# Can we consider non-inertial frames in Lagrangian dynamics formulated through d'Alembert's principle?

When we derive Euler-Lagrange equations from an action principle, there is no explicit mention of a reference frame, so I assumed that the formulation is correct even in non-inertial frames (is this true?).

But I have trouble in accepting this when we derive Lagrange equations from d'Alembert's principle.

The principle states that the sum of the differences between the forces acting on a system of mass particles and the time derivatives of the momenta of the system itself projected onto any virtual displacement consistent with the constraints of the system is zero. Thus, in symbols d'Alembert's principle is written as following, $\sum _{ i=1 }^{ n }{ ({ F }_{ i }-m{ a }_{ i }).\delta { r }_{ i }=0 }$

Now I see that we implicitly assume Newton's equation hold, but for this to be true the frame must be inertial.

• So is it possible to formulate lagrangian in non-inertial frames using d'Alembert's principle?
• The term applied force is ambiguous to me, I am aware that we cannot consider dissipative forces are there any other forces we should disregard?

• It is not necessary. Nowhere in the formulation of D'Alembert principle inertial frames are used. The only requirement is that the reactive forces given by the differences of all forces not due to constraints and $ma$ satisfy the identity you wrote for every choice of virtual displacements $\delta r$. I could point out my (700 pages) lecture notes but they are written in Italian :( – Valter Moretti Jan 14 '18 at 17:44