Newton’s Second Law in Spherical Coordinates

Newton’s Second Law gives a relationship between the total force an object and that object’s acceleration. I’m going to write this equation as I would in a classical mechanics course — I mean, let’s be real for second. If you are using spherical coordinates then you aren’t going to be doing the simple version of Newton’s 2nd Law.

Where the acceleration is the second derivative of the position vector (r) with respect to time. In cartesian coordinates, we can write this position vector as:

Where x-hat and the other “hats” are the unit vectors in the x, y, and z directions. Normal stuff. But what if I take the first derivative of r? Let’s do that.

There are some important things here. First, it’s common to write a time derivative with the dot-notation. So, x-dot is the first derivative of x with respect tot time. Second, notice that I have a product of x and x-hat. That means I have to use the product rule and take the derivative of both terms (that’s why we get an x*d-x-hat/dt term). But the unit vectors don’t change with time. They are constant such that the time derivatives are zero. That makes much nicer.

Now we can take the derivative again to get the acceleration.