Note: this answer was migrated from physics.SE, so I've had to re-render the formulas as images. The answer is also specific to learning physics, though it might work for other fields too.
This looks like a classic XY problem. I don't know of any practicing physicists that use a special memorization scheme 'without understanding', and I think such a thing would be counterproductive. Here are a few tips that might help instead.
1. Use conceptual handles
Here are formulas for the speed of sound in a gas, the speed of waves on a string, and the frequencies of oscillation of a mass on a spring and a physical pendulum.
The intuition for all of them is the same: the numerator is some measure of a restoring force, whether that's how hard the gas pushes back, how hard the rope pulls, the strength of the spring, or the torque of gravity. The denominator is always some measure of inertia, proportional to the mass of the system. Thanks to this intuition I don't have to remember anything, except that a square root is involved.
As another example, consider all the annoying conversions between wave quantities,
along with many others. To remember the conversions between (ω,k) and (T,λ), I just use
which is the fundamental definition of the wavenumber four-vector. This fixes the factors of 2π, since that's the change in phase of one cycle. There's no place all these little tricks are written down; everybody has their own and it's best if you make them yourself as you go.
2. Don't worry about judgment
I was just watching a summer school lecture where a renowned theoretical physicist took a good 30 seconds to flip a fraction. This is completely typical and not embarrassing at all. Some people function better rearranging symbols in their heads and some function better using chalk or paper. Personally I can't do anything in my head; I have to use paper or write it out in the air, but I have never felt judged for doing this. If your colleagues are being judgmental, they are being rude and you shouldn’t let them get you down.
When I see somebody able to remember or rederive something much faster than me, I often ask them what their conceptual handles are. Unless the person is exceptionally rude, they’re typically happy to explain. (This is especially true in physics, where mere memorization is uncool.)
3. Chunk concepts, separately from tools
If you're having trouble writing a grant proposal, the solution is not to memorize the exact sequence of muscle activations needed to write every individual letter. Similarly, if physics feels too 'big', the very worst thing you can do is to make it even bigger, by unpacking every equation into eight separate equations. Learning is instead done by chunking things together.
For example, take the derivation of the wave equation, which is a full two pages of math written out. You don't want to store every line in your head. Instead, you just want to store a general intuition that "curvature means a restoring force because strings under tension straighten out", which gives you the ∂²y/∂x² term. To get from that to the final result you need to know Newton's second law (giving the ∂²y/∂t² term), the small angle approximation, and the binomial approximation. But none of these are specific to the wave equation -- they're just general tools.
As another example, I was lost when first exposed to tensor notation. It looked like there was an enormous amount of stuff to memorize! But it faded away once I sat down and wrote out all the allowed manipulations. It turns out there aren't that many, ten common ones at most. All tensor calculations up to graduate level are just using the same ten steps over and over again, so in a technical sense it's actually easier than high school algebra, which has many more allowed manipulations. Once you have this understanding, many derivations get shorter; they get chunked into "use the standard steps, plus this one trick in the middle".
Then all you have to do is remember the trick, ideally with a conceptual handle.
4. Construct your own understanding
As I emphasized above, the best way to get this kind of understanding is to construct it yourself! There's no shame in revisiting a subject that's "basic" and rebuilding it yourself from the ground up; I've done this with calculus and mechanics several times whenever I felt I was getting rusty. Make a formula sheet, or if you think visually, try drawing a mind map or a dependency diagram. Challenge yourself to rederive key equations without a reference. If you do this a lot, you'll naturally construct the necessary conceptual handles and chunks, and get better at recognizing which tool to use.