Question

I heard that Lagrange mechanics can be derived from Newtonian mechanics, and Newtonian mechanics can be derived from Lagrange mechanics. I've heard many times that they have equal explanatory power. But I encountered that there is a tricky point in deriving the law of conservation of angular momentum in Newtonian mechanics.

On the other hand, theories that use the principle of least (or stationary) action derive each conservation law from the Noether's theorem by assuming each symmetry.

So, is my understanding appropriate? That is, while Newtonian mechanics itself derives conservation of linear momentum without any additional assumptions, and requires some manipulation to derive conservation of angular momentum, but Lagrangian mechanics cannot derive both without the assumption of symmetry, or derive both if the assumption is made.
Then, it seems that it can be explained by Lagrangian mechanics, but there is something that is not correct in Newtonian mechanics, or it seems that more assumptions are made in Newtonian mechanics than in Lagrange mechanics. Which of the two makes sense?
I'd also like to ask if Newton's third law conflicts with Lagrange mechanics, and if it's possible to find a set of Newtonian laws that have the same explanatory power as Lagrange mechanics. However, I want to know (even short) to just the above two questions.

Answer 1

Lagrangian mechanics and Newtonian mechanics are two different approaches to describing the motion of objects. While Newtonian mechanics is based on Newton's laws of motion, Lagrangian mechanics is based on the principle of least action. Both approaches can derive the conservation laws, including the conservation of linear momentum and angular momentum.

Step-by-step explanation:

Lagrangian mechanics and Newtonian mechanics are two different approaches to describing the motion of objects. While Newtonian mechanics is based on Newton's laws of motion, Lagrangian mechanics is based on the principle of least action. Both approaches can derive the conservation laws, including the conservation of linear momentum and angular momentum.

In Newtonian mechanics, the conservation of linear momentum can be derived directly from Newton's laws without any additional assumptions. However, the derivation of the conservation of angular momentum requires some manipulation and additional assumptions.

In Lagrangian mechanics, the conservation laws can be derived from the principle of least action using Noether's theorem, which relates symmetries in the system to conservation laws. To derive the conservation of angular momentum in Lagrangian mechanics, one needs to assume a rotational symmetry in the system.

Regarding Newton's third law, it is compatible with Lagrangian mechanics. Newton's third law states that for every action, there is an equal and opposite reaction. This principle can be incorporated into Lagrangian mechanics to describe systems with interactions.

Both Newtonian mechanics and Lagrangian mechanics have their strengths and limitations. Newtonian mechanics is more intuitive and easier to apply in certain cases, while Lagrangian mechanics provides a more general framework and allows for more elegant derivations of the equations of motion.