# When were fictitious forces introduced

Some problems in mechanics are simplified by considering the action of "fictitious forces". These appear in accelerated frames, such as circular movements. When were these first considered so? Did Newton himself deal with these types of problems?

• Is a non-inertial force the same as a fictitious force? – HDE 226868 Jan 17 '15 at 19:35
• @HDE226868 Yes, it is a force that arises due to being in a non-inertial frame. This is sometimes called a 'fictitious' force as well. – Danu Jan 17 '15 at 20:44
• These are usually associated with the name of d'Alembert, but it is always difficult to tell who was the first. – Alexandre Eremenko Jan 17 '15 at 20:48
• The term is "inertial forces", not "non-inertial forces". They appear in non-inertial frames because of the inertia effect postulated by Newton's first law. Also called fictitious forces, pseudo forces, and d'Alembert forces. – Conifold Jan 18 '15 at 2:30
• I've changed it to "fictitious" hoping to avoid more confusion. Thanks. – Mark Fantini Jan 18 '15 at 10:44

It depends on what you mean by "introduced". The centrifugal and the Coriolis forces are not the only inertial forces, or even the simplest ones. In classical mechanics inertial forces appear in any accelerating frame. The simplest one would be attached to a uniformly accelerated body.

In full generality inertial forces appear in D'Alembert's Traite de Dynamique of 1743, where he clarified what happens under Newtonian laws in non-inertial frames. Since any accelerating body is at rest in a frame attached to itself the sum total of all forces acting on it must be balanced by an additional force, which he called "the force of inertia". This is now known as the D'Alembert's principle, and Lagrange, who gave its more common variational formulation, quipped that "D'Alembert had reduced dynamics to statics by means of his principle". In recognition, inertial forces are occasionally called D'Alembert forces today. Traite de Dynamique was a landmark treatise also because it freed classical mechanics from geometric straightjacket of Newton's Principia, and systematized its basic notions ending terminological disputes of early 18th century.

The centrifugal force was already known in antiquity. There is a passage in Plutarch (1st century AD) describing the motion of a rock in a sling when it is rotated before the discharge, he even compares it to the motion of the Moon around the Earth. This passage is analyzed (perhaps overanalyzed) by Lucio Russo in The Forgotten Revolution, see also his paper, who suggests that the original source was likely Hipparchus (2nd century BC). He goes on to speculate about the law of inertia and "Hipparchian mechanics", but not very convincingly. Most likely, this was still understood in the Aristotelian context, where forces were proportional to induced speeds. Medieval impetus under some interpretations can also be construed as a version of a linear inertial force for a decelerating frame in a resisting medium. For example, Avicenna's self-degrading "impressed force" for a flying arrow almost matches a modern description if we relate it to such a frame.

In the 17th century the centrifugal force was more or less commonly accepted after Descartes gave it a prominent place in his physics, curiously using the same example of rock in a sling. But Huygens was first to derive a formula for it. Today ironically, essentially his very derivation is presented in modern textbooks as the derivation for the centripetal force instead. Newton himself originally used centrifugal force in that sense, but eventually switched (under Hooke's influence) to the centripetal reinterpretation more consistent with the absolute frame of Principia. But when Leibniz wrote the equation for the radial inertial force acting on a planet in the attached frame (with the inverse cube term) he pretended not to understand the context that he himself used before, and declared it wrong quoting the third law (out of place). More on this episode in Aiton's paper from Learn from the Masters.

I suppose linear inertial forces did not receive more attention because observers rarely found themselves in accelerating frames until recently, but overloads during take offs and nose dives are very real effects of these "fictitious" forces. Beyond linearly accelerating and rotating frames however expressions for the inertial forces are typically too cumbersome to be useful. The linear inertial force received more attention after Einstein proposed his famous elevator thought experiment. Ironically again, Einstein's conslusion, the equivalence principle, effectively eliminates it from relativistic physics, because there is no physical way to distinguish it from gravity.

I've found some pretty old usages:

• Coriolis force: From here:

The Coriolis deflection of moving objects seen from within a rotating frame of reference—important in physics, meteorology, and oceanography—was described by Italian scientists Giovanni Battista Riccioli (1598–1671) and his assistant Francesco Maria Grimaldi (1618–63) nearly two centuries before Gaspard-Gustave Coriolis (1792–1843).

Riccioli and Grimaldi give a detailed description in Riccioli’s 1651 Almagestum Novum of how Earth’s rotation should cause a rightward deflection in a projectile fired toward the north.

• Centrifugal force: I can't find a non-Wikipedia citation that doesn't end in a "404" link that is pre-Newton, but I did find this, a translation of Newton's Principia:

The preceding Proposition may be likewise demonstrated after this manner. In any circle suppose a polygon to be inscribed of any number of sides. And if a body, moved with a given velocity along the sides of the polygon, is reflected from the circle at the several angular points, the force, with which at every reflection it strikes the circle, will be as its velocity: and therefore the sum of the forces, in a given time, will be as that velocity and the number of reflections conjunctly ; that is (if the species of the polygon be given), as the length described in that given time, and in creased or diminished in the ratio of the same length to the radius of the circle ; that is, as the square of that length applied to the radius ; and therefore the polygon, by having its sides diminished in infinitum, coincides with the circle, as the square of the arc described in a given time applied to the radius. This is the centrifugal force, with which the body impels the circle ; and to which the contrary force, wherewith the circle continually repels the body towards the centre, is equal.

But above that - aha! It might have been Huygens!

And by such propositions, Mr. Huygens, in his excellent book De Horologio Oscillatorio, has compared the force of gravity with the centrifugal forces of revolving bodies.

I'm not positive (in the latter case) precisely who was first, though. But it wasn't Newton.

Ingoli's letter to Galileo and Galileo's reply give a pretty clear snapshot of one point in the very slow development of this idea over the centuries. See Graney, http://arxiv.org/abs/1211.4244 , esp. pp. 29ff. Mach's critique in The Science of Mechanics shows that as late as 1883, nobody had really thoroughly addressed the logical problems with defining the distinction between inertial and noninertial frames. Einstein didn't even get it quite right; modern physicists don't agree with his interpretation of GR as a theory of accelerated frames. In answer to the question, "When were fictitious forces introduced," I think they were partially understood as early as 1616, and fully understood as late as ca. 1960, when relativists developed what is now considered the correct interpretation of GR. In fact, if you ask a relativist today whether gravity is a fictitious force, you will probably get a nuanced answer. Some might say it's not a force, some might say it's a fictitious force, and still others might say that it's hard to say, because the equivalence principle resists attempts at precise formulation (see Sotiriou et al, "Theory of gravitation theories: a no-progress report," http://arxiv.org/abs/0707.2748 ).

• Interesting, didn't know Einstein's interpretation is controversial. However, inertial arguments in Ingoli's letter, perhaps less sophisticated, like we'd be thrown off if Earth rotated, date to much earlier. At least to Ptolemy and possibly to Aristotle. And to clarify, 1616 is the date of the latter, and the relevant part is Ch.5. – Conifold Jan 21 '15 at 1:29
• @BenCrowell may you elaborate/cite what you mean: "1960, when relativists developed what is now considered the correct interpretation of GR." What do you mean by "correct"? – N. Steinle Dec 31 '18 at 4:28