@ecterrab
Short version:
There is no seamless derivation top-down from the principle of least action
to the principle of d’Alembert in its classical form
(Seamless means without deliberately multiplying at some point by -1)
Long version:
It took several iterations to get it to the point. I hope it is understandable and sufficiently consistent.
My intention is to use the new function Physics:-LagrangeEquations for d’Alembert’s principle (and maybe for other principles) for systems with non-constraint forces (external forces excluding constraint forces). This is relevant in some non-conservative (e.g. with friction) or in non-holonomic systems (e.g. with control forces) where no energy potential can be defined for non-constraint forces (in order to account for them in a Lagrangian).
In your first reply you pointed out that Physics:-LagrangeEquations can change the sign of the output. This lead to the wrong conclusion (from my side) that this is the reason for the wrong sign in my example worksheet Lagrange_Equation.mw. This false conclusion got further support from the comparison of a second case (the first example of ?dsolve,numeric,DAE) where
VariationalCalculus:-EulerLagrange returned - A - B = 0 and
Physics:-LagrangeEquations returned A + B = 0
For this case VariationalCalculus:-EulerLagrange produces the correct sign when used to calculate the left-hand side of the d’Alembert’s principle. However, neither Physics:-LagrangeEquations nor VariationalCalculus:-EulerLagrange produce the correct sign in my example worksheet. So, in one instance Physics:-LagrangeEquations with the new option keeprelativesign can be used, in another not. The reason for that is simple, once noticed: The left-hand side of d’Alembert’s principle has an opposite sign than the Lagrange Equations. Only when Physics:-LagrangeEquations for the d’Alembert’s principle is used this way the sign is correct:
Physics:-LagrangeEquations(-L,…,keeprelativesign)=Q; #note the sign
I do not see another way for a generic use of the command Physics:-LagrangeEquations for d’Alembert’s principle (which is my intention). For this reason (and for compatibility to VariationalCalculus:-EulerLagrange) I consider an option to keep the sign relevant.
Naming a command option keeprealitivesign explains what the command does. Naming it non-holonomic on the other hand describes the context, for those who know what non-holonomic is. I therefore think that keeprealitivesign is a good choice and the description could refer to cases where this option should be used. Thank you for adding this option.
In the below, I have tried to identify cases where a mismatch of sign can occur when deriving one principle from another. It basically can (but does not have to) happen on the way up from Newtons equations to the principle of least action or on the way down.
Comparing some references
D’Alembert’s principle from https://en.wikipedia.org/wiki/Lagrangian_mechanics but not named as such
The right-hand side matches the left-hand side from Goldstein’s book that you cite (assuming T is replaced by L):
Also: One of my old textbooks derives the Lagrange’s equations for holonomic systems from d’Alembert’s principle and is in line with the left-hand side of Goldstein’s book above.
However, https://en.wikipedia.org/wiki/Lagrangian_mechanics chooses a different path to derive Lagrange’s equations from d’Alembert’s principle:
This is done by substituting Q__j into d’Alembert’s principle an doing (lhs-rhs)(…) on it instead of (rhs-lhs)(…). Both operations are correct but only the second allows for what I did in my worksheet: replacing T by L in d’Alembert’s principle.
This deliberate choice of sign is also possible when going backwards from the principle of least action to the Lagrangian’ s equations. You state
Strictly speaking, this is Hamilton’s principle but https://en.wikipedia.org/wiki/History_of_variational_principles_in_physics#action_principle_names states that Feynman called Hamilton’s principle the principle of least action. According to https://en.wikipedia.org/wiki/Principle_of_stationary_action, distinct principles, "the principle of least action" is a common name for many principles.
Anyway, in https://en.wikipedia.org/wiki/Hamilton%27s_principle the same result is derived in more detail by defining the change in the action functional this way
Since the change is stationary (not necessarily minimal) in the principle it is equally possible to change the sign of the integrand (i.e. doing L(q, q_dot)- L(q+eps, q_dot+eps_dot)). By doing so the result will match the left-hand side of the Goldstein book above. Both ways are mathematically correct. Again, the selection of one of two possible ways to define the change of action functional in a stationary context is a deliberate choice of sign. I would have probably applied the same definition as in the functional above because it is in line with the often-used convention: error = measured value minus true value. Another reason to stay with the above functional is that the variation of the action (and the variation of the Lagrangian) can be written in the form of a total differential which leads literally via integration by parts to
---> 
As long as we are dealing with conservative systems the sign of the left-hand side of the Lagrangian’ s equations is not a problem. But for non-conservative systems (where no disspative energy term for the Lagrangian can be defined for the non-conservative forces) and for non-holonomic systems sign matters. For those cases https://en.wikipedia.org/wiki/Lagrangian_mechanics has to change the sign of the left-hand side to correctly account for the sign of forces derived from a dissipative term:
This uncommented change of sign of the left-hand side within this otherwise well-made reference is unfortunate and makes the reference somehow inconsistent.
P.S.:
My “flat” statement was more a whish towards principles that Maple has not covered extensively yet (like d’Alembert’s principle) and not towards the principle of least action.
D’Alembert’s principle leads directly to Newtons equations of motion and is closely related to the principle of virtual work. Only for this, d’Alembert’s principle has a place in physics. In combination with non-constraint forces it has a unique position with its closest cousin being Lagrange’s equations of the first kind. However, it looks to me that the principle of virtual work and d’Alembert’s principle are more used in engineering applications whereas in modern physics the principle of least action dominates.
In classical mechanics: D’Alembert’s principle, Lagrange’s equation of the first kind, and modified Lagrangian’s (by constraints) all deal with forces. Why can’t they be combined into one principle that fits all? On the one hand I have not found good references connecting these dots. On the other hand, I have not found a reason why d’Alembert’s principle cannot be changed to a modified principle using the Lagrangian or even a modified Lagrangian (see P.P.P.S).
I am not sure if a consistent description including Lagrange’s first equation, d’Alembert principle and the principle of least action without a deliberate change in sign can be found. At the moment I only see a flip of sign for the variation of the action (which I think is out of discussion) or a reformulation of the common form of d’Alembert’s principle (It would be a great relief if someone could demonstrate the opposite).
So, if there is no elegant unification, Maple can, at best, help physics and engineering users applying principles as they are making aware of signs when they matter.
P.P.S.: Talking about engineering applications. My old textbook is consistent in itself but does not mention the principle of least action. However it is mentioned that the Hamiltonian principle states that the action int(L(t),t=t0..t1) becomes stationary which is according to the book “An insight of philosophical but not of technical value”. It also considers the Lagrangian Equations of the second kind as unnecessarily complicated to perform calculations and therefore uses for the practical part of the book only d’Alembert’s principle and Jourdain’s principle to solve problems. It also mentions the Gaussian principle that had little technical relevance. The author clearly had to do all calculations by hand, which must have been laborious at the time. These times have changed with computer algebra. I wonder if he would have refrained from making such statements today.
P.P.P.S.:
Reformulating d’Alembert’s classical principle from my worksheet (omega stands for alpha_dot)
to
would allow the use of Physics:-LagrangeEquations without minus sign.
If the Lagrangian accounts for external forces by defining potential and dissipative energy terms, the above equation becomes
where Q__NP stands for all forces that are not included in potential or dissipative energy terms. This equation should be usable with a modified Lagrangian accounting for constraint forces.
