Lie derivative of a function (of a point) with respect to a vector field

Question

Say $f$ is a function of point on $M$, we define $L_Xf$ to be $\lim_{h\rightarrow 0} \frac{f(\phi_h(p))-f(p)}{h}$, where $\phi_h(p)$ is like (but is not) '$p+h$': moving $p$ in manifold $M$ for a displacement 'proportional' to $h$ along vector field $X$. It's similar to differential $y'(x)$ of $y(x)$, except that at the same time we fix the path along which $h$ 'travels' to 0.

With such definition, we bypass the difficulty of defining differential of a function of a point on $M$, namely we can't divide the change of dependence variable by the difference $p-p'$ between two points $p, p'$ on a neighbourhood; the latter, when defined, will often approximate a vector which is not divisible, here it seems with $\phi_t$ we change the 'vector' $p-p'$ to a scalar.

1. Is my intuiitive understanding of Lie derivative correct?
2. What's the motivation behind such definition of differential? It seems some concepts in differential geometry originates from physics, is there any physical context here as well?

(BTW, compared with another way of defining derivative, where we simply eliminate the 'division': we define $df$ as a map from spaces of tangenct vectors (which locally approximate $p-p'$) of $f$'s domain to spaces of tangent vectors of $f$'s image.)

Answer 1

To address your questions:

1. Yes, I'd say your intuitive understanding is reasonable. $L_X f$ is the rate at which $f$ changes along the integral curves of $X$.

2. One answer to your question might be the explanation you gave in your post. It does not make sense to subtract two points on a manifold ($p - p'$), so this definition of Lie derivative gets around that issue.

Also, in question #2, you mentioned that maybe there is a physics interpretation. I can think of one that makes sense. Often the time-evolution of a physical system is described by differential equations. If a moving object has position coordinates $(x_1,x_2,\dots,x_n)$ and its motion is described by differential equations of the form $\frac{dx_i}{dt} = f_i(x_1,\dots,x_n)$, then the path of motion is an integral curve of the vector field whose coordinates are the $f_i$'s. You might be interested in how some scalar quantity (for example temperature or something) changes as the object moves. If that quantity is desribed by a scalar function $g(x_1,\dots,x_n)$, then the Lie derivative $L_X g$ tells you the rate at which that quantity changes as the object moves around according to the differential equation.