Equations are of central importance in all of science and engineering, but especially so in mechanics.
Even leaving aside algebraic equations, handbooks on PDEs alone list hundreds of equations. However, a few of these do stand out, either because they encapsulate some fundamental aspect of physics/science/engg., or because they serve as simpler prototypes for more complex situtations, or simply because they are so complex as to be fascinating by themselves. There might be other considerations too... But the fact is, some equations really do stand out as compared to others.
If so, what equations would you single out as being most important or interesting? To make the matters more interesting, first, please think of making a short list of only 5 equations. Then, if necessary, make it one of 10 equations---but no more, please! :)
As to me, here is my list, put together in a completely off-hand manner:
Top five:
(1-3) The linear wave-, diffusion- and potential-equations.
(4) The Schrodinger equation
(5) The Navier-Stokes equation
Additionally, perhaps, these equations:
(6) The Maxwell Equations
(7) The equation defining the Fourier transform
(8) Newton's second law (dp/dt = F)
(9) The Lame equation (of elasticity)
Am I already nearing the limit or what... Hmm... But, nope, I am not sure whether I want to include E = mc^2. ... I will give this entire matter a second thought some time later on.
But, how about you? What would be your choices for the top 5/10 equations? Why?
--Ajit
PS: Also posted today in the Computational Scientists & Engineers group at LinkedIn, and also will post soon at my personal blog.
[E&OE]
Last time, when I saw one blog of Biswajit Banerjee, I thought it would be much more convenient if we can input some equations in iMechanica for discussion. When I make discussion about mechanics with colleagues, I would like to write down some equations on the blackboard or paper. For discussion based on the web blog, LaTeX equations will be helpful. And it would be better if one can input equations directly instead of uploading one picture after another.
To be honest, I have no idea how to input equations directly in iMechanica (maybe there is already some ways). But I found some simple ways to input LaTeX equations in my MSN Space for making some notes. The Basic idea is to use the functions provided by WordPress.com .
Here is how (probably most of you knew already).
(1) Add a picture
Try to insert one picture in the place where you want to add your equation. Use the editor or just modify the HTML file of your poster.
(2) Edit your link
The link of the inserted 'picture' should look like
http://l.wordpress.com/latex.php?latex= Your LaTeX Code &bg=ffffff&fg=000000&s=0
Some extra modifications of the LaTeX code are needed since it is in the link:
- change space into '+'
- change '+' into '%2B'
For example, 
(http://l.wordpress.com/latex.php?latex= f=x%2By&bg=ffffff&fg=000000&s=0 )
The last part is the information of the color and size (s=0) of the equation. If one leaves nothing, the equation is set to the default setting.
So...done.
(3) Try something else
Original equation:
Colors:
Change the size of the equations:
s=1 
Slightly more complex equations:
......
It is not that difficult to input equations when blogging, at least for myself. This should also work when one wants to make comments.
Yixiang Gan
2007.11