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Hello, this will be my first post on the physics forum, so i wanted to make it decent :P
I've been trying recently to derive for myself a path integral formulation (not quantum mechanical or anything feynman like but for finding the length of a curve on a given interval). Heres my attempt at deriving it for a simple curve, but its a very time consuming tedius process to actually calculate. I know real path integrals involve Reinman sums, but after trying to learn them on my own, I decided to give it a shot on my own, here goes:
f(x) = x^2, from x1 = 3 to x2 = 4 using a small interval dx = 0.01
distance between two points on a function is defined as:
d = √(x2  x1)^2 + (f(x2)  f(x1))^2
so breaking the curve down into infentesibly small portions, (or in this case our small interval dx), and adding the distances together, we should get a reasonably good approximation for the distance on the curve... right?
so in this case, between x1 = 3 and x2 = 4, we have several peices of the curve, ex, the distance between 3  3.01, 3.01  3.02, 3.02  3.03.........3.99  4.0. i've worked it out so that the number of these peices is defined by (x2  x1)*(1/dx). so in this case we have:
1*100 , so 100 individual peices we have to sum up. now this is where i've run into some problems. i've defined the sum as:
Ʃ(i = 1 to 100) = √((3 + i*dx)  (3 + (i1)*dx))^2 + (f(3 + i*dx)  f(3 + (i1)*dx))^2
which gives a reasonable approximation. now my problem is, i want to evaluate this sum as dx tends to zero, so we would have an exact answer. so we could write this as:
lim(dx > 0) Ʃ(i = 1 to ∞) = √((3 + i*dx)  (3 + (i1)*dx))^2 + (f(3 + i*dx)  f(3 + (i1)*dx))^2
but i am stuck here, i feel like i should take an integral, but i don't know how and that probably isn't correct (or even doable). which leads me to my question, what now?
thank you very much
EDIT: using 0.01 i got ~6.94, and continuing to decrease it it seems to converge at ~7.074
I've been trying recently to derive for myself a path integral formulation (not quantum mechanical or anything feynman like but for finding the length of a curve on a given interval). Heres my attempt at deriving it for a simple curve, but its a very time consuming tedius process to actually calculate. I know real path integrals involve Reinman sums, but after trying to learn them on my own, I decided to give it a shot on my own, here goes:
f(x) = x^2, from x1 = 3 to x2 = 4 using a small interval dx = 0.01
distance between two points on a function is defined as:
d = √(x2  x1)^2 + (f(x2)  f(x1))^2
so breaking the curve down into infentesibly small portions, (or in this case our small interval dx), and adding the distances together, we should get a reasonably good approximation for the distance on the curve... right?
so in this case, between x1 = 3 and x2 = 4, we have several peices of the curve, ex, the distance between 3  3.01, 3.01  3.02, 3.02  3.03.........3.99  4.0. i've worked it out so that the number of these peices is defined by (x2  x1)*(1/dx). so in this case we have:
1*100 , so 100 individual peices we have to sum up. now this is where i've run into some problems. i've defined the sum as:
Ʃ(i = 1 to 100) = √((3 + i*dx)  (3 + (i1)*dx))^2 + (f(3 + i*dx)  f(3 + (i1)*dx))^2
which gives a reasonable approximation. now my problem is, i want to evaluate this sum as dx tends to zero, so we would have an exact answer. so we could write this as:
lim(dx > 0) Ʃ(i = 1 to ∞) = √((3 + i*dx)  (3 + (i1)*dx))^2 + (f(3 + i*dx)  f(3 + (i1)*dx))^2
but i am stuck here, i feel like i should take an integral, but i don't know how and that probably isn't correct (or even doable). which leads me to my question, what now?
thank you very much
EDIT: using 0.01 i got ~6.94, and continuing to decrease it it seems to converge at ~7.074
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