&gt; The Lagrangian is great for solving many problems. But it is definitely not more general than Newtonian mechanics<p>Sure it is.
It&#x27;s true that for certain systems, you can derive one from the other so in that sense they might be &quot;just different notations&quot; for the same physics (I&#x27;m disregarding non-conservative forces, which aren&#x27;t really considered in fundamental physics).
But the Lagrangian formalism really consists of two parts: Hamilton&#x27;s principle and a choice (postulate&#x2F;guess) of Lagrangian (or Lagrangian density).
When we say that Lagrangian formalism is more general than Newtonian mechanics, it means we can describe physics using the Lagrangian which we can&#x27;t get to via Newton&#x27;s laws.
For example, you would be hard-pressed to derive general relativity from Newton&#x27;s laws, but if you start with the Einstein-Hilbert action, you can derive Einstein&#x27;s field equations.<p>&gt; A better example of a fundamental law would be action-reaction or something.<p>This is another example where the Lagrangian formalism is more general.
In this case, a translation-invariant Lagrangian implies conservation of (canonical) momentum.
But in more complex systems, Newton&#x27;s third law might may fail when canonical momentum is still conserved (the prototypical example is the Lorentz force law).<p>&gt; Using Lagrangians to derive it then begs the question &quot;but why must electrons obey the Lagrangian?&quot;.<p>This isn&#x27;t circular as much it is one less level of indirection.
It&#x27;s like if you say the reason a dropped object accelerates to the Earth is gravity, you have just shifted the question to &quot;why must gravity behave as an inverse squared law&quot;. You can go further and say that Newtonian gravity is not fundamental, but is an approximation of from general relativity, where the dropped object isn&#x27;t really accelerating. Again, the question is shifted to &quot;why is general relativity described by the Einstein field equations&quot;. At each level of reduction you describe one phenomenon by something more fundamental. That you don&#x27;t have a further explanation doesn&#x27;t logically preclude that it&#x27;s more fundamental.