Finding the equation of a straight line in 3 dimensions.

seeingstars63
Messages
8
Reaction score
0

Homework Statement


Prove that the shortest path between two points in three dimensions is a straight line. Write the path in the parametric form:

x=x(u) y=y(u) z=z(u)

and then use the 3 Euler-Lagrange equations corresponding to ∂f/∂x=(d/du)∂f/∂y'.


Homework Equations


Stated them above:]


The Attempt at a Solution


I found all of the answers in relation to the Euler-Lagrange equations, but I am not sure where to go from there. For each coordinate, ∂f/∂x,∂f/∂y,∂f/∂z, they all equal 0 so that means that d/du(∂f/∂x,y,z) are all also zero. As a result, I get constants for each and hence don't know how to implement these constants into a straight line equation.

The constants are :
∂L/∂x'=x'(u)/(x'(u)^2 +y'(u)^2 +z'(u)^2)^(1/2)=C_1
∂L/∂y'=y'(u)/(x'(u)^2 +y'(u)^2 +z'(u)^2)^(1/2)=C_2
∂L/∂z'=z'(u)/(x'(u)^2 +y'(u)^2 +z'(u)^2)^(1/2)=C_3
 
Physics news on Phys.org
seeingstars63 said:
The constants are :
∂L/∂x'=x'(u)/(x'(u)^2 +y'(u)^2 +z'(u)^2)^(1/2)=C_1
∂L/∂y'=y'(u)/(x'(u)^2 +y'(u)^2 +z'(u)^2)^(1/2)=C_2
∂L/∂z'=z'(u)/(x'(u)^2 +y'(u)^2 +z'(u)^2)^(1/2)=C_3

What's the equation for a straight line? Can you find for example dx/dy from these equations?
 
Thanks for the reply, clamtrox. I get what the equation of a straight line is: y=mx+b, but I'm not sure what you mean for finding dx/dy from those equations. There is also a dz.
 
seeingstars63 said:
Thanks for the reply, clamtrox. I get what the equation of a straight line is: y=mx+b, but I'm not sure what you mean for finding dx/dy from those equations. There is also a dz.

Yes, but perhaps a more useful way to write that equation is y = \frac{dy}{dx}x + y(0).
 
seeingstars63 said:
Thanks for the reply, clamtrox. I get what the equation of a straight line is: y=mx+b, but I'm not sure what you mean for finding dx/dy from those equations. There is also a dz.

Yes sorry, that was bad notation. I of course mean partial derivatives: ∂x/∂y = x'(u)/y'(u)
 
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?
Back
Top