# What exactly is the Action? (Learning lagrangian)

I have been trying to wrap my head around lagrangian mechanics but I find some parts confusing. For example, what exactly is action and why is it defined by the Kinetic energy minus the potential energy? This equation seems weird, wouldn't $E_k + E_p$ make more sense?

Is it some sort of work done? I sort of get that it works and derives newtons equations(?), but I cannot connect it to the real world.

So what exactly is action? Wikipedia does not do a good job of explaining or I just can't grasp it.

The only reason to define things as they are in classical mechanics is that they give rise to the correct equations of motions that can be directly measured and observed. Given a field theory described by $\phi(x)$, its action is defined as $$S(\phi,\dot{\phi})=\int_{\mathcal{D}}d^4x\,\mathcal{L}(\phi(x),\dot{\phi}(x),x)$$ and the equations of motions for the fields are $\delta S=0$. This is an experimental result that can be observed at any level: classical mechanics, optics, electromagnetism, classical field theory, quantum mechanics, string theory and so on and so forth. In the case of a point particle in classical mechanics, the function $$\int d^3x\,\mathcal{L}(\phi(x),\dot{\phi}(x)) = L(\phi(t),\dot{\phi}(t),t)$$ (referred to as the Lagrangian) turns out to be $L=T-V$ because in this way it generates the correct Newton's equations of motion, no other reason than that.

This equation seems weird, wouldn't E_k + E_p make more sense?

Why would that make more sense? That is the total energy of the particle, which is a quite different thing (although it can be related).

Is it some sort of work done? I sort of get that it works and derives newtons equations(?), but I cannot connect it to the real world

There is plenty of literature deriving the Lagrange equations starting from the Newton's laws, passing through the principle of virtual works taking into account constraints and degrees of freedom and so on. Usually the course in analytical mechanics covers extensively such topics; however, the action principle does come from the real world indeed, making use of a little formalism and generalisation.

• So would it be correct to say "objects move through space along the path in which the sum of the incremental differences between kinetic and potential energy at each point of the path is minimized" ? In a sense the sum of incremental Lagrangians? Why does this determine the path? Are conservation laws the root of these differences collectively being minimized? – docscience Nov 21 '18 at 18:59
• Notice that the Lagrangian needs not necessarily correspond to $T-V$: in classical mechanics the above is one of the infinitely many possible Lagrangians. I wouldn't try to phrase physics into English language in order to give it a practical interpretation (but maybe I just don't understand it deeply enough :p). – gented Nov 21 '18 at 20:01
• but isn't all physics practical in the sense of understanding nature? Anything beyond that you are just doing math I believe. "not necessarily correspond to $T - V$ " But if the Lagrangian corresponds to a physical system, then indeed $L = T - V$. Right? Otherwise, again we are just talking math. – docscience Nov 22 '18 at 17:36
• "But if the Lagrangian corresponds to a physical system, then indeed $L=T−V$" not really. The concept of kinetic versus potential energy is a classical concept: take any quantum field theory as an example, where Lagrangian are essentially just any function such that we can derive the equations of motion (provided some constraints hold, of course). – gented Nov 23 '18 at 9:35
• Thanks! I understand ultimately quantum mechanics, a more generalized approach to modeling, predicting physical systems - however without necessarily knowing the inner structure, mechanisms. So given a quantum mechanical system is the Lagrangian unique given known equations of motion? Or does the 'collapse of the wave function', uncertainty allow an infinity of possible Lagrangians? – docscience Nov 25 '18 at 18:47

I'm not aware of a deeper explanation of action from a classical mechanics point of view. In the classical view, the action is just a functional of the trajectory which is minimized (or maximized) at the physical trajectory.

From a quantum mechanical point of view, and particularly the path integral formulation, there is a more physical interpretation: the action is essentially the accumulated quantum phase along a trajectory. Trajectories whose phase (action) is not an extremum (maximum or minimum) destructively interfere with nearby trajectories and so the only surviving trajectory is that with and extremum in the action.

From this point of view, Euler, Leibniz, et al presumably could have postulated quantum mechanics based on the principle of least action, but I'm sure no one would have believed them.

• Fermat's principle gives, in a sense, a first taste of that, does it not? As soon as diffraction was discovered they could have made the big step... but they didn't. Newtonian mechanics was probably so powerful that it seemed to make more sense to try to force everything in a mechanistic framework, instead. Please note how we are trying to do the same thing with gravity and quantum field theory today... QFT is so successful that many people can't even imagine the possibility that it may not be able to explain gravity. – CuriousOne Jul 1 '15 at 5:55