial independent variables.

 

[Pushoam]

In some contexts, 1D means that the problem can be described using 1 spatial independent variable, 2D means that the problem can be described using 2 spatial independent variables, and 3D means that the problem must be described using 3 spatial independent variables.

I want to know these contexts.
Because for circular motion,2 spatial independent variables are needed in Cartesialn Cordinate system, while only one spatial independent variable is needed in Polar Coordinates.

Consider a particle having circular motion in X-Y plane.
In Cartesian Coordinate system, both x and y are changing with respect to time and hence the motion is two dimensional.
But in Polar coordinate system(r,θ) ,only one coordinate i.e. θ is changing with respect to time and hence the motion is one dimensional.
So, does the dimension of motion depend on the coordinate system

 

[DennisN]

while only one spatial independent variable is needed in Polar Coordinates.

Hmm, you may want to rethink that...

But in Polar coordinate system(r,θ) ,only one coordinate i.e. θ is changing with respect to time and hence the motion is one dimensional.

Who says that r is fixed?

Spoiler

r is of course not necessarily fixed to a specific value, and it is independent of θ, thus the motion can also occur in two dimensions (assuming polar coordinates).

EDIT: Also, when r is fixed and θ is changing (using polar coordinates), both x and y are changing in the rectangular (Cartesian) coordinate system.

Can you please give me some reference for it?

See e.g. Coordinate Systems (HyperPhysics)

 

[Pushoam]

Who says that r is fixed?

In circular motion , r remains constant.

 

[DennisN]

In circular motion , r remains constant.

Ok... then, who says the motion has to be circular? Note: I promise I am not trying to annoy you .

 

[Pushoam]

Ok... then, who says the motion is circular? Note: I promise I am not trying to annoy you .

I have taken circular motion as an example.

Consider a particle having circular motion in X-Y plane.

See e.g. Coordinate Systems (HyperPhysics)

This reference doesn't relate coordinate system and dimension.

 

[phinds]

@Pushoam perhaps it would be helpful to you to think of things from the point of view of a traveler. A traveler on a circle can go forward and backward but although he is not able of his own volition to vary the DEGREE to which he moves left/right, he does none-the-less have to also move left/right AS he goes forward/backward, thus 2D motion. In a straight line, this is not the case, so 1D motion. In a helix, he would have to move left/right, forward/backward, and up/down, thus 3D motion.

 

[Pushoam]

@Pushoam perhaps it would be helpful to you to think of things from the point of view of a traveler. A traveler on a circle can go forward and backward but although he is not able of his own volition to vary the DEGREE to which he moves left/right, he does none-the-less have to also move left/right AS he goes forward/backward, thus 2D motion. In a straight line, this is not the case, so 1D motion. In a helix, he would have to move left/right, forward/backward, and up/down, thus 3D motion.

From this ,too, the following can be concluded:

no.of dimension of motion = no.of Cartesian coordinates (needed to describe the motion) changing with respect to time

Isn't the above statement right?
Can you please give me an example where the above statement won't work?

 

[hilbert2]

A vector $\mathbf{x}$ or a set of vectors $\mathbf{x}_i$ in 2D can be obtained with a projection operation from higher-dimensional vectors of arbitrary dimensionality, so it isn't really useful to debate whether circular motion happens in 1D or 2D.

 

[phinds]

A vector $\mathbf{x}$ or a set of vectors $\mathbf{x}_i$ in 2D can be obtained with a projection operation from higher-dimensional vectors of arbitrary dimensionality, so it isn't really useful to debate whether circular motion happens in 1D or 2D.

I don't know about others here but I have no idea what you just said. Is it possible to express it in English?

 

[hilbert2]

Well, a vector $(x,y)$ can be obtained from $(x,y,0)$, $(x,y,0,0)$, $(x,y,0,0,0)$ and so on by just ignoring the dimensions after the second.

 

[phinds]

Well, a vector $(x,y)$ can be obtained from $(x,y,0)$, $(x,y,0,0)$, $(x,y,0,0,0)$ and so on by just ignoring the dimensions after the second.

I still have no idea how this relates to the subject at hand.

 

[hilbert2]

I mean that it doesn't really have any practical significance whether we call motion of a particle, constrained on a 2-dimensional plane of a 3-dimensional space, two- or three-dimensional.

 

[phinds]

I mean that it doesn't really have any practical significance whether we call motion of a particle, constrained on a 2-dimensional plane of a 3-dimensional space, two- or three-dimensional.

OK, thanks. That at least I do understand but find it hard to agree with. By that logic, motion along a straight line (in 3D space) could be considered 1D, 2D, or 3D (or perhaps you mean it should always be considered 3D if it's in a 3D space. I've always understood that motion on a straight line is considered 1D motion.

 

[pixel]

I think in the case of a particle constrained to move in a circle, we would say there is one degree of freedom, but the motion is 2-dimensional.

 

[Pushoam]

I mean that it doesn't really have any practical significance whether we call motion of a particle, constrained on a 2-dimensional plane of a 3-dimensional space, two- or three-dimensional.

By this ,you meant :
There is no need to worry about the dimension of the motion.
When working in polar coordinates system, there will be only one equation of motion as only one coordinate is changing w.r.t. time.
When working in Cartesian coordinates system, there will be two equations of motion as two coordinates are changing w.r.t. time.
Solve the equation of motion, get the answer and forget all about the dimension of motion.

This approach simply removes the problem

Isn't the above statement right?
Can you please give me an example where the above statement won't work?

, and at the same time it makes the concept of dimension useless, doesn't it?

 

[Nugatory]

OK, thanks. That at least I do understand but find it hard to agree with. By that logic, motion along a straight line (in 3D space) could be considered 1D, 2D, or 3D (or perhaps you mean it should always be considered 3D if it's in a 3D space. I've always understood that motion on a straight line is considered 1D motion.

It might be better to say that we don't have to consider more than one dimension if we can describe the motion with one number, don't have to consider more than two dimensions if we can describe the motion with two numbers and so forth.

However, that's "don't have to", not "must not". Depending on the problem, I may find it easier to think of uniform circular motion as one number on a (curved) one-dimensional path or two numbers in the Euclidean plane with the constraint that #x^2+y^2## is constant.

 

[Pushoam]

However, that's "don't have to", not "must not". Depending on the problem, I may find it easier to think of uniform circular motion as one number on a (curved) one-dimensional path or two numbers in the Euclidean plane with the constraint that #x^2+y^2## is constant.

Then, can it be concluded that dimension of motion will depend on the no. of variables changing w.r.t.time in a given co-ordinate system?

 

[Dale]

Then, can it be concluded that dimension of motion will depend on the no. of variables changing w.r.t.time in a given co-ordinate system?

I am in the "it doesn't matter" camp.

If you want you can use Newtonian mechanics and analyze it using 2 variables. If you want you can use Lagrangian mechanics and analyze it using 1 variable.

In the end you get the same answer.

 

[phinds]

However, that's "don't have to", not "must not".

Fair enough

 

[Pushoam]

O.K.
Thanks to all.