Solving Work Along a Force Field Path - A Math Question

[marschmellow]

This might be more of a mathematical question, but the other day in Physics my teacher said that work along a curved path is a line integral, which made perfect sense to me. But then I wondered how one determines the path of travel if the force varies at each point x, y, and z. So how would you find the path of travel of a particle given a vector field of forces, an initial position and an initial velocity?

 

[Dale]

This is exactly what the Lagrangian formulation of classical mechanics is for. Basically you write the Lagrangian, solve the Euler-Lagrange equation, plug in your initial conditions and you have your answer.

 

[marschmellow]

This is exactly what the Lagrangian formulation of classical mechanics is for. Basically you write the Lagrangian, solve the Euler-Lagrange equation, plug in your initial conditions and you have your answer.

Okay good, that sounds hard. I'm glad the answer wasn't something really obvious, because I would be embarrassed for asking.

 

[dgOnPhys]

This might be more of a mathematical question, but the other day in Physics my teacher said that work along a curved path is a line integral, which made perfect sense to me. But then I wondered how one determines the path of travel if the force varies at each point x, y, and z. So how would you find the path of travel of a particle given a vector field of forces, an initial position and an initial velocity?

For a point-like (constant) mass all you need is to solve Newton's equation:
$\mathbf{F}(\frac{d \mathbf{r}}{dt},\mathbf{r},t)=m\frac{d^2 \mathbf{r}}{dt^2}$
which is a system of 3 differential equations, the unknown is $\mathbf{r}$, the 'path of travel' (parametrized by time)

 

[Idoubt]

This maybe slightly off base with what you are talking about but incidentally, if your force field is conservative ( ie the force at any point depends on a function of position, like gravitational and electrostatic forces do ) then the line integral will be a constant for any path you choose and will depend only on your initial and final points.