# During which period did the name Work-Energy theorem enter physics discourse

Throughout the centuries the label 'principle' has been applied liberally. That blurs the lines, it devaluates the word 'principle'.

As an example of a good use of the word 'principle' the Principle of relativity of inertial motion comes to mind.

An example of improper use of the word 'principle', I submit, is d'Alembert's principle of virtual work.

Of course, the designations 'principle' and 'law' have never been sharply defined (nor will they ever be). In effect it is used as a general way of saying: this is important, this has widespread validity.

But the designation 'theorem', I feel, expresses a certain deference. In mathematics, when there is an axiomatic system, you have the axioms and everything that can be derived from those axioms is referred to as 'theorem'. I assume that using the designation 'theorem' carries a message: acknowledgement that the Work-Energy relation derives from $$F=ma$$. The choice of name expresses how the Work-Energy relation is perceived.

Someone must have been the first to use the name 'Work-Energy theorem'. Is it known who was the first? Did it gain wide adoption quickly?

A google search reported 170 million hits for 'work-energy principle' and 40 million hits for 'work-energy theorem'.

Have the two names always coexisted? Or has there been, in the past decades, a shift in the naming?

It may be that a shift in naming indicates a general loss of physics understanding. I just checked a prominent educational website. It is stated there that "the work-energy principle is derivable from conservation of energy."

I feel this is an area where knowledge of history of physics can be used as an instrument to try and counteract erosion of physics understanding.

For completeness I give the derivation:

the following two relations will be used:

$$ds = v \ dt \qquad (1)$$

$$dv = \frac{dv}{dt}dt \qquad (2)$$

The integral for acceleration from a starting point $$s_0$$ to a final point $$s$$

$$\int_{s_0}^s a \ ds \qquad (3)$$

Use (1) to change the differential from ds to dt. Since the differential is changed the limits change accordingly.

$$\int_{t_0}^t a \ v \ dt \qquad (4)$$

Rearrange the order, and write the acceleration $$a$$ as $$\tfrac{dv}{dt}$$

$$\int_{t_0}^t v \ \frac{dv}{dt} \ dt \qquad (5)$$

Use (2) for a second change of differential, again the limits change accordingly.

$$\int_{v_0}^v v \ dv \qquad (6)$$

Putting everything together:

$$\int_{s_0}^s a \ ds = \tfrac{1}{2}v^2 - \tfrac{1}{2}v_0^2 \qquad (7)$$

Combining $$F=ma$$ and (7) gives:

$$\int_{s_0}^s F \ ds = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2 \qquad (8)$$

The Work-Energy theorem is narrower in scope than the principle of conservation of Energy. Kinetic energy is galilean invariant, so the Work-Energy theorem is applicable only for forces such that the change of velocity is independent of the current velocity.

(Also, the Work-Energy theorem is applicable only if an unambiguous integration of the force exists.)

The principle of conservation of Energy, on the other hand, is an unconditional assertion. The scope is unlimited.

• The same period as "work - energy principle" judging by Google ngrams. Both likely represent naming tendency of textbook authors to feed material to students in small digestible portions. – Conifold Apr 6 at 5:40
• @Conifold Thank you for pointing out the 'Google ngrams' resource. From the start more or less the same ratio; no shift. In the wake of the ngrams I noticed that Google search offers narrowing down the search to content of books. And, like all search, this books search can be narrowed down to a time frame. Up until 1960 Google finds only 10 or so books with the expresssion 'work-energy theorem', the first in the list from 1934. So: while the concept is much older; the expressions 'work-energy theorem' and 'work-energy principle' are quite recent; second half of the 20th century. – Cleonis Apr 7 at 16:56