What is D'alemberts principle?

Question

How does it relate to newtons laws? specifically f=ma?

Answer 1

d'Alembert's principle, also known as the Lagrange-d'Alembert principle, is a statement of the fundamental classical laws of motion. It is equivalent to Newton's second law. It is named after its discoverer, the French physicist Jean le Rond d'Alembert.

The principle states that the sum of the differences between the generalized forces acting on a system and the time derivative of the generalized momenta of the system itself along an infinitesimal displacement compatible with the constraints of the system (a virtual displacement), is zero.

Answer 2

D'Alembert's Principle comes about most simply(###) by re-writing Newton's 2ND LAW, F_net = ma (here F_net is the net force operating on the system), in the following way:

F_net - ma = 0 [DP] (!)

This might just seem to be mathematically trivial, but it is actually quite profound.

What it says is that while a body or system is accelerating, it may still be viewed under many (most, all?) circumstances as "at rest" in an associated accelerating frame, PROVIDED that the REVERSED value of "ma", i.e. ' - ma ' is treated as one of the "forces" under which the system is "at rest" in the accelerated frame.

To use this idea effectively, it needs to be very clear that the "forces" in the DP consist of "real" forces, as employed in an INERTIAL FRAME in Newton's equations, PLUS "pseudo-forces" consisting of the reversed acceleration(s).

EXAMPLES OF THE USE OF D'ALEMBERT's PRINCIPLE:

1. Centrifugal Force. The use of the concept of "centrifugal force" is just one example of using a "pseudo-force" to solve a problem involving acceleration.

In inertial space, an object circling at a uniform speed is always accelerating inwards, perpendicularly to its path. That problem can be solved in inertial space by writing:

F = mv^2/r.

HERE, mv^2/r is indeed m times the acceleration in inertial spce v^2/r, and ' F ' is the real inward force needed to provide this inward acceleration, e.g. tension in a string, or force provided by some curved track etc.

The problem can also be solved by writing it as completely a "force balance" (which really means a "balance" between a "real" and a "pseudo-" force):

F - mv^2/r = 0

The ' F,' being a "real force" is just the same as before. But, because of the ' - ' sign, the "pseudo-force" points OUTWARD, not INWARD. Hence its name, "centrifugal force," which literally means "centre-fleeing force" (from the Latin, fugare, to flee).

One should NEVER make the mistake of thinking of "centrifugal force" as a "real" force. It only exists because YOU have chosen to work the problem in a certain conceptual framework, namely in an INWARDLY ACCELERATING FRAME.

Without this clear understanding of what one is doing, all manner of false conclusions can be reached. Believe me, in over 40 years of university teaching, AND on this Yahoo! service, I've seen all manner of confused mistakes made by people who SHOULD know better.

Here is a good (which means REALLY BAD!) example, the so-called Best Answer to the question "A car moving at 19 m/s drives over the top of a hill.....?' posed by 'taniesimmons.' You will find a VERY extended discussion there of the errors made, both in my answer and in the further "Comments" link.

2. AN ASTROPHYSICAL APPLICATION.

A shock wave may travel into a DENSER or a LESS DENSE medium. One case results in a relatively smooth, advancing shock front; the other involves a developing irregularity and then breaking up and mingling of the shock front with the material it is entraining. Which case does which, and why?:

Whichever case holds, the addition of new material to the travelling shock wave medium will DECELERATE it. So as it travels into any other medium, the shock wave is ACCELERATING backwards towards the source of the shock wave. If we now go into that decelerating frame and travel WITH the shock wave, that's like having a "pseudo-body-force" pointing OUTWARD from the source, just like a pseudo-gravity. Now, in a gravity field, light material overlying denser material is STABLE; but DENSER material overlying lighter material is UNSTABLE. (This is one of several instabilitiues associated with Kelvin's name.) And that is the basic explanation for what is often observed: a shock wave is STABLE if less dense material advances into denser material; but if DENSER material advances into lighter material, the shock-front is UNSTABLE.

### Comment on both the first point made in my answer and the shock wave application just described:

The latter example of the advancing shock wave is an instance that can be understood from what most dynamical applied mathematicians would properly call d'Alembert's Principle. The formulation referred to below, with its talk of generalised momenta and infinitesimal displacements, simply doesn't fit it. In fact this latter formulation, whatever the writer of the Wikipaedia article might say, is more properly called the "Principle of Virtual Work." That applies to problems rather idealised and therefore antiseptic compared to the messy business of understanding the asymmetries between stably or unstably advancing shock wave fronts!

3. A CORRECTION TO THE ANSWER FOLLOWING MINE:

While we're discussing this topic, both the Wikipaedia article AND the extract from it reproduced by the responder below are seriously deficient. They both assert the following:

"The principle states that the sum of the differences between the generalized forces acting on a system and the time derivative of the generalized momenta of the system itself along an infinitesimal displacement compatible with the constraints of the system (a virtual displacement), is zero."

The phrase "along an infinitesimal displacement" contained in the preceding paragraph of mumbo-jumbo is utterly meaningless, out of context. What has been forgotten, and SHOULD have been written, is that this formulation involves the fact that the GENERALISED SCALAR PRODUCT OF [the sum of ... ... momenta of the system] WITH A GENERALISED infinitesimal VECTOR displacement ... , is zero. With the addition of the words that I have just written and capitalised, what was previously dynamical and mathematical gibberish has been converted into a sound and understandable mathematical statement.


One of the best introductions to the sheer power of d'Alembert's Principle is a wonderful if old book by Cornelios Lanczos, entitled "The Variational Principles of Mechanics." This book truly opened my eyes! Among other things in it, one can find an incredibly insightful "missing link" between the behaviour of a rotating body in Newtonian mechanics and the use of geodesics in General Relativity --- simply amazing!

Live long and prosper.