Invariance of Newton's second law

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SUMMARY

The invariance of Newton's second law, specifically F=dp/dt, under Galilean transformations within a system of variable mass is confirmed through the principles outlined in Goldstein's mechanics text. The discussion emphasizes that momentum, denoted as p, remains invariant (p=p') when forces are derived from a Lagrangian framework, aligning with Hamilton's principle of least action. Additionally, Noether's theorem is referenced as a foundational concept supporting this invariance.

PREREQUISITES
  • Understanding of Newton's second law and its mathematical formulation
  • Familiarity with Lagrangian mechanics and Hamilton's principle of least action
  • Knowledge of Galilean transformations in classical mechanics
  • Basic comprehension of Noether's theorem and its implications in physics
NEXT STEPS
  • Study Goldstein's "Classical Mechanics" for detailed proofs and examples
  • Explore Noether's theorem and its applications in theoretical physics
  • Investigate the implications of variable mass systems in classical mechanics
  • Learn about the relationship between Lagrangian and Hamiltonian formulations of mechanics
USEFUL FOR

Physics students, educators, and researchers interested in classical mechanics, particularly those focusing on the principles of invariance and the mathematical foundations of motion.

Emanuel84
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Is someone able to proove the invariance under Galilean transformations of F=dp/dt within a system of variable mass? In particular is the momentum invariant? i.e. p=p', as Goldstein states? Please answer me! :wink:
 
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Do you have Goldstein? I think he proves this in the text. Online, try looking up Noether's theorem (and in Goldstein, too for that matter)

http://math.ucr.edu/home/baez/noether.html

Note that you need to know that your forces are derived from a Lagrangian to prove this is true, this is equivalent to assuming Hamilton's principle of least action.
 

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