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.

**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!**