Is pseudo force just an ad hoc number to explain motion in non-inertial frames?

Question

Consider an observer in a non-inertial frame $S$ who observes a particle's motion with a relative acceleration $\vec a_s$ and further calculates (or was told by his fellow observer in an inertial frame) the net real forces acting on it as $\vec F$.

Now the observer adds a pseudo force $\vec f_s$ to the net real forces acting on the particle to explain relative acceleration $\vec a_s$.

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Question:

Is what I described above the "proper" method to calculate pseudo forces?

In other words, is a pseudo force just an arbitrary constant that arose out of our desperation to explain non-inertial motion?

Answer 1

Is what I described above the "proper" method to calculate pseudo forces?

I wouldn't say that it is a method to calculate the inertial forces, but it is the result that methods for calculating inertial forces are designed to achieve. I.e. I would treat it more as a goal than as a method.

In other words, is a pseudo force just an arbitrary constant that arose out of our desperation to explain non-inertial motion?

It is not generally constant and "desperation" is an unnecessarily emotional word, but otherwise, yes.

Consider a general reference frame, either inertial or non-inertial. If you write the Lagrangian of a free particle in that reference frame, and solve the Euler equations then you will get an expression for the acceleration of a free particle in that frame. Multiply that expression by $m$ and you have the inertial forces in that frame. That is the general and systematic approach for discovering such forces.

Be aware that if your generalized coordinates include non-Cartesian spatial coordinates then some of the terms from the Lagrangian will address the spatial coordinates and not just the acceleration. Some authors do not consider those to be fictitious forces, but if you take that approach then you must identify those terms "by hand" which I don't prefer.

For example, consider the Lagrangian of a free particle in a Cartesian frame that is rotating about the $z$ axis at an angular velocity of $\Omega$. The Lagrangian is $$L = \frac{1}{2}m \dot{\vec r}^2+ m \dot{\vec r} \cdot (\vec \omega \times \vec r) + \frac{1}{2} m ( \vec \omega \times \vec r)^2$$ where $\vec r=(x,y,z)$ is the position in the rotating frame and $\vec \omega = (0,0,\Omega)$.

Solving the Euler Lagrange equation we get: $$\ddot x = \Omega^2 x + 2 \Omega \dot y$$ $$\ddot y = \Omega^2 y - 2 \Omega \dot x$$ $$\ddot z = 0$$ and we immediately recognize the first terms (multiplied by $m$, of course) as being the centrifugal force and the second terms as being the Coriolis force. The same approach can be taken for other frames.

Answer 2

In other words, is pseudo force just an arbitrary constant that arose out of our desperation to explain non-inertial motion?

Suppose we have a motion happening somewhere in space, now we can either observe it with a frame attached to the person in motion or one which is outside and is inertial. According to newton's laws, the acceleration measured by the person outside is given as:

$$ F_{net} = ma \tag{1}$$

Now, let's see the frame for the guy with accelerating frame, of course, it is naive to apply newton's laws here as they are only stated for inertial frames. However, there is a 'fix' by introducing a new function $G$ such that sum of forces minus the 'perceived' acceleration by a particle in the frame.

$$ F_{net} -ma'=G \tag{2}$$

Now, this is cool and all but how would we calculate G? Well that's easy, you see equation (1) and plug that information into (2):

$$ m(a-a') = G$$

Now what is $a-a'$ physically? This means the difference between acceleration as measured from an inertial frame and the acceleration measured in the accelerating frame of reference.

Here is a famous example, suppose you are in a car accelerating to the right and there is a pendulum attached to the roof, let's say there was someone standing on the ground outside and looking at the car. The discussion on this example is excellently done by Bob D in this post and linked.

Answer 3

Inertial forces aren't desperate attempts to explain motion in non-inertial frame, but rather correct explanation of motion in non-intertial frames. In physics, everything is about frames of reference.

For observer at inertial frame looking at someone at merry go round, centrifugal force doesn't exist, however this force is real for someone on merry go round since from his perspective it is imposible to correctly explain its motion (equation of motion) without it.

There is no absolute frame of reference in physics. Electromagnetism manifests itself as electric or magnetic force in different reference frames. Theory of relativity is fundamentally derived from differences which arise when the same phenomenon is observed in different frames of reference only here one observer moves relatively to another at speeds close to light speed.

Sometimes physicists say that inertial forces are pseudo forces because they don't exist in all frames of reference while some other forces do (like centripetal force).