# Can analytical mechanics be deduced from Newtonian mechanics?

• dRic2

#### dRic2

Gold Member
As you can see from the very last line of my post, this whole post may come from the fact that I don't get sarcasm

Hi, reading the above mentioned book I ran into the following footnote:
pag 77 said:
Those scientist who claim that analytical mechanics is nothing but a mathematically different formulation of the laws of Newton must assume that Postulate A is deducible from the Newtonian laws of motion. The author is unable to see how this can be done. Certainly the third law of motion, "action equals reaction", is not wide enough to replace Postulate A

Postulate A was earlier stated as:
The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints

An alternative, but equal, version of Postulate A is given the page before
The principle of virtual work assets that the given mechanical system will be in equilibrium if, and only if, the total virtual work of all the impressed forces vanishes:
$${\delta w} = \sum_i^{n} \mathbf F_i \cdot \delta \mathbf R_i = 0$$

The importance of Postulate A is later reinforced:
This postulate is not restricted to the realm of statics. It applies equally to dynamics, when the principle of virtual work is suitably generalized by means of d'Alembert principle. Since all the fundamental variational principles of mechanics, the principles of Euler, Lagrange, Jacobi, Hamilton, are but alternative mathematical formulations of d'Alembert principle, Postulate A is actually the only postulate of analytical mechanics

Going back to the original question: do you think there is a way to deduce Postulate A from Newton's laws? Because I didn't understand if the author left it as an open question or if he was just being sarcastic.

Thanks
Ric

No. The problem here is "are Newtonian mechanics and and analytical mechanics the same thing?". Because the answer seems to be negative.

Historically people inferred the principles of least action from the Newton's equation of motion. Now thus found mathematics of principle of least action provides Newton's equation of motion. As for your point of "the same thing", I have no idea whether there exists another different principle that also derives the Newton's equation of motion.

Last edited:
Historically people inferred the principles of least action from the Newton's equation of motion. Now thus found mathematics of principle of least action provides Newton's equation of motion. As for your point of "the same thing", I have no idea whether there exists another different principle that also derives the Newton's equation of motion.
Sorry I do not know why, but I just saw you post. Well, to infer something is not the same as deduce. For example you can infer that men evolved from primates but you can't prove it by turning a man into a monkey. On the other end, once you made the hypothesis that men come from primates you can take a population of primates and wait long enough to see if it turns into men. I hope I made myself clear

We can solve the problems of mechanics and get the same answers either by Newton's equation of motion or by Lagrange's equation which is founded on the principle of least action.

Lagrange's equation is easier to handle in many practical and complicated systems with the prescription of "Find the Lagrangean and put it into Lagrange's equation"

The both work. We can get answers immediately. We do not have to wait as you say primates become men.

I am not trained and confident in discussing the concept order of the two equations, e.g. "which comes first? which is more fundamental? are they same or not?".

I may say analytic mechanics is an advanced version of Newton's equation. I would add that in 20th century words of analytical mechanics have been used for quantum mechanics and relativity. Quantization is expressed as
$$[p,q]=i\hbar$$
Hamiltonian, which is another quantity in analytic mechanics, appears most familiar in QM.
General relativity adds new term in action thus in Lagrangean.

Last edited:
vanhees71
I just wonder why people who say that Lagrangian mech is not deduced from the Newtonian one, never bring an example of a classical mechanics problem such that its Lagrangian equations can not be obtained by the Newtonian means.

Last edited:
Well, if you have a Newtonian-mechanics equation of motion you can either derive it with the Lagrangian method, which is often much easier, or by using the Newtonian equation from the beginning and struggle with all the forces adding up to the total force acting on each particle.

You can, e.g., derive the Euler equations for the spinning top using solely Newtonian (or rather Eulerian) methods, but that approach made at least me hate this very beautiful topic when I first heard about it. Using the Hamilton principle and taking variations of the rotation group to derive them is just pure esthetics, and I loved the topic again. I've not seen this approach in any textbook, although I think it's the most straightforward way to derive the Euler equation of the spinning top. I guess there's some textbook out there which uses this approach, but I've not found one. Maybe one should write an AJP article about it...

The Lagrangian approach is much nicer and easier. Everybody agrees that I guess.
Using the Hamilton principle and taking variations of the rotation group to derive them is just pure esthetics, and I loved the topic again
There are the Euler-Poincare equations on Lie algebras but the references I know are all in Russian unfortunately

upd: this book is translated I did not know
book by Kozlov

Last edited:
I just wonder why people who say that Lagrangian mech is not deduced from the Newtonian one, never bring an example of a classical mechanics problem such that its Lagrangian equations can not be obtained by the Newtonian means.
That's not the point. The point is that Newton's third law is *not* an axiom of analytical mechanics, yet is an axiom in Newtonian Mechanics. That's a pretty big difference to me.

Actually, the Postulate A from the initial post is not a postulate it is a definition of ideal constraints. And yes it can not be deduced from anything else because definitions do not need to be deduced.
Once this definition is introduced we deduce the Lagrange–d'Alembert principle and then we deduce the Lagrange equations.

The Lagrangian approach is much nicer and easier. Everybody agrees that I guess.

There are the Euler-Poincare equations on Lie algebras but the references I know are all in Russian unfortunately

upd: this book is translated I did not know
book by Kozlov
I referred to the Euler equations for the spinning top.

Now I understand what you meant
Nevertheless look at Euler-Poincare equations in that book I think you will be interested

vanhees71
That's not the point. The point is that Newton's third law is *not* an axiom of analytical mechanics, yet is an axiom in Newtonian Mechanics. That's a pretty big difference to me.

As for three Newton laws, the second one, equation of motion obviously corresponds with Lagrange's equation.
The first one, the law of inertial motion and the third one, the law of action-reaction or conservation of momentum also have correspondence in behavior of Lagrangean with thought of homogeneity of time and space. If you are interested in it, you will find Noether's theorem to describe it.

vanhees71
I don't want to seem rude, and I really appreciate the effort you all are putting in trying to make me understand, but I feel you are "circling around" the point. What you say is indeed very true (even thought I haven't explore Noether's theorem, I am aware of its consequences), but I don't think all of this is required to "answer" Lanczos' question about Postulate A, that is why I marked it as B-level thread.

As I said, Lanczos states that Postulate A is really all you need to know about analytical mechanics. It is all contained in that sentence. So the problem is really "simple": Is that sentence equivalent to Newton's laws (not only on practical/empirical grounds -i.e they give the same result-, but rather in an "fundamental" sense)?

Since you can deduce Newton's mechanics from d'Alembert's principle it's clear that d'Alembert's principle is at least as or more comprehensive than Newton's formulation of mechanics. Since d'Alembert's principle is also correct if you have anholonomous constraints, for which I think Newton's mechanics is not sufficient, d'Alembert's principle is the most comprehensive formulation of mechanics. Hamilton's principle can be deduced from d'Alembert's principle if the forces are gradients of scalar fields (including the case that these fields are time dependent).

dRic2
Ah, now I see what you meant. I must be really dumb... It took a lot of messages to make me understand what you were saying (it's not your fault, it's mine of course).

I have to thank you all for the patient.

if you have anholonomous constraints, for which I think Newton's mechanics is not sufficient,
please bring an example of a nonholonomic system that can not be described by equations of Newtonian mechanics

How do you treat any anholonomic system without d'Alembert (or equivalently the action principle when applicable)? I've no idea to be honest!

For example consider a homogeneous ball on a perfectly rough horizontal floor. By ##S## denote the center of the ball; by ##P## denote the contact point.
The equations of motion are
$$m\boldsymbol {\dot v}_S=\boldsymbol T+m\boldsymbol g\qquad (1)$$
$$J\boldsymbol{\dot\omega}=\boldsymbol{SP}\times \boldsymbol T.\qquad (2)$$
Here ##T## is a reaction force.
The non-slippery condition (nonholonomic constraint) is
$$\boldsymbol v_S+\omega\times\boldsymbol{SP}=0.\qquad (3)$$
System (1)-(3) is easy to solve

Is this really nonholonomic. Can't (3) be integrated?

(3) can not be integrated.

vanhees71
please bring an example of a nonholonomic system that can not be described by equations of Newtonian mechanics
Ignoring for the moment the discussion about non-holonomic constraints I thought that what @vanhees71 meant is the following:

If you work on a classical problem, of course Newton's law and analytical mechanics are the same, but if you now try to move you attention somewhere else (for example SR), the principle of least action now yields a different result from Newton's law. In this sense they are not equivalent. They are equivalent only for a sub-set of cases, but they are not absolutely equivalent.