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I am taking a 3rd year course in analytical mechanics, taught by a professor of mathematical physics.

One of the important results of analytical mechanics is d'Alembert's principle. According to our teacher, this "principle" is a "theorem" which he states as follows:

On a surface $\Sigma$ of equation $G(\vec x_j)=0$ such that the constraint forces $\vec R_i$ are perpendicular to it, Newton's equations of motion for a system composed of $N$ particles are equivalent to saying that the sum of the virtual works is zero for any virtual displacement $\delta\vec r_i$ compatible with these constraints: $$ \begin{cases} \vec R_i=\sum_q\lambda_q(\vec x_j, t)\nabla_iG^q(\vec x_, t)\\ \vec F_i=\vec F_i^{(a)}+\vec R_i \end{cases} \Leftrightarrow \begin{cases} \forall\delta\vec r_i,\sum_i\delta\vec r_i\cdot\nabla_iG^q(\vec x_j)=0\\ \implies\sum_i(m_i\vec a_i-\vec F_i^{(a)})\cdot\delta\vec r_i=0 \end{cases} $$ where $\lambda_q$ is an unknown parameter.

In all the references I have been able to consult, d'Alembert's principle corresponds, to the point of formalism, to this statement. A little later, in the course, we saw the "d'Alembert principle", which he states as follows:

The equations of motion of a system subjected to external forces $F_a^{(a)}$ and subjected to $k$ linear constraints $K^\alpha_\beta$ in velocities, $0=K^q_a\dot x^a+K^q_0\equiv K^q_0\dot x^A$, are given by $m_a\ddot x^a=F_a=F_a^{(a)}+\lambda_q(x^b, t)K^q_a$.

According to our teacher, d'Alembert's theorem allows to find Newton's equations for dynamical systems subjected to holonomic constraints, this theorem can be elevated to the rank of a principle, which allows to have access to the constraint forces for any dynamical system subjected to constraints expressible by linear relations in velocities.

I can admit that terminologically we are talking about a theorem (after all, it can be proved), but our teacher insisted very well on the fact that we "erect the theorem in principle". In no reference did I see such a statement and our teacher did not justify himself on this subject.

In what case can a mathematical result be elevated to the rank of principle even though the principle in question evokes a different statement from the theorem?

I am quite confused about this, I would appreciate being enlightened.

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    $\begingroup$ I'm not sure that this terminology really matters, so I'm not sure what the instructor is making a fuss about. Perhaps the instructor is trying to make the distinction between a theorem and a useful fact. That is to say, a theorem might or might not allow you to do calculations, or it might or might not simplify some problem, or something like that. Perhaps what the instructor is emphasizing, then, is that the theorem is important because it is useful for actually doing things like predicting the trajectories of particles in classical systems... $\endgroup$
    – march
    Nov 18, 2022 at 19:00
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    $\begingroup$ ...If it were me, I would use that language instead of just declaring the theorem a principle. I don't know. $\endgroup$
    – march
    Nov 18, 2022 at 19:00
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    $\begingroup$ Related: physics.stackexchange.com/q/68599/2451 and links therein. $\endgroup$
    – Qmechanic
    Nov 18, 2022 at 19:50
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    $\begingroup$ To reopen this post (v2), consider to limit the question to d'Alembert's principle in body and title. $\endgroup$
    – Qmechanic
    Nov 18, 2022 at 19:55

1 Answer 1

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It all depends from which angle you approach the problem. For instance, the Lorentz transformation was initially derived by Einstein from the principle of the constancy of the speed of light c, but you sometimes you can see it derived from certain symmetry principles with the constancy of the speed of light resulting as a theorem from this.

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