Teaching mechanics: Between rigor,applications,and misconceptions

Introduction

Classical mechanics, rational mechanics, theoretical mechanics are equivalent designations for university subjects and teachings that seem to have long since lost their appeal in the framework of the departments of mathematics and physics all around the world. Even in engineering science departments, the situation is not much better. It is often thought that classical mechanics topics are exhausted by their treatment in college physics courses or that their contents can be incorporated straightforwardly into some applied mechanics courses.

The reasons for this situation are complex and have several sociological angles to be explored, but by sure there is a widespread idea that nowadays we have all these theories and the new challenge is to see how existing theorems, theories, and methodologies can be used to delve into new areas. So, it seems to be convenient to leave theory aside and focus on what is needed in applications with a consumer attitude or in the worst cases with a shoplifters attitude.

This point of view began to take hold especially in Anglo-Saxon countries and then spread a little everywhere. Clearly in countries such as France and Italy resistance to it has been stronger because in these places the term rational mechanics is still connected to good historical memories that, although long gone, still have an epic dimension linked to fundamental textbooks such as the Traité de mécanique rationnelle by Paul Emile Appell ( 1855 − 1930 ) ¹ and the Lezioni di meccanica razionale by Tullio Levi-Civita ( 1873 − 1941 ) and Ugo Amaldi ( 1875 − 1957 ). ² A similar situation to the Franco-Italian framework can be found in Germany considering the famous volume Theoretische Mechanik by Georg Karl Wilhelm Hamel ( 1877 − 1954 ). ³

These classics volumes played a fundamental role not only as textbooks or scientific reference books, but because they classified and rigorously delineated the unique characteristics of this discipline, rooting them in national academic systems for a long period. It is no coincidence that in Italy, even an experimental physicist like Enrico Fermi ( 1901 − 1954 ) taught this discipline and published a book of Lezioni di meccanica razionale that follows the framework of the above treatises. ⁴

In the Anglo-Saxon tradition the situation has been different because a systematic organization of the mechanical science discipline has never been proposed, although volumes of the highest level on specific topics of theoretical mechanics have been offered even in these countries. Notable examples are the volume A treatise on the analytical dynamics of particles and rigid bodies by Edmund Taylor Whittaker ( 1873 − 1956 ) ⁵ and the volume A treatise on analytical dynamics by Leopold Pars ( 1896 − 1985 ). ⁶ A partial exception to this state of affairs is perhaps the encyclopediac article Classical Dynamics by John Lighton Synge ( 1897 − 1995 ) in the Springer’s Handbuch der Physik. ⁷ It is interesting to note that in the United States the term rational mechanics, although largely monopolized by Clifford Truesdell ( 1919 − 2000 ) in the framework of continuum mechanics, has also been used by Robert Catesby Taliaferro ( 1907 − 1989 ) for a very interesting course in Notre Dame. ⁸ Taliaferro’s volume does not follow the classics mentioned above, it is much more unbalanced towards the historical aspect and lacks a complete treatment, even going so far as to cover topics that do not fully fall within the scope of classical mechanics but it shares with these treatments the level of rigor and abstraction.

The reflections developed in the present paper are motivated by two main considerations. The first is that, in several contexts, professional associations, universities, and industrial organizations perceive a progressive decline in the average level of preparation of a portion of students, and in particular of engineering graduates. The second motivation, albeit of a more secondary nature, stems from the publication of the English version of a classical textbook on rational mechanics by Biscari, Ruggeri, Vianello, and myself, primarily intended for engineering courses. ⁹ As already mentioned, we do not want to address the sociological problem, although certainly interesting, of how there has been a growing disaffection towards this discipline but, instead we want to analyze the reasons why even today a good training in classical mechanics is fundamental in many scientific and technological studies and why this discipline cannot be replaced by the notions imparted in a college physics course or relegated to a simple introduction in an applied mechanics course.

After presenting the characteristics of classical mechanics in Section “Rational mechanics”, Section “College physics and mechanics” shifts to examining some fundamental college physics texts to highlight how mechanical topics are addressed. This analysis allows for an emphasis on the limitations of presenting these subjects within a general physics course. Section “Theoretical and applied mechanics”, on the other hand, addresses the issue of the relationship between applied and theoretical mechanics. This discussion is naturally more pertinent to engineering curricula, and for this reason the research evidence concerning the interplay between contemporary engineering education and the marginalization of rational mechanics is explored in greater depth in Section “Rational mechanics and the formation of engineering thinking”. The final Section, dedicated to the concluding remarks, briefly also addresses the case of mathematics and physics curricula.

From this perspective, the present paper adopts a conceptual and epistemological approach to engineering education. Its aim is not to provide empirical evidence, but rather to clarify the role of rational mechanics in the intellectual formation of engineering students, and to discuss the educational implications of its marginalization in contemporary curricula.

Rational mechanics

The first step to take in our journey is to understand well and in depth what is meant by rational mechanics ¹ . Surely Truesdell is the one who more than any other historian of science has allowed us to find a solution to this question. In Truesdell ¹⁰ we read

Rational mechanics was a science of experience, but no more than geometry was it experimental. While some great mechanical experiments were done in the Age of Reason, they had only occasional bearing on the growth of the theories we now regard as classical. Experiment and theory result from different kinds of reaction to experience. If, ideally, they should complement and check one another, yet even today, with all our superior knowledge not only of facts but also of scientific method, it is difficult enough to relate them, so why should it have been easier 300 years ago? It was not. A factual view of the history of mechanics must concede that rational mechanics and experimental mechanics, both arising from human beings’ intelligent reaction to mechanical experience, grew up separately.

This is a crucial point that is often confused especially because of Galileo Galilei’s ( 1564 − 1642 ) role in this story. In fact, the multifaceted figure of Galileo and his many technological inventions have created confusion about his role as the founder of modern science. Galileo cannot be reduced only to the first to have used the experimental method in a systematic way. In Galileo mind was clear that to advance in sciences it is not sufficient just a collections of facts, but we need the pre-order of a theory and this ordering often depends upon mathematical reasoning in defiance of common sense. ¹¹ This mathemathical reasoning is the ground of Galileo’s inertia principle and the ground rational mechanics (let’s continue with Truesdell’s quote from Truesdell ¹⁰ ):

The method used in the great researches was entirely mathematical, but the result was not what would now be called pure mathematics. Experience was the guide; experience, physical experience and the experience of accumulated previous theory. If we were to seek a word for what was done, it would not be physics and it would .not be pure mathematics; least of all would it be applied mathematics: it would be rational mechanics.

Here we come to face another land of great confusion: many people think that rational mechanics is one of the many applications of mathematics. The confusion arises for two reasons. First, because it is often not understood as Richard Courant ( 1888 − 1972 ) said applied mathematics it is not a definable scientific field but a human attitude. Second, because the relationships between mathematics and rational mechanics are much more complicated than they may seem at first glance.

In pure mathematics we treat the relations between quantities irrespective of the objects they measure or count and therefore we do not need experience. In rational mechanics we deal with quantities, and their relations, as they come out from experience of the real world, i.e. as they occur in natural objects, For these reasons, rational mechanics is declined as the axiomatic mathematical science of mechanics and there is a a clear taxonomic distinction between mathematics and a mathematical science.

The fact that all the authors of the previously mentioned classic volumes were mathematicians (modulo Enrico Fermi) by training is not coincidental: all the great eighteenth-century mathematicians, the Bernoullis, Clairaut, d’Alembert, Euler, Lagrange, and others, worked both in mathematics and in the mathematical sciences, and did not think these fields distinct.

David Hilbert ( 1862 − 1943 ) in the lecture delivered before the International Congress of Mathematicians at Paris in 1900 says:

Surely the first and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena. Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization, just as the child of today learns the application of these laws by empirical methods. The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle; also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential—to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics.

But it is in Ernst Mach ( 1838 − 1916 ) theory of the development of a scientific discipline that we find a deeper epistemological reason for this attitude that today is perceived as a nineteenth-century remnant. In Mach’s thought there are three stages in the development of a scientific theory. In the first phase, we have the observations of the phenomena. Then, the deductive period in which the scientists learn to deduce logically and theoretically certain phenomena from others. Finally, we record the formal development of the discipline in which certain observations are recognized as the basic principles of the science and in which the deductions are organized in a theoretical unity. The old school of the history of mathematics and science ascribe this strong union between mathematics and mechanics during the nineteenth century precisely to the fact that this discipline being in its third phase could afford to become rational. ¹² Thus, the twentieth-century crisis of this discipline can be attributed to new experimental discoveries that have plunged back mechanics in its first phase.

This interpretation could be the explanation for the widespread view that rational mechanics is a somewhat ancient subject that no longer has anything to say, but it is believed that this view completely overlooks two fundamental points.

The first one is that the first phase does not mean that we are able to experiments and measure. The observation we are talking about uncovers the underlying structures of conscious experience. Experiments are the product of theory, not vice versa. The second point is a fundamental insight of Mach’s thought :the principle of economy of thought that demands that the deduction of phenomena from basic principles should be as easy and straightforward as possible.

Certainly the experience of the infinitely large and the infinitely small developed in the twentieth century changed the paradigm of the mechanical sciences, but the creation of rational mechanics made it possible to realize with such perfection the economy of thought evoked by Mach, that no one could halt its success in forming the theoretical basis of the new technology that would enlighten the past millennium. Proof of this is that all scientific disciplines, and others field of knowledge, in the early twentieth century, infused with naive positivism, tried to retrace the path that led the mechanical sciences to become a mathematical science (obviously unsuccessfully). One example for all is that of the economic sciences, starting with Vilfredo Pareto’s ideas ( 1848 − 1923 ) that have persisted through time to the present day.^13–15

It should therefore be clear from the discussion on these pages that:

Rational mechanics is a science of experience and not based on experiments and measures, therefore its place cannot be confined to a physics course.

Clearly, one could argue that despite all this we fail to see the importance of this discipline. Such considerations cannot be answered objectively but, even once one can find contextual and historical reasons. Rational mechanics explains how our model of the reality that surrounds us (or that seems to surround us) is formed. Thanks to this discipline, we can practice this construction in a concrete way with real exercises and this because of the great economy of thoughts inherent to its nature. No other discipline succeeds in this intent. Truesdell ¹⁰ noted that:

Not only private, individual experimental researches were performed in the eighteenth century; there were also large, cooperative projects. As today, they cost more than real science, and they attracted administrators. But the effect of all this expense on what we now consider the achievement of the period was nil.

This might seem like an extreme view, but it is substantially aligned with reality and clearly explains the significance and importance of rational mechanics.

We know well that it is possible to find a place for this discipline in various academic curricula. Physics, mathematics, and engineering have been mentioned, but the general part of Taliaferro’s course ⁸ demonstrates how even the history or philosophy of science can substantially benefit from rational mechanics.

As the final point of this paragraph, we must take a moment to reflect on the boundaries of this discipline. In fact, the naive view of the positivists has sometimes been adopted, perhaps unconsciously, even by some scholars of rational mechanics, and there has been a temptation, perhaps even the arrogance, to see this discipline as the foundation for all knowledge. Obviously, this is an untenable and even ridiculous point of view. For this purpose, one can adopt Synge’s point of view as presented in the introduction ⁷ and limit ourselves, considering the purpose of this note, to what is usually indicated as Newtonian and Relativistic mechanics, even though we will only briefly mention relativity (and maybe quantum mechanics) in the conclusions. Obviously, there are other fields that could fall under this designation, but for the sake of simplicity and to avoid unnecessary disputes, this will suffice for now.

College physics and mechanics

The purpose of this paragraph is to show that the mechanics contained in college physics courses is not adequate to replace a rational mechanics course. Note that we are not talking about the fact that the treatment is not sufficient because it is too narrow: we are actually discussing the quality of the exposition of this content in college physics books.

We must start from the evidence that for some academic curricula the fundamental laws of mechanics are a fundamental part of the student’s cultural background. If, for example, we focus on engineering courses we know very well that there are branches of this discipline that cannot do without these concepts and that it must be accepted that this content must certainly be addressed. For other academic curricula the discussion is broader and for that reason we postpone it.

Our discussion is based mainly on Halliday and Resnick’s volume, ¹⁶ but first we want to comment on Feynman’s text, which is a true cult work. ¹⁷ In fact, Feynman’s book is a seminal work the physicist’s attitude toward mechanics but does not turn out to be a sufficiently widely adopted textbook. We read ¹⁷

…The lectures form only part of the complete course. …The special problem we tried to get at with these lectures was to maintain the interest of the very enthusiastic and rather smart students coming out of the high schools and into Caltech. They have heard a lot about how interesting and exciting physics is—the theory of relativity, quantum mechanics, and other modern ideas. By the end of two years of our previous course, many would be very discouraged because there were really very few grand, new, modern ideas presented to them. They were made to study inclined planes, electrostatics, and so forth, and after two years it was quite stultifying. The problem was whether or not we could make a course which would save the more advanced and excited student by maintaining his enthusiasm. The lectures here are not in any way meant to be a survey course, but are very serious. I thought to address them to the most intelligent in the class and to make sure, if possible, that even the most intelligent student was unable to completely encompass everything that was in the lectures by putting in suggestions of applications of the ideas and concepts in various directions outside the main line of attack.

This sentence shows that the aim of the Feynman lectures is completely out of the actual pedagogical needs but it is a good evidence of the attitude to consider the topics of mechanics none exciting and age old.

In the Physics Today April 1972 issue George H. Bowen ( 1925 − 2009 ) reviews ¹⁶ and he writes

This is a calculus-level textbook designed for use in the standard introductory course for students of science and engineering that is taught in the majority of colleges and universities. It is not at all innovative, and indeed is not meant to be, but it is well organized and well written, and it should prove very satisfactory for its intended purpose. In fact it is surely one of the best of the currently available books of its kind.

This review shows that ¹⁶ is adequate for our purposes. However, it remains interesting to read to the end of this review:

A number of physics textbooks that appeared during the 1960 ’s contained far more sophisticated material than most students could absorb in a typical beginning course, as if the authors had sometimes written more to impress their colleagues than to help their students. Perhaps we are now seeing a healthy move away from that extreme. Fundamentals of Physics is one of several recently appearing texts aimed at slightly more modest goals and which may actually accomplish more because of it. Unfortunately none of them appears likely to stimulate the student’s interest and imagination as effectively as one would wish. There still is a need for a book that not only serves as an effective basic teaching instrument but also leaves the student with some real appreciation of the beauty, wonder and excitement of physics.

This teaching attitude that finds its hyperbole in Leighton and Sands ¹⁷ but is creeping throughout the college physics literature is already incompatible from the outset with the purpose of a discipline that belongs to the mathematical sciences because it sacrifices rigor and thoroughness of treatment to a pandering sensationalism that should focus only on the best students while leaving out the masses.

There is no discussion of whether this point of view is not beneficial and/or necessary. Here, we only want to make it clear that this attitude distinguishes a priori a college physics course with a rational mechanics course also from a pedagogical point of view. A rational mechanics course being a course in a mathematical discipline is based on abstraction and rigor. It starts with a clear and rigorous definition of the various quantities involved and fully deduces the consequences that follow from these definitions. If in some passages it grants exceptions to this rigorous approach, it does so for serious pedagogical reasons and not to implement the student’s attention.

Just to have an idea let us take the definition of a rigid body in Halliday et al. ¹⁶ : a rigid body is a body that can rotate with all its parts locked together and without any change in its shape.

The effect of such an imprecise and vague definition is naturally dangerous because of its consequences. In fact, on page 272 of Halliday et al. ¹⁶ after this definition we read

A fixed axis means that the rotation occurs about an axis that does not move. Thus, we shall not examine an object like the Sun, because the parts of the Sun (a ball of gas) are not locked together. We also shall not examine an object like a bowling ball rolling along a lane, because the ball rotates about a moving axis (the ball’s motion is a mixture of rotation and translation).

Firstly, this definition of a fixed axis is not sufficient to identify a rigid motion of simple rotation, as the axis must be fixed both in the fixed reference system and in the reference attached to the body, but by sure second definition has never been proposed. Moreover, this wording suggests that a fixed axis is required to define a rigid body. It is not true that the Sun cannot be seen as a rigid body: celestial mechanics is able to predict many interesting fact looking the sun just a single point! The misture of rotation and translation means just nothing and it’s a questionable image from a metaphorical standpoint. The result of this confusion is that the definition of angular velocity is reduced to the derivative of an angle, with an endless series of arguments that could have been completely avoided by introducing Poisson’s formulas.

One could continue to highlight many more of these statements, which contain a level of imprecision that, in turn, leads the reasoning into dead end where confusion reigning supreme.

For example to introduce the reaction force we read: When a body presses against a surface, the surface (even a seemingly rigid one) deforms and pushes on the body with a normal force F n that is perpendicular to the surface.

Another example, is at page 230 : We state (and shall prove next) that the vector equation that governs the motion of the center of mass of such a system of particles is F n e t = M a c o m .

About the problems associate with this statement you can read ¹⁸ where all the technical details and the non-understandings associated with the motion of the center of gravity are considered into details. It should in fact be recalled that the equation F n e t = M a c o m , in general, is not a closed equation: the left-hand side cannot, in fact, be evaluated without prior knowledge of the positions and velocities of all the particles in the system. Nevertheless, there exist certain exceptional cases of particular importance. One example is the case of the gravitational force under the standard approximation.

In conclusion, it becomes clear that for a civil, mechanical, or aerospace engineering student, where the equations of rigid body dynamics and deformable systems form the scientific foundation of their technical and professional skills, the approach proposed by college physics on these topics is insufficient. A more rigorous and comprehensive reconsideration of these subjects is necessary.

This does not mean that mechanics should be removed from a college physics course. It simply means that the motivations and utility of a college physics course are entirely different from those of a course in rational mechanics. For this reason, the approach in Leighton and Sands ¹⁷ seems far more appropriate than that in Halliday et al. ¹⁶ In the end, Halliday et al. ¹⁶ may cover more pages and topics, but the educational outcome is no better than the provocations of Leighton and Sands. ¹⁷ Provocations that in a sensationalist view of education where it is essential to grab the student’s attention at all costs may have valid marketing reasons. After for winning the sympathy of the audience one must remember that the important thing is to educate and train in substance in a universal way trying to leave behind the least number of students

It becomes clear that with this approach, students cannot truly learn mechanics; they can only be introduced to some physics problems related to mechanics. On the other hand, with a good course in rational mechanics, students receive a detailed and comprehensive introduction to mechanics, and much more.

Theoretical and applied mechanics

While a direct comparison between rational mechanics and introductory college physics is relatively straightforward–since a long tradition has essentially standardized the corresponding textbooks–the situation becomes more nuanced when applied mechanics in engineering curricula is considered.

Indeed, two distinct issues arise. First, applied mechanics is not a single homogeneous discipline: civil, mechanical, and structural engineering traditions emphasize different aspects and methodologies. Second, the boundary between theory and application is, to a significant extent, culturally and historically determined. As a consequence, in academic systems where rational mechanics is clearly established as a foundational discipline, the distinction is well understood, whereas in other contexts it becomes less sharp, and even the terminology may lose precision. This is illustrated, for instance, by certain modern textbooks such as Beatty,^19,20 Udwaida and Kalaba, ²¹ or Papastavridis, ²² which combine theoretical mechanics with stylistic elements typical of engineering manuals.

To clarify the discussion, it is useful to refer to a classical applied mechanics text such as Timoshenko, ²³ which still represents a standard reference in many engineering curricula. Compared with this type of presentation, more recent textbooks tend to emphasize graphical exposition, a larger number of worked examples, and a selective coverage of topics, typically restricted to subsets of classical mechanics. In this sense, Timoshenko’s treatise can be regarded as a reference model for identifying the essential structure of traditional applied mechanics teaching.

The organization of Timoshenko and Young ²³ reflects a classical pedagogical structure based on the dichotomy between statics and dynamics. However, it does not explicitly develop kinematics as an independent foundational discipline. This omission, which is common in many engineering-oriented treatments, indicates that kinematics is often treated implicitly as a tool rather than as a structural component of the theory. From a rational mechanics viewpoint, this choice has important consequences. In fact, a meaningful treatment of relative motion requires a precise formulation of kinematics, including the concept of velocity fields and their geometric interpretation. Moreover, classical results such as the composition of angular velocities are most naturally understood within a fully developed kinematical framework.

A similar remark applies to constrained motion. In most engineering mechanics courses, constraints are introduced as problem-specific conditions, whereas in rational mechanics they play a central role as geometric restrictions on the configuration space of the system. The lack of a systematic treatment of constrained kinematics in many engineering physics courses leads to a fragmented understanding of relative dynamics and rigid body motion.

Another important issue concerns the logical status of fundamental principles. In rational mechanics, the principle of determinism and the structure of state spaces provide a unifying framework that connects statics, dynamics, and stability theory. In many engineering presentations, however, such connections remain implicit. As a result, classical results such as d’Alembert’s principle may appear as computational tools rather than as consequences of a deeper variational or structural formulation.

A further limitation of many applied treatments is the absence of a systematic introduction to Lagrangian mechanics. In modern engineering practice, particularly in vibration theory, structural dynamics, and computational mechanics, Lagrangian methods are ubiquitous. They are also the basis of most modern simulation tools used in engineering analysis. For this reason, it is no longer reasonable to exclude them from a general mechanical education. Nevertheless, their presentation in engineering curricula is often algorithmic rather than structural.

The key point, however, is not the presence or absence of specific topics, but rather their role within the conceptual architecture of the discipline. In standard engineering mechanics, topics such as rigid body kinematics, constrained motion, and Lagrangian equations are typically introduced as tools for solving classes of applied problems. In rational mechanics, by contrast, these same elements are developed within a unified theoretical structure in which the configuration space of the system, the representation of constraints as geometric objects, and invariance principles constitute the primary foundations of the theory.

In this sense, rational mechanics should not be viewed as an alternative collection of topics, but rather as a structural reorganization of classical mechanics based on a higher level of abstraction and logical coherence. This viewpoint allows one to clarify the relationships between apparently disparate subjects and to identify the minimal set of principles from which classical results can be systematically derived.

Historically, attempts to bridge theory and application at a very high level can be found in the works of Giulio Krall, ²⁴ whose treatment of vibration theory combines analytical depth with engineering relevance. However, such works, while intellectually stimulating, are often too demanding for standard undergraduate education. A similar observation applies to Levi-Civita’s lectures, ² where advanced theoretical ideas are embedded in technical discussions, sometimes at the expense of pedagogical clarity.

In the Italian tradition of civil engineering education, two influential approaches can be identified. The treatise by Odone Belluzzi ²⁵ emphasizes a strongly example-driven methodology, where theoretical concepts are embedded in a large number of applied problems. By contrast, Riccardo Baldacci ²⁶ adopts a more axiomatic and theory-oriented presentation, largely avoiding computational exercises. These two approaches can be viewed as representing opposite pedagogical extremes. In both cases, a background in rational mechanics plays a crucial role: in the first case, it provides conceptual unification and prevents purely empirical accumulation of techniques; in the second, it offers the structural foundation necessary to understand abstract theoretical developments.

More generally, mechanics plays a dual role in scientific and engineering education. On the one hand, it provides a unifying framework that organizes a wide range of physical phenomena under a small set of principles. On the other hand, it serves as a laboratory for the application of mathematical concepts in a physically meaningful context. In this respect, it constitutes an essential bridge between mathematical theory and engineering practice.

What distinguishes rational mechanics is therefore not the list of topics it covers–many of which also appear in engineering mechanics, vibrations, or dynamics courses–but rather its systematic and hierarchical organization. The discipline is characterized by:

a formulation based on general principles rather than problem-specific rules;

From this perspective, rational mechanics should be understood not as an additional subject, but as a foundational framework that structures and integrates the various components of classical mechanics into a coherent theoretical system.

Ultimately, the most reasonable approach is to recognize the complementarity between rational mechanics and applied mechanics. The former provides the conceptual and structural foundations, while the latter focuses on problem-solving techniques and engineering applications. This synergy has historically proved effective in many educational systems where a clear separation of roles between theory and application has been maintained, ensuring both conceptual depth and practical competence.

Rational mechanics and the formation of engineering thinking

After having attempted, in the preceding sections, a rapid comparison between rational mechanics and introductory mechanics courses in physics, as well as between rational and applied mechanics, I would now like to return to the issue of the progressive decline in the average level of preparation of a portion of engineering students and graduates.

In the previous sections we emphasized two main points. First, there exists a profound didactic and methodological difference between the treatment of mechanics typically encountered in physics curricula and that developed within rational mechanics courses. This difference becomes immediately evident through a simple comparison of the corresponding textbooks (see, for instance, Biscari et al. ⁹ ) and has important consequences for the intellectual formation of students. In our opinion, it is not reasonable to leave future engineers with only an approximate and fragmentary introduction to basic and theoretical mechanics.

Second, we argued that a cooperative interaction between rational mechanics and applied mechanics represents the most fruitful educational approach. Admittedly, the discussion here necessarily becomes more qualitative. For this reason, it is useful to recall the celebrated remarks contained in the preface to Timoshenko’s classical text: ²³

The importance of mechanics in the preparation of young engineers for work in specialized fields cannot be overemphasized. The demand from industry is increasingly for engineers who are soundly grounded in their fundamental subject rather than those with narrowly specialized training. There is a good reason for this trend: the industrial engineer is continually confronted with new problems, which do not always yield to routine methods of solution. The engineer who can successfully cope with such problems must have a sound understanding of the fundamental principles involved and be familiar with various general methods of approach, rather than being proficient in the use of any single one. It seems evident, then, that university training in such a fundamental subject as mechanics must seek to build a strong foundation, to acquaint students with as many general methods of approach as possible, to illustrate the application of these methods to practical engineering problems, and to avoid routine drill in the manipulation of standardized methods of solution.

Even today, it is difficult to disagree with these observations. Engineers working in modern industrial environments are increasingly confronted with problems that cannot be solved through routine procedures alone. What is required is not merely operational competence, but rather the capacity to formulate models, identify the relevant mechanisms, and adapt general methods to new situations.

It is within this context that one may reasonably ask whether the progressive marginalization of rational mechanics within engineering curricula has contributed, at least in part, to some widely perceived weaknesses in engineering education.

Let us be clear: the issue is not a generic decline in the “quality” of engineers. Rather, the concern involves a set of specific phenomena frequently discussed in the engineering education literature:

increasing heterogeneity in students’ preparation;

A substantial literature has documented similar concerns over several decades. Reports from professional engineering bodies frequently mention:

deficiencies in technical communication;

In other words, operational capability often appears to increase while conceptual understanding weakens.

Both recent and earlier studies in engineering education have documented similar concerns. Indeed, as early as the 1980s, the National Research Council ²⁷ highlighted these issues, which have since been placed in historical perspective by, for example, Chatzis ²⁸ and Akera and Seely. ²⁹ In any case, the decline in analytical and problem-solving skills is documented in Eleri et al., ³⁰ Volk ³¹ as well as in more recent works such as Litzinger et al., ³² Carberry and Baker. ³³ Even more relevant to the present paper are studies such as Bajracharya and Thompson, ³⁴ which found that engineering students frequently struggle with the mathematical foundations of mechanical concepts, even after completing standard dynamics courses. Furthermore, Brouwer and Jansen ³⁵ reported that many students lack a coherent conceptual framework for understanding rigid body motion and constrained systems. While these studies do not directly address rational mechanics, they point to broader issues of mathematical and conceptual preparation that a discipline like rational mechanics is particularly well-suited to address.

The Italian case is particularly interesting because it intersects directly with the institutional role historically played by rational mechanics. Before the reforms associated with the Bologna Process ² (1999), rational mechanics was included in the national curricular framework and was therefore systematically present in civil and mechanical engineering programs. Following the reform, universities progressively acquired greater freedom in defining curricula, making it possible to reduce or even eliminate rational mechanics courses.

In many cases the discipline did not disappear immediately, but its role was progressively reduced during the early 2000s, eventually becoming marginal in several engineering programs.

Various reports published by the National Council of Italian Engineers (CNI)^36,37 suggest that, during the ”State Examination” required for professional registration ³ , candidates trained under the new system often exhibited weaker familiarity with extended analytical procedures and with the preparation of complete technical projects ⁴

Obviously, such phenomena cannot be attributed solely to the reduction of rational mechanics. Engineering education is shaped by many interconnected factors. Nevertheless, it seems reasonable to reflect on whether the disappearance of a discipline specifically devoted to analytical modelling, abstraction, and structured problem solving may have contributed to this broader transformation.

Indeed, more than thirty-five years of teaching rational mechanics have convinced me that the remarkable difficulty students experience with mechanics examinations is not accidental. The reason, in my opinion, is relatively clear: mechanics exercises are, in essence, genuine engineering problems.

Even when presented in abstract form, each exercise possesses its own internal structure. The variety of possible dynamical systems — even in planar mechanics alone — is so large that no purely algorithmic scheme can adequately capture them.

This feature sharply distinguishes mechanics from many standard mathematical exercises. Consider, for instance, the classical study of a function in calculus. No matter how technically involved the computation may become, there still exists an essentially fixed procedural structure guiding the solution process.

Mechanics problems are fundamentally different. Each problem requires the student to construct a strategy involving kinematics, constraints, geometry, force transmission, and analytical reasoning. In this sense, every exercise constitutes a partially open problem rather than the execution of a predetermined algorithm.

Precisely for this reason, mechanics exercises represent an extraordinary educational laboratory for developing abstraction, creativity, modelling ability, and intellectual flexibility — exactly the qualities emphasized by Timoshenko.

It is difficult to identify another discipline capable of reproducing this pedagogical function with the same continuity and effectiveness. Project-based learning may partially fulfill this role, but it is typically episodic and limited in scale. More advanced engineering sciences, such as elasticity or fluid dynamics, involve mathematical complexities that often make the construction of large collections of genuinely original exercises impractical. Physics courses are frequently too broad, whereas mathematics courses, although rigorous, usually lack direct interaction with concrete mechanical systems.

For these reasons, it seems plausible that the weakening of rational mechanics within engineering curricula may have indirectly contributed to some of the difficulties discussed above. Clearly, no single course can determine the quality of an engineer. However, the disappearance of a discipline that historically provided systematic training in analytical modelling and semi-open problem solving may have amplified pre-existing weaknesses in engineering education.

More importantly, the loss of synergy between rational mechanics and applied mechanics may force the latter to devote increasing time to theoretical preliminaries, thereby reducing the space available for genuinely engineering-oriented training.

Finally, Timoshenko’s remarks point toward a broader issue: the relationship between creativity, innovation, and theoretical understanding. Genuine innovation rarely emerges from pure serendipity alone. The history of science and technology repeatedly shows that the recognition of new possibilities presupposes a sufficiently deep conceptual framework. A strong theoretical education does not oppose innovation; rather, it constitutes one of its essential preconditions.

Concluding remarks

The purpose of these remarks is not to advocate a nostalgic return to older educational systems, nor to idealize the past. Engineering education must evolve together with technology, industry, and society. Nevertheless, in this process of transformation, it is important not to lose those disciplines whose primary educational value lies precisely in training students to think structurally, analytically, and creatively about complex systems.

The considerations developed in this work suggest several implications for engineering education. First, the progressive marginalization of rational mechanics may encourage the development of problem-solving techniques detached from a unified conceptual framework. As a consequence, students may experience increasing difficulty in transferring knowledge across different domains and in approaching genuinely unfamiliar problems.

Second, the absence of a rigorous and unified treatment of mechanics may weaken the development of abstraction and modelling abilities, which remain essential for advanced engineering practice. Rational mechanics, precisely because of its emphasis on structure, generality, and logical coherence, provides a particularly fertile environment in which such intellectual skills may be cultivated.

More generally, the relationship between theory and application should not be interpreted as a dichotomy, but rather as a continuum in which foundational disciplines play an essential organizing role. From this perspective, the presence of rational mechanics within engineering curricula may contribute to a more balanced and intellectually coherent educational framework.

A few additional remarks may be made concerning mathematics and physics curricula. In mathematics programs, the role of rational mechanics naturally shifts from applications toward methods and structures. Mathematics cannot remain confined within a purely self-referential world. As Eric Temple Bell famously observed, mathematics is both the Queen and the Servant of science. In this respect, rational mechanics offers a particularly valuable example of how concrete experience may progressively evolve into an autonomous mathematical theory while preserving contact with physical reality.

The situation in physics is even more delicate and complex. Modern physics has progressively moved toward highly abstract theoretical frameworks, and one may wonder whether future developments could eventually motivate forms of “rational quantum mechanics” analogous, at least in spirit, to the classical tradition of rational mechanics. At the same time, works such as Taliaferro’s book ⁸ demonstrate how historical and philosophical reflection may still provide important perspectives for contemporary mechanics education.

Ultimately, the educational importance of rational mechanics does not lie solely in the specific technical contents it transmits, but rather in the distinctive intellectual attitude it cultivates: the ability to move continuously between abstraction and reality, between mathematical structure and physical interpretation, and between general principles and concrete problems.