Cadabra Software

Cadabra software is Field theory motivated approach to Computer Algebra Systems (CAS for short). It can be downloaded from https://cadabra science/


There are some very nice tutorials and user notebooks in this site. I installed cadabra on opensuse linux (leap). I could only use their interface cadabra2-gtk. Had issues downloading other interfaces because of incompatibilities of boost library versions Leap is supporting.


cadabra2-gtk launches Cadabra notebook which behaves like Jupyter notebook. There is supposed to be command completion which didn't work for me.



Nice thing about Cadabra is its elegant latex support. Latex is built into commands directly. A sample session is shown below:

Cadabra SW: Applying to some exercises in Nakahara

Tests on differential forms:


Test 1: If $\omega$ is a differential form of odd dimension - say 3, then $\omega  \wedge \omega = 0$. This is the attempt to let Cadbra solve this wedge form.


Nakahara eqn 5.67a

Cadabra code:

-----------------

{a,b,c,l,m,n}::Indices.

{e^{a}, \omega^{a}_{b}}::DifferentialForm(degree=3);

$\displaystyle{}\text{Attached property DifferentialForm to }\left[e^{a}, \omega^{a}\,_{b}\right].$

eq1 := \omega^{a}_{b} ^ \omega^{a}_{b};

0

Nice! Solves this. Test 2: Now we want to show explicitly - that is using numeric indices for $q,r$ the following expression:

$\eta \wedge \nu = (-1)^{qr} \nu \wedge \eta$.

Cadabra code:

-----------------

def post_process(ex):

sort_product(ex)

canonicalise(ex)

collect_terms(ex)

{ \eta^{a}_{b}}::DifferentialForm(degree=3);

{ \nu^{a}_{b}}::DifferentialForm(degree=5);

$\displaystyle{}\text{Attached property DifferentialForm to }\eta^{a}\,_{b}.$

$\displaystyle{}\text{Attached property DifferentialForm to }\nu^{a}\,_{b}.$

eq2 := \eta^{a}_{b} ^ \nu^{a}_{b}; eq3 := \nu^{a}_{b} ^ \eta^{a}_{b};

$\displaystyle{}\eta^{a}\,_{b}\wedge \nu^{a}\,_{b}$

\eta^{a}_{b} ^ \nu^{a}_{b}

$\displaystyle{}-\eta^{a}\,_{b}\wedge \nu^{a}\,_{b}$

-\eta^{a}_{b} ^ \nu^{a}_{b}

Cadabra code:

---------------

eq2 + eq3;

combine(_);

$\displaystyle{}\eta^{a}\,_{b}\wedge \nu^{a}\,_{b}-\eta^{a}\,_{b}\wedge \nu^{a}\,_{b}$
\eta^{a}_{b} ^ \nu^{a}_{b}-\eta^{a}_{b} ^ \nu^{a}_{b}

$\displaystyle{}0$

0





Exercise 5.15: Let $\xi \in \Omega^{q}(M)$ and $\omega \in \Omega^{r}(M)$.

Show that $d(\xi \wedge \omega) = d\xi \wedge \omega + (-1)^{qr} \xi \wedge d\omega$.

For simplicity, we shall set $q=3$ and $r=5$ - thus inducing a negative in the expression.
Using the following link is nice: https://cadabra.science/notebooks/exterior.html

Cadabra code:

-----------------

\xi::DifferentialForm(degree=3);

\omega::DifferentialForm(degree=5);

$\displaystyle{}\text{Attached property DifferentialForm to }\xi.$
$\displaystyle{}\text{Attached property DifferentialForm to }\omega.$


Add definition of exterior derivative.

Cadabra code:

-----------------

d{#}::ExteriorDerivative;

d{#}::LaTeXForm("{\rm d}").


$\displaystyle{}\text{Attached property ExteriorDerivative to }d{\#}.$
ext1 := d{ \xi ^ \omega };
$\displaystyle{}{\rm d}\left(\xi\wedge \omega\right)$

Cadabra code:

-----------------

d(\xi ^ \omega)
product_rule(_);


$\displaystyle{}{\rm d}{\xi}\wedge \omega-\xi\wedge {\rm d}{\omega}$
d(\xi) ^ \omega-\xi ^ d(\omega)


This demonstrates equation $5.69$ in Nakahra for even and odd indices.