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## Main Question or Discussion Point

**momentum, position vector dot (scalar) product "action"**

Hello,

I was playing with single mass point classical mechanics, when I realized that the dot product of the position vector and momentum vector, p.r , has action dimension. Furthermore, its time derivative, d/dt(p.r) = F.r + p.v, has dimension (and looks like) some kind of "Lagrangian" (p.v suggests double of cinetic energy). I tried to look around the web if I could find some info about this (in classical mechanics), but without success.

What could be the trouble with this "action" / "lagrangian"?

Why nobody mentions this "action" or the quantity p.r, even if only to discard it?

(I also find the quantity F.r interessting, but can't attribute it a general meaning, it suggests to be somekind of double "minus potential energy" ("-2U"), but for potential forces it generally "deforms" the intial potential U by the operator -x∂U/∂x-y∂U/∂y = F.r = "-2Udeform", which keeps invariant the harmonic oscillator potential (in 2-D: U(x,y)= 1/2(x^2+y^2)), but that is a special case, generally we'll get a different "deformed" potential "Udeform" from starting potential U (e.g. central gravitational potential in 2-D Ug(x,y) = -1/(x^2+y^2)^1/2 leads to "Udeform"= -1/2Ug.)

Thank you

I wish a pleasant day