The three formalisms of classical mechanics, i.e. the Newtonian, the Lagrangian (analytical mechanics) and the Hamiltonian (canonical formalism) are generally not equivalent to each other -at least not in some strict sense of the word "equivalent" which could be considered valid for the totality of physical systems. But in fact they are, for lots of systems of interest. In order to be more specific, let me try to sketch their logical interconnections:
$\blacktriangleright$ If we start from Newtonian mechanics and the assumptions:
- there are holonomic constraints only, i.e. constraints of the form
$f(x,y,z,t)=0$, which are independent of the velocity.
- arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces, i.e. the constraint forces do no work (this is equivalent to saying that the constraint forces are normal to the hypersurface determined by the constraints at a given instant $t=t_0$).
If the constraints are scleronomic then the actual displacements $d\mathbf{s}$ are virtual displacements $\delta\mathbf{s}$.
($\Sigma$ is the hypersurface determined by the constraints $f(x,y,z)=0$, $\ R \ $ the constraint force, $F$
is the resultant non-constraint force and $\mathbf{s}$ the position vector).
then Newton's $2^{nd}$ law is equivalent to the D'Alembert's principle which in turn implies the Euler-Lagrange's equations of motion i.e. the analytical mechanics formalism.
(Note that, systems sattisfying the second of the above assumptions are sometimes called "mechanical" or "pure mechanical" systems in the literature.
On the other hand, one does not have to go far in order to find systems violating one or both of the above assumptions: rolling without slipping is a common system with non-holonomic constraints and generally systems with resistance forces -various friction forces for example- violate the second of the above assumptions).
The converse implication, that is starting from the Euler-Lagrange equations and deriving D'Alembert's principle and thus Newton's $2^{nd}$ law is relatively straightforward.
$\blacktriangleright$ If we start from a Lagrangian and its Euler-Lagrange equations of motion, under the assumptions:
there are holonomic constraints only
the Lagrangian function is "standard" or "regular" in the sense that $\det\Big|\frac{\partial^{2}L}{\partial\dot{q}_{i}\partial\dot{q}_{j}}\Big|\neq 0$, i.e.the Hessian of the Lagrangian w.r.t. to the generalized velocities is non-degenerate (see: When does a Lagrangian dynamical system have an equivalent Hamiltonian description? for more details on this point)
then we can derive the Hamiltonian and the canonical formalism of Hamilton's equations through a Legendre transformation. Note that Legendre's transformation, transforms functions on a vector space to functions on the dual space. In this case, it transforms the Lagrangian function (on the tangent bundle of the configuration space manifold) to the Hamiltonian function (on the cotangent bundle of the configuration space manifold).
Summarizing the above discussion:
$$
\small{
\left\{
\begin{array}{c}
\text{Newtonian} \\ \text{mechanics}
\end{array}
\right\}
\underset{\begin{array}{c}
\text{pure} \\ \text{mechanical} \\ \text{system}
\end{array}}{\overset{\begin{array}{c}
\text{holonomic} \\ \text{constraints}
\end{array}}{\mathbf{\leftrightsquigarrow}}}
\left\{
\begin{array}{c}
\text{d'Alembert's} \\ \text{principle}
\end{array}
\right\}
\leftrightsquigarrow
\left\{
\begin{array}{c}
\text{Lagrangian} \\ \text{mechanics}
\end{array}
\right\}
\underset{\det\Big|\frac{\partial^{2}L}{\partial\dot{q}_{i}\partial\dot{q}_{j}}\Big|\neq 0}{\overset{\begin{array}{c}
\text{holonomic} \\ \text{constraints}
\end{array}}{\mathbf{\leftrightsquigarrow}}}
\left\{
\begin{array}{c}
\text{Hamiltonian} \\ \text{mechanics}
\end{array}
\right\}
}
$$
Remarks:
(a). in the framework of Newtonian mechanics, forces may depend on positions and velocities but not on accelerations and
(b). throughout the preceding disussion, we consider all non-constraint forces to be derivable from generalized scalar potentials depending on coordinates and -at most- linearly on velocities: $V=U(q_i,t)+A_j(q_i,t)\dot{q}_j$. Such systems are more general than conservative systems and fall into the class of monogenic systems.
Now, regarding your first question: I don't think it would be suitable to speak about "superiority" or "richer structure". From a technical aspect, Euler-Lagrange's equations are a system of $n$, second order, ODE and the Lagrangian function involved "lives" on the tangent bundle while The Hamiltonian formulation comprises of a system of $2n$, first order, ODE, involving the Hamiltonian function which "lives" on the cotangent bundle (of the configuration space manifold).
If we consider the passage to quantum mechanics, then both formalisms are suitable to handle the elementary aspects of the quantisation problem (at both levels, first and second quantization as well): Hamilton's equations have almost the same typical form in quantum mechanics with their classical counterparts, although their interpretation is quite different in the quantum case -but this is an apparently different story. The road to quantisation through the hamiltonian formalism is generally refered to as canonical quantisation while the road through Lagrangian formalism is known as path integral quantization.
Regarding your second question, and since you are asking for references, this article appears to discuss examples of classical Hamiltonian systems possessing no Lagrangian formulation. On the other hand, the Hamiltonian and the Lagrangian formulation of geometrical optics -as has already been mentioned in another answer- is a well known classical system possessing no meaningful description at the level of Newton's laws.
From one point of view, one may say that the Hamiltonian formalism is "wider" in the sense that it includes several systems of ODE not being directly or obviously related to classical mechanics (an example is the lagrangian/hamiltonian description of Maxwell's laws of the electromagnetic field) or to physics at all. In practise, the hamiltonian is frequently considered to be some abstract function while the Lagrangian is more intimately related to the concepts of kinetic and potential energy of some system. I think that you might also find some interest in the article Classical Mechanics Is Lagrangian; It Is Not Hamiltonian by Erik Curiel.
However, from another point of view, Newtonian formulation is "wider", in the sense that if one is to consider, frictions, dissipative forces, non-holonomic constraints etc then -although there are various formal generalizations of Lagrange's and Hamilton's equations in order to deal with such systems- usually one turns to the fundamentals that is Newton's laws.
So, concluding, i agree that there are no simple proper subset relationships between these three formulations of classical mechanics.
Some further References:
- Mathematical methods of classical Mechanics, by V.I. Arnold, and
- A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, by E.T. Whittaker,
are invaluable -imo- sources, delving into the foundations, not trying to hide the pitfalls under the carpet, with an eye on details (either in the computations or the arguments).
- This: https://physics.stackexchange.com/q/89035/130499 question in physics.stackexchange has a natural overlap with the OP and has lots of answers. You might be interested in taking a look there as well.
P.S.: One last thing: the above disussion and the diagram provided, aimed at emphasizing the logical interconnections of these three formalisms and the usual assumptions upon which one is being derived from or imply the other(s). However, it should be noted that both the Euler-Lagrange's and the Hamilton's equations, can be derived -each one of them seperately and independently of Newton's laws- in an "axiomatic" (or should I say "ad hoc" ?) manner through suitable variational principles: these are the Hamilton's principle (from which euler-lagrange's eqs are derived) and the modified Hamilton's principle (from which the canonical eqs of hamiltonian mechanics can be derived). They both determine the corresponding equations of motion and thus the evolution of the system, through the demand that the trajectories are such that the corresponding action-functional $S$ gets a stationary value (its $\delta$-variation becomes zero):
$$\delta S=0$$
In the former case $S=\int_{t_1}^{t_2} Ldt$ and the principle is applied in the configuration space, while in the later case $S=\int_{t_1}^{t_2} (\dot{q}p-H)dt$ and the corresponding principle is applied in the phase space.
It is actually this possibility of independent foundation of these formalisms that inspired/enabled the extension of these formalism to non-mechanical systems such as classical fields.
