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i) Starting from the variation(s) of the action (or Lagrangian), derive the Euler-Lagrange equation(s). ii) If we assume that the Lagrangian is invariant under a transformation of the coordi- nates, derive Noether's theorem.

i) Starting from the variation(s) of the action (or Lagrangian), derive the Euler-Lagrange equation(s). ii) If we assume that the Lagrangian is invariant under a transformation of the coordi- nates, derive Noether's theorem.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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solve the secon i) and ii) and don't use chatgpt thank you!

We consider a system with two objects moving in one dimension. We assume that the corresponding Lagrangian depends on the positions and on the velocities of the objects. We also assume the Lagrangian has no explicit time dependence. i) Starting from the variation(s) of the action (or Lagrangian), derive the Euler-Lagrange equation(s). ii) If we assume that the Lagrangian is invariant under a transformation of the coordi- nates, derive Noether's theorem. iii) Using Noether's theorem, assuming specifically that the Lagrangian is invariant by translation, find the corresponding conserved quantity (ies). We consider the same situation as in the previous question but with two objects moving in two dimensions. We work under the same assumptions (the corresponding Lagrangian depends on the positions and on the velocities of the objects and has no explicit time dependence). Expanding on your previous answers: i) Derive the Euler-Lagrange equation(s). ii) Assuming that the Lagrangian is invariant by translation, find the conserved quan- tity (ies).
Transcribed Image Text:We consider a system with two objects moving in one dimension. We assume that the corresponding Lagrangian depends on the positions and on the velocities of the objects. We also assume the Lagrangian has no explicit time dependence. i) Starting from the variation(s) of the action (or Lagrangian), derive the Euler-Lagrange equation(s). ii) If we assume that the Lagrangian is invariant under a transformation of the coordi- nates, derive Noether's theorem. iii) Using Noether's theorem, assuming specifically that the Lagrangian is invariant by translation, find the corresponding conserved quantity (ies). We consider the same situation as in the previous question but with two objects moving in two dimensions. We work under the same assumptions (the corresponding Lagrangian depends on the positions and on the velocities of the objects and has no explicit time dependence). Expanding on your previous answers: i) Derive the Euler-Lagrange equation(s). ii) Assuming that the Lagrangian is invariant by translation, find the conserved quan- tity (ies).
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