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When learning Lagrangian mechanics in a standard mechanics course, it is typically introduced as an alternative formulation of classical mechanics which can be derived from Newtonian mechanics. Despite its limitations, Newtonian physics has the benefit of being more intuitive so this is a natural progression. It is not at all clear why action should be minimized, or why the Lagrangian should have the form it does without seeing these derivations.

But once you get to quantum mechanics and general relativity, our Newtonian foundations get pulled out from under us, and is reduced to a special case. Despite this we retain the Lagrangian (or Hamiltonian) formulation, implying it is somehow more foundational. My question is what reason do we have for thinking that this formulation is generally true, given that it was originally created for a special case? Obviously it works very nicely, so it must be valid in those particular cases. But why should action always be stationary? Its not as if you can measure this quantity to check.

Qmechanic
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Alexander Shook
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2 Answers2

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There are a lot of things in physics that we use because they work. Lagrangian mechanics gets used a lot because it works a lot. It's just a tool, like addition or subtraction or Fourier transforms.

I recommend Vertasium's video, The Closest We've Gotten to a Theory of Everything. He did a great job of showing how these concepts evolved and got selected over time. It's a good background to start from.

In my own research into the history of Lagrangian mechanics, I note that Lagrangian mechanics did not start out as an attempt to solve everything. Lagrange used it to solve a few problems like the brachistochrone curve. His approach was more elegant than past attempts. In doing so, he constructed the calculus of variations and he and Euler developed the ever important Euler-Lagrange equation:

$$\frac{\partial L}{\partial q} - \frac{d}{dt}\frac{\partial L}{\partial \dot q} = 0$$

This was an equation that held true for every stationary path.

It was Hamilton that noticed that if you use the now famous Lagrangian $L=T-V$, the Euler-Lagrange equation happens to give you $F=ma$, which means any system whose behavior is defined by $F=ma$ can be rephrased as a variational problem finding a stationary path.

After that, Lagrangian mechanics became popular because it had some very nice properties, particularly that it made it very easy to transform coordinate systems and to use curvilinear coordinates. This decoupled us from the cartesian coordinate systems needed to make Newton's laws tractable (they can be written in curvilinear coordinate systems, but they're much uglier than the elegant $F=ma$).

This made Lagrangian mechanics a popular go-to tool. If I'm exploring a new physical property, I'm going to try my existing useful tools first before branching out into developing new tools. And Lagrangian mechanics has held up astonishingly well. The assumptions of the universe around us (such as conservation of energy) play nicely into the Lagrangian formalism.

Thanks to Cleonis's comment below, I can end this the way I intended, with a quote from the forward of "Applied Differential Geometry":

To all those who, like me, hve wondered how in hell you can change q-dot without changing q

Cort Ammon
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    About the origin of that 'for all those who wondered' quote: there is a 2011 answer by contributor Kostya that gives the origin of that quote: 'Applied Differential Geometry' by William L. Burke. A screenshot of the dedication is uploaded as image. So you have the option of re-using that image. – Cleonis Jun 21 '25 at 06:33
  • About wondering why $q$ can be varied without varying $\dot q$. There is a related issue, discussed by Grant Sanderson in his 2017 'Essence of Calculus' series. Part 2 of that series is titled: The paradox of the derivative. Quote: "You see, it's common for people to say that the derivative measures an instantaneous rate of change, but when you think about it, that phrase is actually an oxymoron. [...] when you blind yourself to all but just a single instant, there's not really any room for change." It seems: differentiation is itself already paradoxical. – Cleonis Jun 21 '25 at 07:05
  • Take the case of differentiation of the volume of a square base rectangular solid with length $x$ and basis area $y^2$. Let the volume be $V$. When $x$ and $y$ are independent of each other the expression we get for differentiation of the volume with respect to $x$: $\frac{\partial V}{\partial x}=y^2$ and with respect to $y$: $\frac{\partial V}{\partial y}=2y \cdot x$. Now define a variable $u$ with $x=u$ and $y=2u$, so that $x$ and $y$ are locked to each other. The thing is: in the locked case the partial differentiation expressions are the same as in the independent case. – Cleonis Jun 21 '25 at 12:33
  • I think what bugged me was that $\dot q$ is only associated with dq/dt on the true (stationary) path. At all other points not on that path, it's whatever is needed to make the math work. – Cort Ammon Jun 21 '25 at 14:38
  • I think that whether being on the stationary path or not being on the stationary path is collateral rather than the core of the issue. The core, in my opinion, is that it is a property of differentiation generally. That is: in my opinion the scope isn't specifically calculus of variations; the scope is differentiation generally. It just so happens that you have run into it in the context of application of calculus of variations (to Hamilton's stationary action). – Cleonis Jun 21 '25 at 15:06
  • For completeness: to verify that William Burke is indeed the source: while the Google books preview skips the page with the dedication, the Amazon preview shows all of the first pages. As of this writing (June 2025) the Amazon preview shows the page with the 'to all those who have wondered' line. – Cleonis Jun 21 '25 at 15:13
  • I think we have to differentiate between calculus of variations and Lagrangian mechanics. If you want to minimize a functional, then yes the Euler-Langrange equation is how you do that. It follows from geometry. But if you claim that systems evolve according to the principle of least action, that is a physical claim. The original formulation was fully equivalent to Newtons laws, so there is no new claim. But in the modern context it has to be assumed at the beginning. – Alexander Shook Jun 21 '25 at 16:14
  • About your remark in the answer: "This decoupled us from the cartesian coordinate systems needed to make Newton's laws tractable". It could be that you are under a impression that the Newtonian formulation is limited to using cartesian coordinates. The Newtonian formulation of mechanics generalizes to using generalized force. For reference: discussion by Richard Fitzpatrick: Generalized Forces Example: the product of torque and corresponding angle has dimensions of work, (while torque itself doesn't have dimensions of force) – Cleonis Jun 21 '25 at 17:37
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We aren't born knowing that mechanical systems minimize an action or obey Newton's laws. When someone says Newton's laws seem more intuitive, this is probably because he or she has been seeing them for years before the Lagrangian is introduced. A high school which isn't afraid of an integral sign might teach Newton for the first time in term 1 and Lagrange for the first time in term 2. And then there might be students thinking that both previously unfamiliar concepts are equally intuitive.

But this is hypothetical. What I want to object to is the statement that you can't measure the action. With a classical system, you can measure its full trajectory. This is enough information to deduce the equations of motion and then either find a corresponding action or prove that one does not exist. I think it's useful to reinterpret all pre-industrial results about rigid bodies, planetary motion and so on in this way. In each case, people were doing experiments to determine the Lagrangian of a system because there is no way to determine it otherwise.

It is liberating to think of the Lagrangian as a fundamental experimental result and not "kinetic minus potential energy" (except when it's not which causes endless confusion). If it happens to be a function of $\dot{q}$ minus a function of $q$ with the $\dot{q}$ part being quadratic then Lagrangian mechanics after the fact will show you that it's valid to think of those functions as being kinetic and potential energy respectively. The repeated appearance of this splitting in systems we look at is simply a statement of reductionism since composite systems are able to retain some of the most basic properties of the individual particles they contain.

To emphasize

All models we use in this business are models that we have because of experiment. Measuring $q$ at one time and $\dot{q}$ at one time is not enough to determine an energy or an action when you don't already have a model but this doesn't mean that no measurement can determine both things. Instead, you have to measure $q$ and $\dot{q}$ at many times until you get a sense of the equations of motion that govern them. We have been trained to think of this as a way to get $K$ and $U$ from the ground up and then take $L = K - U$. This is bad. The above process is actually enough to get $L$ from the ground up and it is then clear how to get $K$ and $U$ (or prove that they don't exist) from pure thought.

Connor Behan
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