# Higher dimensional relation between angular momentum, moment of intertia and angular velocity

In 3 dimensions we have the well known relation (summation convention is being used)

$$L_i = I_{ij} \omega_j$$

However, as is well known the angular momentum and angular velocity are not vectors but tensors and their Hodge duals are what are used in the above expression. So using the actual 2-forms $$\tilde L_{ij} = \epsilon_{ijk} L_k~,$$

and likewise for $\tilde \omega$, we get the above relation as

$$\tilde L_{ij} = \frac{1}{2} ~I_{kl} ~\epsilon_{kij}~\epsilon_{lmn} ~\tilde \omega_{mn}$$

My question is how do I generalize this to higher dimensions? The angular momentum, moment of inertia and angular velocity will remain second order tensors. However, in higher dimensions the Levi-Civita tensor will be of higher order and I cannot seem to find an unambiguous way to get a $3^rd$ order Levi-Civita tensor from a higher order one.

• Related: physics.stackexchange.com/q/9864/2451 and links therein. – Qmechanic Jan 5 '17 at 12:28
• I think you need to consider that angular momentum in an arbitrary location then you use a cross product in 3D to do the parallel axis theorem. But in higher dimensions you would need to transition to the wedge product. Then you can decompose it into components using tensor notations. – ja72 Jan 5 '17 at 13:49
• @Qmechanic while the link you mentioned is related in that it talks about angular momentum in higher dimensions, I don't think deals with relation between angular momentum and angular velocity. If you think there is some connection to angular velocity could you please point it out? Thanks. – Borun Chowdhury Jan 6 '17 at 10:35
• @ja72 You are talking about the definition of angular momentum $\tilde L_{ij}=x_i p_j - x_j p_i$. That is included in my question when I mention the 3-d angular momentum 'vector' is a Hodge dual of $\tilde L_{ij}$. My question is not about this topic. My question is how to express angular momentum in terms of angular velocity. – Borun Chowdhury Jan 6 '17 at 10:37

It turns out the answer is rather simpler in higher dimensions. It gets complicated when specializing to 3-dimensions.

If a rigid body has angular velocity $\omega_{ij}$ then given the location of a point $r_i$ its velocity is given by

$$v_i = -\omega_{ij} r_j$$

and its angular momentum is given by

$$L_{ij} =\sum m(r) ( -r_i v_j +r_j v_i )\\ = \sum m(r)( r_i r_m \omega_{mj} -r_j r_m \omega_{mi}) \\ = 2 [\sum m(r) r_i r_m] \omega_{mj}$$ where in the last expression one must remember the anti-symmetry of $L_{ij}$.

Specializing to 3D we get $$\tilde L_k = \frac{1}{2} \epsilon_{kij} L_{ij}\\ =\epsilon_{mjt}\epsilon_{ijk} [\sum m(r) r_i r_m] \tilde \omega_t \\ = -\delta^{tm}_{ik} [\sum m(r) r_i r_m ] \tilde \omega_t \\ = [\sum m(r) (-r_k r_i +r^2 \delta_{ki})] \tilde \omega_i \\ = I_{ki} \tilde \omega_i$$

Thus we see that in arbitrary dimensions the relation between angular momenta, moment of inertia and angular velocity is given by

$$L_{ij} = 2 \tilde I_{im} \omega_{mj}$$

where $\tilde I_{ij} = \sum m(r) r_i r_j$.

## 3D common way

Definition of angular momentum vector ($$m = 1$$) $$$$L_l = \epsilon_{lij}x^iv^j$$$$ ($$L_l$$ dual to $$L^{ij} = x^iv^j - x^jv^i$$ tensor)

Rortational velocity of particle $$$$v^j = \epsilon^{jrk}\omega_rx_k.$$$$ ($$\omega_r$$ dual to $$\omega^{jk}$$ antisymmetric tensor of angular velocity $$\epsilon^{jrk}\omega_r = \omega^{jk}$$)

Substitute in the angular momentum definition $$$$L_l = \epsilon_{lij}x^i\epsilon^{jrk}\omega_rx_k.$$$$

Let us use the property of the Levi-Civita tensor: $$$$\epsilon^{lij} = -\epsilon^{jli}$$$$

then

$$$$L_l = - \epsilon_{jli}\epsilon^{jkr}x^ix_k\omega_r.$$$$

Let us use another property of the Levi-Civita tensor: $$$$\epsilon_{jli}\epsilon^{jkr} = \delta_l^k\delta_i^r - \delta_l^r\delta_i^k.$$$$

$$$$L_l = \left( \delta_l^r\delta_i^k - \delta_l^k\delta_i^r \right) x^ix_k\omega_r.$$$$

Expand the brackets and take into account that $$\delta_i^r x^i = x^r$$, $$\delta_l^k x_k = x_l$$ and $$\delta_i^k x^i = x^k$$, we get $$$$L_l = \left( \delta_l^r x^kx_k - x^rx_l\right) \omega_r,$$$$

or

$$$$L_l = \left( \delta_l^r x^2 - x^rx_l\right) \omega_r,$$$$

where inertia tensor $$$$I_l^r = \delta_l^r x^2 - x^rx_l$$$$

## Arbitrary number of dimensions

For Higher dimensions we can use directly angular momentum tensor $$$$L^{ij} = x^iv^j - x^jv^i$$$$ and Rortational velocity of particle we can express via angular momentum tensor $$$$v^j = \omega^{jm}x_m.$$$$

Then

$$$$L^{ij} = x^ix_m\omega^{jm} - x^jx_m\omega^{im}$$$$

Now we can lower indices near $$\omega$$ with metric tensor: \begin{align} \omega^{jm} = g^{jn}g^{mr}\omega_{nr} \\ \omega^{im} = g^{in}g^{mr}\omega_{nr} \end{align}

So, we get

$$$$L^{ij} = \left( x^ix_mg^{jn}g^{mr} - x^jx_m g^{in}g^{mr}\right) \omega_{nr}$$$$

So, we can conclude $$L^{ij} = I^{ijnr} \omega_{nr}$$, the inertia tensor is

$$$$I^{ijnr} = x^ix_mg^{jn}g^{mr} - x^jx_m g^{in}g^{mr}$$$$