# When can a "theorem" be raised to a "principle"? [duplicate]

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?