Not quite what you asked, but I found this video visualizing Lagrange multipliers quite helpful. Plausibly it’ll help clarify Euler–Lagrange as well.
Your units question is easy: you get the same dimensionless quantities whatever units you choose. Instead of thinking of units as dimensions, I’d think of them as basis vectors in a five-dimensional space (length, mass, time, current, temperature[1]): you should have exactly five of them, and they need to be “linearly” independent, but beyond that, you can choose any set you like: you could instead have something like the natural units of (speed, gravitationality[2], angular momentum, entropy, charge). In this conception, this 5D space is fundamental: you need at least five dimensions (the ones spanned by c, G, h, k and e), and if you want more, you need to find some new as-yet-unknown independent dimension (maybe baryon/lepton number counts?).
Of the seven SI units, the other two are Candela, which is some “human visual perception” bullshit masquerading as fundamental, and amount of stuff/Avogadro’s constant, which I don’t think meaningfully constitutes a “dimension.”
Not quite what you asked, but I found this video visualizing Lagrange multipliers quite helpful. Plausibly it’ll help clarify Euler–Lagrange as well.
Your units question is easy: you get the same dimensionless quantities whatever units you choose. Instead of thinking of units as dimensions, I’d think of them as basis vectors in a five-dimensional space (length, mass, time, current, temperature[1]): you should have exactly five of them, and they need to be “linearly” independent, but beyond that, you can choose any set you like: you could instead have something like the natural units of (speed, gravitationality[2], angular momentum, entropy, charge). In this conception, this 5D space is fundamental: you need at least five dimensions (the ones spanned by c, G, h, k and e), and if you want more, you need to find some new as-yet-unknown independent dimension (maybe baryon/lepton number counts?).
Of the seven SI units, the other two are Candela, which is some “human visual perception” bullshit masquerading as fundamental, and amount of stuff/Avogadro’s constant, which I don’t think meaningfully constitutes a “dimension.”
ykwim: something with the units of G. If you don’t like this, just choose mass instead.