d’Alembert’s Paradox

1. A solid body in a steadily moving, inviscid (and incompressible) fluid does experience no drag. That’s fine! However, a drag must appear when, by reversing the process, we observe the steadily moving body through the fluid at rest. Otherwise, who pays for irreducible non-steady motions which take place locally with the passage of the body? How can these two facts be brought together?
2. The explanation deals with an irreversible process: while a steadily moving fluid spontaneously embeds a body at rest, the reverse requires an external source of work (per unit time). This because, as shown by bernoulli613, the boundary of the body at rest becomes also a source of vorticity for the moving fluid. Can we say alike for Faraday’s Paradox on electromagnetic induction?

Inertia and Irreversibility

1. It’s usually held that Mechanics belongs to the Reversibility domain and the time-symmetry of dynamics’ second law would confirm this belief. Of course, it is maintained, the solution exhibited at the time “t” must show a reversible process. Otherwise the solution at time “-t” would cause the actual inversion of an irreversible process, which cannot happen in nature. In other words, the fact that both solutions are observable would confirm beyond any doubt Mechanics’ Reversibility.
2. Unfortunately, the argument misses the point that solution “-t” might not violate any natural law simply because it exhibits the same irreversible process of solution “t”. As such, both solutions are again observable. An example will clarify that this is indeed the case.
3. In a 1D elastic collision the two bodies move, say, from left to right. In this “t” solution we always observe the faster than average body slowing down and the slower than average one accelerating. This is an irreversible process in which kinetic energy flows spontaneously (since our system is isolated) from the faster body to the slower one, just like heat flows spontaneously from a hotter body to a colder one in contact. And the same irreversible process occurs when we observe the “-t” solution: now our colliding bodies will be moving from right to left, and it’s still true that the faster body slows down while the slower one accelerates.
4. We meet the same situation with inertial forces when the motion of a material point along its path is considered. In a frame aboard the material point we detect the same inertial force for both, “t” and “-t”, solutions as stated by the dynamics’ second law. Here the classic example is the elevator in free fall: the inertial force is directed upwards compensating for “gravitational effects” of bodies inside. The “-t” solution shows a raising elevator, ascending at less rate in order to have the same inertial force still directed upwards.
5. This is what, in my opinion, establishes a deep connection between Inertia and Irreversibility. By the way, the motion of an inertial frame would be reversible. But, does it really exist in nature a frame like that?

Newton’s First Law

Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.

Did you ever know that after this statement, reported with minor differences by all Mechanics textbooks as Principle of Inertia, Newton added the following three examples to better explain his concept of inertial motion? In his words:

Projectiles persevere in their motions, except insofar as they are retarded by the resistance of the air and are impelled downward by the force of gravity. A spinning hoop, which has parts that by their cohesion continually draw one another back from rectilinear motion, does not cease to rotate, except insofar as it is retarded by the air. And larger bodies – planets and comets – preserve for a longer time their progressive and their circular motions, which take place in spaces having less resistance.

Apparently these examples, interpreted as casting doubts rather than light on the initial statement, were dropped by commentators and authors, including those engaged in a chauvinistic campaign against Galileo’s “circular inertia”. I.B. Cohen in the Guide to Newton’s Principia which precedes his new translation (above quote is taken from it) could not avoid the matter. He wrote :

On first encounter, these examples may seem confusing since each involves curved paths and yet the subject of the first law is uniform linear (or rectilinear) motion.

and, by mentioning the first two examples only, vaguely concludes that

… it is only the tangential or linear component that is inertial, not the curved motion.

Also E. Mach deemed convenient to skip the matter. Under this condition, did we really understand Newton’s First Law?

Newton’s Letter

Did Einstein ever read what Newton wrote to R. Bentley on February 1693? Let’s hear it again.
“… It is inconceivable that inanimate brute matter should, without the mediation of something else, which is not material, operate upon and affect other matter without mutual contact, as it must be, if gravitation in the sense of Epicurus, be essential and inherent in it. And this is one reason why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it. Gravity must be caused by an agent acting constantly according to certain laws; but whether this agent be material or immaterial, I have left to the consideration of my readers. …”

(Isaac Newton: Philosophical Writings, CUP 2004, A. Janiak ed., p. 102)

MyEpigrams!

Science, too, has its own modern terrorists.

Darwinism: bodies fall down because the evolution eliminated those going up.

Sex is a male obsession. Usually females respond on demand.

Bastard!

It’s amazing how stupid are intelligent people sometimes.

Bad manners. Now Computing like Statistical Mechanics which started as an aid and ended by questioning Physics.

When something bears a proper name, be sure its discovery belongs to someone else.

Do you really think science was made from scratch in Cambridge at the end of 17th century?

Theory of Relativity depicts a world without Inertia.

At the end, nothing seems to you more universal than what is always with you.

Mozart: just a gift to all of us.

“Physical Mathematics”, unlike Mathematical Physics, is easy and quick until a negative test. Then you must restart, confused, from the very beginning.