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voko
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The Lagrangian you got in polar coordinates can be simplified before you compute derivatives. Do that properly.
Invariant quantities of a lagrangian?
- Context: Graduate
- Thread starter KleZMeR
- Start date
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Discussion Overview
The discussion revolves around identifying invariant quantities in the context of a given Lagrangian, exploring the theoretical foundations of invariance, conservation laws, and the implications of symmetries in Lagrangian mechanics. Participants express interest in both the conceptual understanding and specific applications related to kinetic and potential energy in Cartesian coordinates.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant suggests that invariant quantities might be those that do not depend on position or time, but acknowledges this could be an oversimplification.
- Another participant emphasizes the need to define "invariant" and connects it to time derivatives in the context of Lagrange equations.
- Some participants propose that conserved quantities arise from symmetries in the Lagrangian, specifically mentioning energy and momentum conservation.
- There is a discussion about the specific problem involving a Lagrangian with kinetic and potential energy, where one participant expresses confusion about identifying invariant quantities.
- Another participant refers to Landau's mechanics, suggesting that symmetries lead to conservation laws and that the term "invariant quantities" may not be accurately used.
- One participant expresses uncertainty about the meaning of "invariant quantities" and questions how to derive them without explicit symmetry transformations.
- Participants discuss the explicit transformations related to rotational symmetry and how they relate to the invariance of the Lagrangian.
- There is a request for clarification on the theoretical foundations behind certain steps in the derivation of constants of motion.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the definition of invariant quantities or how to identify them in the given context. Multiple competing views on the interpretation of invariance and conservation laws remain evident throughout the discussion.
Contextual Notes
Participants express limitations in their understanding of the relationship between symmetries and invariant quantities, as well as the need for explicit functions to determine motion. There are unresolved questions regarding the application of the Euler-Lagrange equation and the treatment of partial derivatives in the context of symmetry transformations.
Who May Find This Useful
This discussion may be of interest to students and practitioners of classical mechanics, particularly those exploring Lagrangian mechanics, conservation laws, and the implications of symmetries in physical systems.
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