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From the principle of least (or stationary) action, we get that a classical system will evolve according to Euler-Lagrange equations:

$$\frac{d}{dt}\bigg (\frac{\partial L}{\partial \dot{q_i}}\bigg) = \frac{\partial L}{\partial q_i} .$$

I have often read and heard from physicists that this differential equation encapsulates all of classical mechanics. A glorious reformation of Newton's laws that are more general, compact and much more efficient.

I get that if you plug in the value of the Lagrangian, you re-obtain Newton's second law. But Newtonian mechanics is based on 3 laws, is it not? The law of inertia is a special consequence of the second law, so we don't need that, but what about the third law, namely that forces acts in pairs; action equals minus reaction?

My question is, can we obtain Newton's third law from this form of Euler-Lagrange equation? I understand that Newton's third law for an isolated $2$-body system follows from total momentum conservation, but what about a system with $N\geq 3$ particles? If not why do people say that it's all of classical mechanics in a nutshell?

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    $\begingroup$ related physics.stackexchange.com/q/12122 $\endgroup$ – SRS Jan 1 '18 at 8:35
  • $\begingroup$ I'm not good enough at math to say definitively, but I suspect it does not follow. Consider a system with a single particle and a constant force on it. $\endgroup$ – Owen Jan 4 '18 at 20:18
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Newton's third law is that for every action, there is an equal and opposite reaction. This is a statement of momentum conversation.

In the Euler-Lagrange equation, the last term $$ \frac{\partial\mathcal{L}}{\partial q_i} $$ is a generalized force. Similarly, the generalized momentum is $$ \frac{\partial\mathcal{L}}{\partial \dot q_i}. $$ If the generalized force is zero, then $$ \frac{d}{dt} \frac{\partial\mathcal{L}}{\partial \dot q_i} = 0 $$ Mathematically, this means that the generalized momentum is constant over time, i.e. it is conserved, which is Newton's third law.

We don't even need the Lagrangian to summarize all of Newton's laws. As you likely know, Newton's second law $F=ma$ is a special case of $F = \frac{dp}{dt}$. This generally accounts for all of Newton's laws:

  • Newton's 1st Law - an object will persist in a state of uniform motion unless compelled by an external force: If $F = 0$, then $\frac{dp}{dt} = 0$, and thus $p$ is constant.
  • Newton's 2nd Law - $F = ma$: if m is constant, then $$ F = \frac{dp}{dt} = \frac{d(mv)}{dt} = m\frac{dv}{dt} = ma.$$
  • Newton's 3rd law - see the below derivation.

So, generally speaking, Newton's third laws are slightly redundant in the sense that they can all be described by $$ \vec F = \frac{d\vec p}{dt}. $$

(or by the Euler-Lagrange equation, as you argue.)


Edit: Derivation of N3L with conservation of momentum

Consider a system with total momentum $\vec p_{\rm tot}$ and two particles with momenta $\vec p_1$ and $\vec p_2$ such that $\vec p_{\rm tot} = \vec p_1 + \vec p_2$. If the system is closed, then the total momentum is conserved, so $$ \frac{d\vec p_{\rm tot}}{dt} = 0.$$ Differentiating both sides, you get $$ \frac{d}{dt}(\vec p_{\rm tot}) = \frac{d}{dt}(\vec p_1 + \vec p_2)$$ $$ 0 = \frac{d\vec p_1}{dt} + \frac{d\vec p_2}{dt} = \vec{F_1} + \vec{F_2}$$ $$ \vec F_1 = -\vec F_2 $$

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – rob Jan 2 '18 at 1:37
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    $\begingroup$ Isn't the assumption that total momentum is conserved in a closed system an additional assumption beyond F = ma? $\endgroup$ – Owen Jan 2 '18 at 3:59
  • $\begingroup$ The first law most certainly isn't redundant. It states in which systems the other laws are valid. For example in an accelerated reference systems Newton's laws don't work and that doesn't follow from the definition of force you gave. Also, in your derivation of the third law you assumed that the total momentum is given by the sum of the single momenta, which also needs to be described in an axiom. $\endgroup$ – Jannik Pitt May 19 '18 at 22:08
  • $\begingroup$ Another comment: The statement $F=\frac{dp}{dt}$ does only hold when the total mass is constant, as a simple Galileo transformation shows. $\endgroup$ – Jannik Pitt Apr 20 at 18:18
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Here is a rather different take on the same question by Lanczos who writes in his classic "THE VARIATIONAL PRINCIPLES OF MECHANICS" (see: https://archive.org/details/VariationalPrinciplesOfMechanicsLanczos ) on page 77 that

Those scientists 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.

Here Postulate A is defined on the previous page 76 as:

Since the principle of equilibrium requires that "impressed force plus resultant force of reaction equals zero," we see that the virtual work of the impressed forces can be replaced by the negative virtual work of the forces of reaction. Hence the principle of virtual work can be formulated in the following form, which we shall call Postulate A:

"The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints."

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's principle. Since all the fundamental variational principles of mechanics, the principles of Euler, Lagrange, Jacobi, Hamilton, are but alternative mathematical formulations of d'Alembert's principle, Postulate A is actually the only postulate of analytical mechanics, and is thus of fundamental importance.

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