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Are q and q' dependent variables in Lagrangian or not?

  1. Jan 27, 2016 #1
    Hi.

    I have thought that the variables q and q' in L = L(q, q') are independent. (q' = dq/dt)

    Of course q and q' are functions of time t , but they are only dependent in terms of t .

    However, in the sight of general(or abstract? I mean, not specific) functional L(q, q'),

    q and q' are just independent variables of L .

    Only after L is determined by using calculus of variations, they are dependent.

    Here is more detailed logic of mine.
    -------------------------------------------------------------------------------
    1a) Think about the coordinates (x, y, z). All x, y and z are independent thus they can make the coordinates. And there are infinite possible functions z = f(x,y).

    2a) Let's choose a specific function among those infinite possible functions, say, a line (x, y, z) = (s, s, s)
    where s is a parameter. Then, by considering this function, x and y are no longer independent because
    x=y on this line.

    1b) There is a coordinates (q, q', L). There are infinite possible functions L = L(q, q').
    And We don't know what path the particle will follow. When q=1, q' can be 1 or 100. When q=2, q' can be 1.1 or 500. This means that there are infinite possible paths, or physical laws.
    On each path q and q' are dependent. But considering all the paths, eventually it means q and q' are independent variables.

    2b) In 2a, we chose a specific function. Here, in Lagrangian, it is same to adding a condition, the principle of least action, which yields the Euler-Lagrange equation.
    By this condition, a specific Lagrangian L is chosen, which makes q and q' dependent.
    And this also means that among those infinite possible paths, we chose a specific path, which can be said that we chose a specific law, Newton's 2nd law, among those infinite possible physical laws.
    -------------------------------------------------------------------------------

    Is this logic wrong or not?

    Please point out where I am wrong. Thank you.
     
  2. jcsd
  3. Jan 27, 2016 #2

    BvU

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    Logic is just fine. Lagrangian has independent variables ##q## and ##\dot q## (and possibly others). Euler-Lagrange equations establish a relationship and you get the equations of motion.

    Compare the mechanical energy of a mass on a spring: E is a sum of spring energy, gravitational potential energy and kinetic energy. So the independent variables are ##y## and ##\dot y##.
     
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