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Arnold's book “Mathematical methods of classical mechanics” develops the standard material on mechanics (e.g. the 3 Newton’s laws and the gravity law etc.). But what differs it from all other textbooks on the subject I know is an attempt to connect this material to mathematical structures actively studied by mathematicians. For example he introduces the Hamiltonian equations on $\mathbb{R}^{2n}$ which are traditionally important in physics. Then he introduces symplectic manifolds, and generalizes the Hamiltonian equations to arbitrary symplectic manifolds.

I am wondering if this generality is useful for mechanics? Another question, whether the Hamiltonian equations on abstract symplectic manifolds are considered to be a part of classical mechanics nowadays?

YCor
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user174848
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    I am aware of that question. It is useful, but my question is more specialized. – asv Feb 25 '21 at 14:16
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    Even if you start doing classical Hamiltonian mechanics in $R^{2n}$ you may easily end up in some more exotic symplectic manifold if you start to quotient out the symmetries. By the way, the first symplectic manifold to be considered was not $R^{2n}$, but the space of oriented straight lines in $R^3$ (Hamilton's paper on systems of rays). – alvarezpaiva Feb 25 '21 at 16:05
  • Just to add to alvarezpaiva's comment: 1. when you look at a constrained system you end up working on a symplectic quotient and this is often nonlinear; 2. symplectic quotients also arise in gauge theory (e.g. moduli space of flat connections). – Jonny Evans Feb 25 '21 at 19:32
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    Yes it is useful to classical mechanics. Arnold's point is that a classical mechanics' problem can be translated in a symplectic geometry problem. If you limit yourself to $R^{2n}$ you can study local properties of the system (eg. what happens near an equilibrium point but you may loose the global properties. For example if you study a spherical pendulum your configuration space wouldn't be $R^{2n}$ so you need to consider more general manifolds. Since any abstract manifold can be realized as a submanifold of $R^N$ you don't need abstract manifolds. continues--- – Overflowian Feb 25 '21 at 23:44
  • People deal with them sometimes for doing useful constructions but mostly because it's a beautiful formalism.

    Also another point in adopting the symplectic manifold pov is that is perfect to talk about integrable system -which is now a geometric global property-.

    Unfortunately quantum mechanics kind of lacks this beautiful geometric background. It is inherently Euclidean. And trying to bridge between classic mechanics and QM brings us to the problem of geometric quantisation (but is not needed for QM to work).

     – Overflowian Feb 25 '21 at 23:53

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The difference between mathematics and mechanics is blurred here. Lagrangian and Hamiltonian formulations of mechanics were introduced by mathematicians who had specific, practical problems in mind, namely problems of celestial mechanics. In particular the 3-body problem which had a very concrete practical application (prediction of the Moon motion).

Amazingly, these generalized formulations of classical mechanics also found applications in quantum mechanics and relativistic mechanics (as well as fluid mechanics and statistical mechanics). So in some sense, most of physics, not only classical mechanics, can be formulated in these terms.

Alexandre Eremenko
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  • Do you mean that Hamilton equation on general symplectic manifolds is useful for quantum and relativistic mechanics? A reference to a concrete application would be helpful. – user174848 Feb 25 '21 at 16:59
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    Any book on quantum mechanics, for example Faddeev and Yakubovski. In fact Hamiltonian mechanics played an important role even in the creation of quantum mechanics. – Alexandre Eremenko Feb 25 '21 at 17:04
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    @AlexandreEremenko : The books on QM I am familiar with (3rd vol. of Landau-Lifshitz, Schiff) do use Hamiltonian formalism, but only on $\mathbb{R}^{2n}$, not on general symplectic manifolds. This is the point. – asv Feb 25 '21 at 17:25