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Homework Help: Physical Interpretation of point transformation invariance of the Lagrangian

  1. Jan 13, 2013 #1
    1. The problem statement, all variables and given/known data
    The problem asked us to show that the Euler-Lagrange's equations are invariant under a point transformation q[itex]_{i}[/itex]=q[itex]_{i}[/itex](s[itex]_{1}[/itex],...,s[itex]_{n}[/itex],t), i=1...n. Give a physical interpretation.

    2. Relevant equations
    [itex]\frac{d}{dt}(\frac{\partial L}{\partial \dot{s_{j}}})[/itex]=[itex]\frac{\partial L}{\partial s_{j}}[/itex]

    3. The attempt at a solution

    I proved the invariance.
    I am stumped with the physical interpretation. Except for the fact that the E-L equations are invariant when we change coordinates pointwise, I don't see any other physical interpretation. But this answer seems just repeating their question. :confused:
  2. jcsd
  3. Jan 13, 2013 #2
    Are coordinates something physical to begin with?
  4. Jan 13, 2013 #3
    No they are not. They are something we make in order to do the calculations.
    It was just a wild guess really.
  5. Jan 13, 2013 #4
    That's what one should expect physically. Now you have proved that the Lagrangian formalism does not require any special coordinates, they all work. What does that mean about the formalism itself?
  6. Jan 13, 2013 #5
    This means that the Lagrangian is independant of the coordinate system. And this makes sense, because a coordinate system is nothing physical and is arbitrary.
    So, this means that the Lagrangian is universal? In other words, it works for any coordinate system.
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