Four unit point masses ( each) sit at the corners of a regular tetrahedron

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Scientific programming for physicists

Scientific Computing for Physicists, Ch Arrays in memory: layout, cost and the data structures of physical problems

Problem

Four unit point masses (

m_k=1\,\mathrm{kg}

each) sit at the corners of a regular tetrahedron, at positions (in metres)

\mathbf{r}_1=(1,1,1)

\mathbf{r}_2=(1,-1,-1)

\mathbf{r}_3=(-1,1,-1)

\mathbf{r}_4=(-1,-1,1)

. The inertia tensor is the contraction

I_{ij}=\sum_k m_k\left(r_k^2\delta_{ij}-r_{k,i}r_{k,j}\right).

((a)) Write the second-moment tensor

S_{ij}=\sum_k m_k r_{k,i}r_{k,j}

as a single einsum call on arrays m (shape

(4,)

) and r (shape

(4,3)

), and form

I_{ij}

from

S

and the

3\times3

identity matrix. ((b)) Evaluate

I_{ij}

numerically for the positions given. ((c)) The tetrahedral arrangement has enough rotational symmetry that the inertia tensor should be isotropic,

I_{ij}=I\delta_{ij}

. State whether the computed tensor satisfies this and give

I