Angular momentum in a two dimensional world

[Shubham135]

If we were living in a two dimensional world. would we know about angular momentum of an object?

 

[jbriggs444]

If we were living in a two dimensional world. would we know about angular momentum of an object?

Your concern is that in our three dimensional world, angular momentum is represented as a [pseudo-]vector at right angles to both applied force and moment arm and that people in a two dimensional world could not represent such a quantity?

It would not be a problem. Angular momentum in such a world would be a scalar.

 

[croad]

Imagine the body rotating in a plane about a point. In a circle. The mass is a mystery though to me, because then the body ought to have infinite density, but perhaps the definition of mass would be different in such a world...

 

[jbriggs444]

Imagine the body rotating in a plane about a point. In a circle. The mass is a mystery though to me, because then the body ought to have infinite density, but perhaps the definition of mass would be different in such a world...

Even in our own three dimensional universe, satellites can revolve around the point at the center of a planet [actually the barycenter of the system] without requiring the planet to have infinite density.

 

[croad]

Even in our own three dimensional universe, satellites can revolve around the point at the center of a planet [actually the barycenter of the system] without requiring the planet to have infinite density.

mass = density x volume. In a 2-dim world volume (as it is defined) is zero. How does a body therefore get any mass in a 2-dim world?

 

[jbriggs444]

mass = density x volume. In a 2-dim world volume (as it is defined) is zero. How does a body therefore get any mass in a 2-dim world?

In a 2 dim world, volume is length times width. Height does not come in.

Edit: If you wanted to embed such a two dimensional world in our three dimensional world then one way of proceeding would indeed require using sheets of infinite 3-density. But my understanding is that we are not talking about actually implementing such a world but merely contemplating its logical consequences.

 

[A.T.]

If we were living in a two dimensional world. would we know about angular momentum of an object?

In classical mechanics we often solve problems in 2D only. A 2D object is a collection of point masses, and it's angular momentum is the sum of their angular momenta.

 

[croad]

In a 2 dim world, volume is length times width. Height does not come in.

Edit: If you wanted to embed such a two dimensional world in our three dimensional world then one way of proceeding would indeed require using sheets of infinite 3-density. But my understanding is that we are not talking about actually implementing such a world but merely contemplating its logical consequences.

That was precisely my point. The definition of mass (or volume) would change.

 

[vanhees71]

By (modern) definition angular momentum is the generator of rotations. The most convenient description of rotations is to describe the plane as the complex plane of $\zeta=x+\mathrm{i} y$. Then a rotation is described by
$$\zeta'=\exp(-\mathrm{i} \varphi) \zeta, \quad \varphi \in [0,2 \pi[,$$
where $\varphi$ is the rotation angle. For an infinitesimal translation you get
$$\delta \zeta =-\mathrm{i} \delta \varphi \zeta=\delta \varphi (y-\mathrm{i} \varphi).$$
Now going back to real Cartesian $\mathbb{R}^2$ vectors we have
$$\delta \vec{r}=\delta \tilde{\varphi} \vec{r} \quad \text{with} \quad (\delta \tilde{\varphi})_{ij}=\delta \varphi \epsilon_{ij}.$$
So angular momentum is an antisymmetric tensor or equivalently a pseudoscalar
$$J=\epsilon_{ij} x_i p_j,$$
because then the Poisson bracket gives the correct relation
$$\delta \varphi \{J,x_k\}=\delta \varphi \epsilon_{ij} \{x_i p_j,x_k\}=-\delta \varphi \epsilon_{ij} x_i \delta_{kl} \delta_{jl}=\delta \varphi \epsilon_{ki} x_i.$$

 

[Shubham135]

In classical mechanics we often solve problems in 2D only. A 2D object is a collection of point masses, and it's angular momentum is the sum of their angular momenta.

in our 3D world suppose an object is rotating in X-Y plane we say that it has angular momentum whose direction is Z direction.In a 2D world how will they give direction to such rotation in fact they won't know about any axis of rotation.

 

[Shubham135]

By (modern) definition angular momentum is the generator of rotations. The most convenient description of rotations is to describe the plane as the complex plane of $\zeta=x+\mathrm{i} y$. Then a rotation is described by
$$\zeta'=\exp(-\mathrm{i} \varphi) \zeta, \quad \varphi \in [0,2 \pi[,$$
where $\varphi$ is the rotation angle. For an infinitesimal translation you get
$$\delta \zeta =-\mathrm{i} \delta \varphi \zeta=\delta \varphi (y-\mathrm{i} \varphi).$$
Now going back to real Cartesian $\mathbb{R}^2$ vectors we have
$$\delta \vec{r}=\delta \tilde{\varphi} \vec{r} \quad \text{with} \quad (\delta \tilde{\varphi})_{ij}=\delta \varphi \epsilon_{ij}.$$
So angular momentum is an antisymmetric tensor or equivalently a pseudoscalar
$$J=\epsilon_{ij} x_i p_j,$$
because then the Poisson bracket gives the correct relation
$$\delta \varphi \{J,x_k\}=\delta \varphi \epsilon_{ij} \{x_i p_j,x_k\}=-\delta \varphi \epsilon_{ij} x_i \delta_{kl} \delta_{jl}=\delta \varphi \epsilon_{ki} x_i.$$

Yes ..the generator of rotation tells us about rotation of an object in a plane about an axis, my point is that in 2D world we won't know about any axis of rotation in fact to extend it further in any space with even number of dimentions .It is ony logical to talk about rotation in space with odd number of dimentions. Fortunately we live in 3D world so the concept of rotation seems logical.IN fact you can imagine roation about an axis in 1D also as the point is rotating and the axis of the rotation is the dimension.

 

[jbriggs444]

Yes ..the generator of rotation tells us about rotation of an object in a plane about an axis, my point is that in 2D world we won't know about any axis of rotation in fact to extend it further in any space with even number of dimentions.

One does not need for the "axis of rotation" to be associated with a direction vector in order for it to be meaningful.

 

[A.T.]

they won't know about any axis of rotation.

Right, they just need a point.

It is ony logical to talk about rotation in space with odd number of dimentions.

Nonsense. See:
http://ocw.mit.edu/courses/aeronaut...fall-2009/lecture-notes/MIT16_07F09_Lec21.pdf
https://en.wikipedia.org/wiki/Angular_momentum#Scalar_.E2.80.94_angular_momentum_in_two_dimensions

 

[Shubham135]

Right, they just need a point.Nonsense. See:
http://ocw.mit.edu/courses/aeronaut...fall-2009/lecture-notes/MIT16_07F09_Lec21.pdf
https://en.wikipedia.org/wiki/Angular_momentum#Scalar_.E2.80.94_angular_momentum_in_two_dimensions

Sorry ..i would like to modify my statement...it only make sense to talk about rotation around AN AXIS in space withh odd number of dimentions for even number we can talk about point of rotation and not the axis.for example in two dimentions the axis of rotation is out of the space.

 

[Shubham135]

Sorry ..i would like to modify my statement...it only make sense to talk about rotation around AN AXIS in space withh odd number of dimentions for even number we can talk about point of rotation and not the axis.for example in two dimentions the axis of rotation is out of the space.

and i read all those...do you have any thoughts of your own?

 

[Hornbein]

If we were living in a two dimensional world. would we know about angular momentum of an object?

In Euclidean 2D angular momentum is a scalar, a plain old real number. There is only one possible plane, so there is no need or use for a normal vector to define the plane. The scalar can be positive if the mass is spinning one direction, negative if spinning in the other sense, or zero with no spin at all.

If 2D space is a Mobius strip then some 2D Bernhard Riemann might come up with a more complicated scheme.