Optimizing Work: Finding the Minimum Path in Non-Conservative Vector Fields

[goffinj]

I was wondering, if you have a non-conservative vector field (so that the line integral of each path from point A to point B isn't the same) that represents some sort of force, then is there a method to find the path that requires the least amount of work from a designated point A to point B? Even if you can't calculate it exactly is there a numerical approximation for such a path? Thanks for any help or suggestions.

 

[lugita15]

The technique you would use to solve this kind of problem is the Calculus of Variations. You just need to determine a Lagrangian and then solve the Euler Lagrange equations.

 

[Fraggler]

What Lagrangian would you set up?

We want to find the path that minimizes W = integral F dot dr. How do you solve for a path?

I apologize for the sloppy notation; this is my first post and I'm not familiar with tricks to get the integral and dot product signs to show up.

 

[HallsofIvy]

Just take your "Lagrangian" to be "1". That way minimizing the value will minimize the path length.

 

[lpetrich]

There's an absurdly simple way to solve this problem.

For position x, vector field v is given by v(x). Solve

dx/dt = v(x)

where t is a parameter: x = x(t).

It's a system of ordinary differential equations, but such equations should be easy to solve numerically. The main problem is keeping the ODE-solver step size in a reasonable range.

 

[elibj123]

The path that takes the least energy, is the path that an actual particle will travel along (a result of minimum action principle) which is the path that abides Newton's Second Law, so it will solve the ODE:

\frac{d^{2}\vec{x}}{dt^{2}}=\vec{F}(\vec{x})

(F being the vector field)