Hodge Laplacian in Polar Coordinates¶

This notebook derives the scalar Hodge Laplacian from the metric and differential-form operators. For polar coordinates $(r,\theta)$ with $r>0$,

$$g=dr^2+r^2d\theta^2.$$

With the codifferential convention used below, the result is the negative of the usual positive-coordinate Laplace operator.

Polar metric¶

Both the covariant and contravariant metrics carry explicit tensor indices. This lets Egison contract them automatically in the Hodge formula.

declare symbol r, θ : MathValue

def N : Integer := 2
def x : Vector MathValue := [| r, θ |]

def g_i_j : Matrix MathValue :=
  [| [| 1, 0 |], [| 0, r ^ 2 |] |]_i_j
def g~i~j : Matrix MathValue :=
  [| [| 1, 0 |], [| 0, r ^ (-2) |] |]~i~j

g_#_#

$\begin{pmatrix} 1 & 0 \\ 0 & r^{2} \\ \end{pmatrix}_{\#\#}^{\;\;}$

Exterior derivative and Hodge star¶

The Hodge star combines the Levi-Civita tensor, the inverse metric, and the volume density $\sqrt{|g|}=r$.

def d (A : Tensor MathValue) : Tensor MathValue :=
  !(flip ∂/∂) x A

def hodge (A : Tensor MathValue) : Tensor MathValue :=
  let k := dfOrder A
   in withSymbols [i, j]
        sqrt (M.det g_#_#) *
        foldl
          (.)
          ((subrefs A (map 1#j_$1 (between 1 k))) .
           (subrefs (ε' N k) (map 1#i_$1 (between 1 N))))
          (map 1#g~(i_$1)~(j_$1) [1..k])

Codifferential and Laplacian¶

The codifferential is the metric adjoint of $d$ and can be written in terms of two Hodge stars. The degree test selects the appropriate endpoint formula for zero- and top-degree forms.

def δ (A : Tensor MathValue) : Tensor MathValue :=
  let k := dfOrder A
   in (-1) ^ (N * (k + 1) + 1) * (hodge (d (hodge A)))

def Δ (A : Tensor MathValue) : Tensor MathValue :=
  match (dfOrder A) as integer with
  | #0 -> δ (d A)
  | #N -> d (δ A)
  | _  -> d (δ A) + δ (d A)

def f : MathValue := function (r, θ)

Applying the operator to an arbitrary scalar function leaves its derivatives symbolic, so the coordinate formula is visible directly.

Δ f

$-\frac{\partial^2 f}{\partial 1^2} - \frac{\partial f}{\partial 1} r^{-1} - \frac{\partial^2 f}{\partial 2^2} r^{-2}$

The displayed expression is

$$ \Delta f=-\left( \frac{\partial^2f}{\partial r^2} +\frac1r\frac{\partial f}{\partial r} +\frac1{r^2}\frac{\partial^2f}{\partial\theta^2} \right). $$

The $1/r$ and $1/r^2$ terms are not inserted by hand: they arise from the determinant and inverse metric inside the Hodge star.

Hodge Laplacian in Polar Coordinates¶

Polar metric¶

Exterior derivative and Hodge star¶

Codifferential and Laplacian¶

Links