Commonly, in textbooks, differential of differential increment vanishes, good example is how Euler-Lagrange equations are derived using the differential (from step 2 to step 3 here, you can see
d[q(t) + e*eta(t)] becomes
eta(t) is infinitesimal change). And it makes sense because such small changes are considered negligible, and that is essential to define derivative rules (to see why I recommend this video at 3:00).
I would like to replicate similar behaviour in Mathematica to check correctness of certain derivations I made. Without it vanishing, my expressions will grow and not exlude negligeable terms.
I used the DifferentialD operator to represent differential increment, however Mathematica doesn't seem to replicate such behaviour.
I started with the following
DifferentialD[x + DifferentialD[y]]
I also tried to check if following expression be equal to zero but it wasn't the case.
Could someone more experienced with Mathematica (or symbolic computation in general) comment on that?
It appears that problem of how to handle operations on infinitesimal quantities is more deep than just simple algebra. For reference I'd like to share this thread, it has a lot of good references and explanations.