What is the bare Minimum Lagrangian\Hamiltonian mechanics

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Discussion Overview

The discussion revolves around the prerequisites for understanding general relativity (GR), specifically the necessity of knowledge in Lagrangian and Hamiltonian mechanics. Participants explore the relevance of action principles in the context of GR and the depth of understanding required for an introductory grasp of the theory.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses uncertainty about the depth of understanding needed in Lagrangian mechanics before studying GR, noting familiarity with basic concepts but not with non-linear dynamics or Hamiltonian mechanics.
  • Another participant argues that knowledge of Lagrangian mechanics is not necessary for understanding GR, emphasizing that the derivation of Einstein's equation from an action principle is not typically included in introductory courses due to its complexity.
  • A different participant acknowledges the common belief that physical theories often have an action formulation, expressing relief at potentially avoiding extensive reading of a classical field theory book.
  • Another participant counters that many GR texts, including older ones, do derive field equations from an action principle, suggesting that while it may be a side topic, it is still relevant and can be introduced without major loss of understanding.
  • This same participant notes that even early texts introduce matter and electromagnetic fields using Hamiltonian mechanics to derive the stress-energy tensor.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the necessity of Lagrangian mechanics for understanding GR. Some argue it is not essential, while others maintain that it plays a significant role in the derivation of field equations.

Contextual Notes

Limitations include varying definitions of what constitutes a necessary understanding of Lagrangian and Hamiltonian mechanics, as well as the differing emphasis on action principles across various texts and courses.

cpsinkule
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I am currently trying to self learn general relativity. I understand you need knowledge of an action principle, but what I am not so sure of is how deep of an understanding I need. I currently have a book on classical field theory and I am at the point of basic Lagrangian's and holonomic systems. I can compute lagrangians and I am very familiar with the E-L equation, but I have yet to come to non-linear dynamics and hamiltonian mechanics. My question boils down to how much do I need for an introduction to G-rel?
 
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GR is the theory that says that spacetime is a smooth manifold with a metric that satisfies Einstein's equation. A derivation of Einstein's equation from an action principle wouldn't even be a part of the theory. That derivation is also not included in most introductory courses, because it requires an understanding of integration of manifolds. So I'd say you don't need any knowledge of Lagrangian mechanics at all.
 
thanks so much, I was worried id have to drudge through the other 600 pages of this book, however I was under the impression that any physical theory usually has a side-kick action formulation (i have several books on grel, but wald's was the only one to mention it)
 
Well, I have GR books ranging from 1921 (two of them) to 1973 (MTW). All describe derivation of the field equations from an action principle. I would agree that in most of this is dealt with as a side topic that could be dispensed with without major loss. However, even an old, first course like Bergmann's 1942 book gets the vacuum field equation without variation, but introduces matter and EM fields via Hamiltonian to get appropriate stress energy tensor.
 

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