Yes, Lagrange's mechanics is used widely in practical application.
I will get back to that, first some historical remarks.
Joseph-Louis Lagrange developed his mechanics, as laid down in the 'Mechanique analytique', by achieving the following two things:
-Systematic use of representing physics taking place in terms of interconversion of potential energy and kinetic energy
-Systematic use of generalized coordinates
Both of those ideas were used before, but opportunistically, as one-off solutions for specific cases. It was Lagrange who recognized the general applicability of those ideas.
Joseph-Louis Lagrange did not use the Euler-Lagrange equations in the context of mechanics. The historian of science Rene Dugas writes about that. In his work on history of mechanics Rene Dugas gives an extensive quote of Lagrange commenting on Maupertuis' least action. Lagrange was of the opinion that Maupertuis' least action wasn't particularly interesting or relevant.
It was in 1834 that William Rowan Hamilton introduced the concept that nowadays is referred to as Hamilton's stationary action. (In Hamilton's original paper he suggested the qualification 'stationary', since he recognized that that is a better description of the mathematics.)
That is: it was Hamilton who introduced the concept of casting the formulation of equation of motion in terms of solving a calculus of variations problem.
My point is:
By the time that William Rowan Hamilton introduced Hamilton's stationary action the mechanics of Joseph-Louis Lagrange had been up and running for several decades.
The properties of Lagrange's mechanics that make it powerful and expressive are independent from the later work of William Rowan Hamilton; those properties had already been established, and had been put to use.
Actual usage
To your question specifically:
Usage for complicated physics problems:
No doubt many physics simulation software packages implement the calculations in terms of interconversion between potential energy and kinetic energy.
Note that in classical mechanics that is what the Euler-Lagrange equation does: in the context of classical mechanics the Euler-Lagrange equation is populated with expressions for potential energy and kinetic energy.
(There is a discussion by me of how Hamilton's stationary action expresses the relation between potential and kinetic energy)
As to generalized coordinates:
The thing is: before the developement of electronic computers: being able to convert a physics problem into a mathematically tractable form was key. If the computing power that you have is your brain then you have to work smarter, not harder. When working with pen and paper: well chosen generalized coordinates can be key to making a problem tractable.
But when an electronic computer is used to do numerical analysis then I think there will be less difference in efficiency between expressing the motion in terms of generalized coordinates, or just ploughing on with cartesian coordinates.
Kinetic energy and Pythagoras' theorem
Let me write some more about the power of expressing physics taking place in terms of potential energy and kinetic energy.
Whatever coordinate system you are using (cartesian coordinates, or any set of generalized coordinates), the directional information that you need in order to solve for the equation of motion is in the expressions for the potential energy. To recover the force vector from the potential energy you compute the gradient of the potential energy.
As we know: the expression for the kinetic energy is such that the directional information of the velocity vector is discarded. As we know: to compute the kinetic energy the computation uses only the magnitude of the velocity vector, squaring it.
We can afford to discard the directional information of the velocity vector because the necessary directional information is expressed by way of the potential energy being a function of position.
Discussion of kinetic energy:
The case of uniform circular motion.
We have the option of decomposing that uniform circular motion in two motion components, perpendicular to each other. As we know those motion components are both harmonic oscillation.
Each component motion proceeds according to $F=ma$
Each component motion has a corresponding component kinetic energy, we can refer to those as x-component and y-component.
$ E_{K,x}$
$ E_{K,y}$
We can express the component kinetic energy, and we can also express the combination of those to component energies: the resultant kinetic energy.
To compute the resultant kinetic energy we apply Pythagoras' theorem to find the resultant velocity $v$:
$$ (v_{resultant})^2 = (v_{x})^2 + (v_{y})^2 \tag 1 $$
In my opinion this is the biggest windfall in the history of physics: kinetic energy and Pythagoras's theorem slotting in with each other.
It is because kinetic energy and Pythagoras' theorem slot in with each other that we can treat kinetic energy as a scalar quantity, instead of having to keep track of it as a vector quantity.
That is what Joseph-Louis Lagrange capitalized on when he developed his mechanics.
David R. Wilkins, Trinity College Dublin, has made new transcriptions of the two essays by William R. Hamilton on Dynamics.
The titles of the essays are:
- On a General Method in Dynamics
- Second Essay on a General Method in Dynamics
William R. Hamilton's works on dynamics, Trinity College Dublin
Direct links to PDF's:
On a General Method in Dynamics
Second essay on a general method in Dynamics
