I recently started reading about Lagrangian Mechanics. I observed that it uses some basic expressions that are derived by taking Newton's laws of motion as fundamental such as kinetic energy, potential energy, momentum, etc.

So how can Lagrangian even be fundamental if it uses expressions that are based on some other fundamentals?

Also, can Lagrangian be used to solve any of the problems out there in mechanics easily?


[Lagrangian mechanics] uses some basic expressions that are derived by taking newtons laws of motion as fundamental such as kinetic energy, potential energy, momentum, etc.

That is a valid viewpoint. However, you could also say, with equal validity, that Newtonian mechanics, taken around Newton's laws of motion, uses some basic expressions that are taken from Lagrangian mechanics, such as kinetic energy, potential energy, momentum, etc.

Where the Newton approach is valid, it is exactly equivalent to Lagrangian mechanics. This means that neither is "more fundamental" than the other in such a setting.

However, Lagrangian mechanics is more general, because it describes a wider range of possible dynamics, and it is much cleaner to work with when dealing e.g. with systems under constraints, which are extremely messy (or impossible) to work with in the Newtonian approach.

Which is to say:

Also can Lagrangian be used to solve any of the problems out there in mechanics easily?

very much so. Go to the problems section of your textbook on the Lagrangian Mechanics chapter, find a problem near the back of the section, and try to solve it using a Newtonian approach. It will quickly become clear just how useful the Lagrangian approach is.

  • $\begingroup$ Can you give some more examples where Lagrangian might prove to be more useful? $\endgroup$ – Arnav Mahajan Aug 20 '20 at 11:00

Both Newtonian and Lagrangian formalisms say the exact same thing about nature, both represent the same set of laws, which are the laws of classical Newtonian mechanics.

But Lagrangian mechanics can be extremely usefull when compared to Newtonian formalism. For example, the motion of the a simple pendulum is difficult to solve using a Newtonian approach since you have to take into account the weight of the ball and the tension on the string at each instant (which are both vector quantities). While in lagrangian mechanics you only have to define this thing, The Lagrangian, and work with it, which is much more simple since it is just a scalar quantity.

For the pendulum, with the Newtonian formalism you have to solve two second order differential equations (Newton's Second Law)

$ \sum\vec{F}= m\ddot{\vec{x}}$

  • 2 equations because of the two vector components of $\vec{F}$ (assuming we have prior knowledge of the fact that the pendulum swings only in one plane).
  • Second order differential equations because of the double derivative of $\vec{x}$ with respect to time. Making two derivatives over the same variable $t$ can sometimes make the final equation much more difficult to integrate.

Meanwhile with the lagrangian formalism you only need to solve one second order partial differential equation (Euler-Lagrange formula)

$\frac{\mathrm{d} }{\mathrm{d} t}\left( \frac{\partial L}{\partial \dot{\theta}}\right) = \frac{\partial L}{\partial \theta}$

  • 1 equation because you only have one degree of freedom, since you really only care about the $\theta$ angle the pendulum makes with the vertical to describe the entire system, and because $L$ is a scalar quantity.
  • Second order partial differential equation. This can also be an advantage in many situations, since you make the second derivative in $L$ over a different variable (first by $\dot{\theta}$ and second by $t$), which is very easily integrated if the Lagrangian has some particular forms that commonly appear in nature.

These points make for advantages in the computation of motion in Lagrangian mechanics against the Newtonian formalism (even if there are a few scenarios where the Newtonian approach can be easier than the other). In certain systems the Newtonian formalism can even become impossible to compute exactly (an analytical solution is unreachable and you can only hope for a numerical solution) while the Lagrangian formalism allows for an analytical solution.

I would also add that, besides the mathematical advantages, the Lagrangian formalism allows for a better comprehension of the physical behaviour of the system as a whole, while the Newtonian formalism cares about the forces involved in each and every part of the system (for the simple pendulum this is not an issue since it is composed of only one part and the system is the part, but for the double pendulum you would see that difference since the lagrangian of the system is still only one scalar function while the number of objects involved rises to two).

You can also get many insights about the behaviour of the system with the Lagrangian formalism even without solving the Euler-Lagrange equations. You get an idea of the general behaviour, the energy balances and can even constraint the evolution of your system inside the phase space. With the Newtonian formalism you can only start to have some knowledge of the qualitative behaviour of your system after you have solved the equations and have gotten some quantitative knowledge about the exact motion the system displays.

Remember what the core difference between the formalisms is: The concept of energy is not part of the Newtonian formalism. From a Newtonian perspective you can only care about forces (energy can be derived from the behaviour of those forces throught space and time but is not necessary to describe the evolution of the system at all), from a Lagrangian perspective, the force could perfectly be an artifact of the human mind, and the only thing that "really exists" are the amounts of energy and how they are exchanged between kinetic and potential throught space and time.

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    $\begingroup$ You example of a pendulum isn't a great one, since it's relatively straightforward to write down Newton's Laws in polar coordinates with $r = {}$ constant. Then you only get one ODE rather than two, your ODE is exactly the same as the one you get from the Euler-Lagrange equations, and there's no real advantage to the Lagrangian formalism. $\endgroup$ – Michael Seifert Aug 20 '20 at 13:56
  • $\begingroup$ @MichaelSeifert Agree, I should have gone with the double pendulum for that. $\endgroup$ – Swike Aug 22 '20 at 12:51

There's more than one answer because there is more than one "dependency" in physics on Lagrangian and Hamiltonian Dynamics. I will tell you the importance of these subjects within my own interests, which I know best.

If you want to learn quantum mechanics and quantum field theory, many of the concepts came from or directly use lagrangian and/or hamiltonian mechanics. The theories are not formulated in terms of Newtonian mechanics, so in that sense there isn't an equivalence. So to get a feeling of what you are actually doing in those subjects, for the aspects of it that depend on the lagrangian/hamiltonian formalism, you will need at least a "blueprint" understanding of the formalism.

The history of how people figured out / guessed quantum mechanics comes directly through hamiltonian dynamics, for example (poisson brackets). And in Quantum Field Theory/The Standard Model, the formalism directly uses the Lagrangian formalism to derive the equations of motion. There is no Newtonian equivalent that I am aware of*, which would mean expressing QFT in terms of forces. So the equivalence between the two formalisms is only really clear for classical mechanics itself. Later on you basically abandon the Newtonian formalism, at least in the standard treatment of QM and QFT.

*There is a lot of work out there. I cannot rule out that this has been done, but I have not come across it in any books.


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