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Consider the following question in classical mechanics

Are Newton's Second Law, Hamilton's Principle and Lagrange Equations equivalent for particles and system of particles?

  • If Yes, where can I find a complete proof?
    Are there certain conditions for this equivalence?

  • If No, which one is the most general one?

I couldn't find the answer of my question in the books since there are lots of sentences and no clear conclusion! Or at least I couldn't get it from the books! Maybe the reason is that physical books are not written axiomatically (like mathematics books). The book which I had my focus on was Classical Mechanics of Herbert Goldstein.

\begin{align*} \text{Newton's Second Law},\qquad\qquad &\mathbf{F}_j=m_j\mathbf{a}_j,\qquad j=1,\dots,N \\[0.9em] \text{Lagrange's Equations},\qquad\qquad &\frac{d}{dt}\frac{\partial T}{\partial\dot q_j}-\frac{\partial T}{\partial q_j}=Q_j,\qquad j=1,\dots,M \\ \text{Hamilton's Principle},\qquad\qquad &\delta\int_{t_1}^{t_2}L(q_1,\dots,q_M,\dot q_1,\dots,\dot q_M,t)dt=0 \end{align*}

where $N$ is the number of particles and $M$ is the number of generalized coordinates $q_j$. Interested readers may also read this post.

Qmechanic
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Hosein Rahnama
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2 Answers2

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The earlier formulation of this question was quite broad. This answer is constructed as as a broad response within classical$^{\dagger}$ theories with some hopefully helpful navigation points:

  1. On one hand, the stationary action principle (= Hamilton's principle) and the Euler-Lagrange equations make sense far beyond the realm of Newtonian mechanics, e.g. in field theory or relativistic point mechanics.

  2. On the other hand, there are dissipative systems in Newtonian mechanics that have no action formulation, see e.g. this Phys.SE post.

  3. One may show that broad classes of Newtonian systems satisfy d'Alembert's principle, such as, e.g. rigid bodies, see this Phys.SE post.

  4. For the validity of d'Alembert's principle, see this & this Phys.SE post.

  5. One may show that d'Alembert's principle leads to Lagrange equations, cf. e.g. this Phys.SE post.

  6. Note that Lagrange equations are more general than Euler-Lagrange equations, cf. e.g. this Phys.SE post.

Within Newtonian mechanics, a comparison of various formulations is also discussed in this Phys.SE post.

$^{\dagger}$ By the word classical theories we will mean non-quantum theories, i.e. $\hbar=0$.

Qmechanic
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The equivalence of Newtons 2nd law with Hamilton's principle and Lagrange's equations means that you can (mathematically) derive Hamilton's principle and Lagrange's equations from Newtons law, and conversely that you can derive Newtons law from Hamilton's principle and Lagrange's equations.

First, the variational Hamilton's principle of stationary action is equivalent to the Euler-Lagrange equations (Lagrange equations of second kind)Hamilton's Principle, i.e., each follows from the other. Second, from Newtons laws follow the Lagrange equations. On the other hand, it can be easily seen that Newtons law follows from the Lagrange equations for cartesian coordinates.See e.g.Equivalence Newton and Lagrange

Thus Newtons law, Hamilton's principle and Lagrange's equations are equivalent, because they can mutually can be derived from each other. However, these equivalences might be restricted to certain conditions, like, e.g., assumption of conservative forces derived from a potential while the validity of the Lagrange equations or Hamilton's principle might be more general.

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