[Math] Hamiltonian, Lagrangian and Newton formalism of mechanics

Question

If my thinking is wrong please let me know. I have little knowledge on beyond-college physics.

For research purposes, I read a few introductions to these three formalisms of classical mechanics [1,2,part of 5](Hamiltonian, Lagrangian and Newton formalism).
It seems to me that Newton formalisms is incapable of describing a quantum system(The uncertainty principle is not well addressed) while both Hamiltonian and Lagrangian are capable of describing a quantum system.So the Hamiltonian and the Lagrangian is simply a more general framework including Newton.

As Ben and Tobias pointed out in their answers, these three formalisms are equivalent their relationship are not simply inclusion but complementary. There are situations one of three systems that are particularly suitable to use.

For my purposes, I see there often seems to be one-to-one correspondence in-between Lagrangian construction and Hamiltonian construction for dynamic systems(OR in the simpler case the derived differential equations w.r.t. a chosen coordinate frame) and (sympletic) geometry when the concern is the dynamics on the manifold, say [3,4].

A curious question in my mind is that if these two physical formalisms are equivalent, then why only Hamiltonian is studied in most cases(say geometric analysis and sympletic geometry)?

(1)What is its(The Hamiltonian formalism) superiority over Lagrangian from mathematical perspective?
Does it lead to a richer structure or more natural structure?(by structure I mean manifold structure over which the system is defined)

(2)Moreover, is there any example that is easily formalized in
Hamiltonian formalism but too complicated/unnatural to formalize in
Lagrangian?

Any comments or reference is appreciated! Please add some reference in your answer to support your claims, thanks a lot!

Reference

[1]http://www.macs.hw.ac.uk/~simonm/mechanics.pdf

[2]http://image.diku.dk/ganz/Lectures/Lagrange.pdf

[3]Boundary conditions and the relationship between Hamiltonian and Lagrangian Floer theories

[4]Koon, Wang Sang, and Jerrold E. Marsden. "The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems." Reports on Mathematical Physics 40.1 (1997): 21-62.

[5]Meyer, Kenneth, Glen Hall, and Dan Offin. Introduction to Hamiltonian dynamical systems and the N-body problem. Vol. 90. Springer Science & Business Media, 2008.

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Motivation of OP

And as for my motivation, I primarily want to figure out why [Mumford&Michor] proposed Hamiltonian approach in (which looks not quite natural to me at first since they are just laying down a framework for dynamics on $Cur(\mathbb{R}^2)$.)

[Mumford&Michor]Michor, Peter W., and David Mumford. "An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach." Applied and Computational Harmonic Analysis 23.1 (2007): 74-113.

From these nice answers below, I feel that the most convincing reason why Hamiltonian approach is preferred in [Mumford&Michor] is that
(1)the space of curves $Cur$ involves $\mathrm{Diff}(C)$ and Hamiltonian formalism is a convenient formalism to incorporate these transformations.
(2)And the infinitesimal generators of $\mathrm{Diff}(C)$ can be used to describe the velocity field along the curves in $Cur$.
(3)Combined with Tobias' answer, [Mumford&Michor] also said symmetry is a reason for Hamiltonian. Now I understand the sentence better.

…The Hamiltonian approach also provides a mechanism for converting
symmetries of the underlying Riemannian manifold into conserved
quantities, the momenta.[Mumford&Michor]

(Unless the authors disagree 🙂

Thanks again for everyone's input, I learnt a lot from you and willing to learn more!

Answer 1

Each of the different formalism of classical mechanics has its advantages and disadvantages. However, in the end all three frameworks tend to be equivalent, and thus the following list is very subjective and there may be exceptions to all points.

Newton:

• Easily includes dissipative systems and is the only formalism that can handle non-potential forces.
• Taken literally only applies to systems without constraints (but can be extended to include some constraints using d'Alembert's principle; nonetheless, it is usual not the right framework to handle constraints).
• Doesn't say much about conserved quantities for symmetries.

Lagrangian:

• Highlights the variational principles underlying classical mechanics by its direct connection with the principle of least action
• as such, it allows for direct extensions to relativistic systems and field theories.
• Noether's theorem provides a direct link between symmetries and conserved quantities.

Hamiltonian:

• There are systems described by symplectic manifolds which are not cotangent bundles and thus have no Lagrangian equivalent (for example, internal degrees of freedom like spin)
• The framework of "momentum maps" is often superior to Noether's theorem and symmetry reduction is better understood in the symplectic case.
• Breaks the covariant nature of the Lagrangian formalism and thus has problems with relativistic theories.
• The Hamiltonian is always conserved along the dynamical evolution. This has advantages in e.g. bifurcation theory but makes it (nearly) impossible to discuss dissipative systems.
• The variational nature is hidden (or on a more positive note encapsulated in the symplectic form).
• Symplectic manifolds are interesting in its own right, even without a Hamiltonian (this is the advantage of having the additional structure given by the symplectic form).

Note that the equivalence of the Lagrangian and Hamiltonian view depends on the Legendre transformation. Sometimes, especially when constraints are present, the Legendre transformation is not an isomorphism and thus both formalism might be not equivalent!