Force Field to Motion Equation

[champion19]

In my mechanics class, we have discussed basics dynamics relationships and equilibria aspects, and we recently considered Newton's Law of Gravitation. In our discussion, we went through how to determine potential energy and used this to perform most of our calculations with respect to velocity at a position, and we discussed Kepler's Laws as well.

However, something which we did not discuss, and which intrigues me is how, given say, an arbitrary force field (such as gravitational) as a function of position $\vec{\mathbf{r}}$ and an initial starting position $\mathbf{r_0}$, how would we find the equation of motion?

(If it is extremely complicated in three dimensions, I would still be interested to see how it is done in one dimension)

 

[Simon Bridge]

You can't, you need two initial conditions to get a unique equation of motion.

Do you know the relationship between potential and force?
Do you know how to get an equation of motion from force?

 

[A.T.]

However, something which we did not discuss, and which intrigues me is how, given say, an arbitrary force field (such as gravitational) as a function of position r⃗ \vec{\mathbf{r}} and an initial starting position r0\mathbf{r_0}, how would we find the equation of motion?

Is the initial velocity zero? The force field gives you the acceleration, which integrated twice gives you the position change. But you need to know the integration constant for initial velocity.

 

[champion19]

Oh I hadn't thought about that but sure, it would be fine to assume that the initial velocity is zero. Yes we have discussed how to convert potential and force as well as how to convert acceleration or force to an equation of motion when it is given as a function of time. What is confusing me though is that in a field equation, it is given as a function of space, and to get the position function, you have to integrate it with respect to time.

 

[Simon Bridge]

Well sure, you end up with a second order differential equation, which you must solve.
$$-\vec\nabla \phi(\vec r) = m\ddot\vec r$$
Since you want $\vec r(t)$, you'll most likely be given $\phi$.
You won't find a general equation of motion though... just like normal really.

You may also want to have a look at "lagrangian mechanics".
Or look up college lectures under the heading "classical mechanics".