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Partial derivatives in analytic mechanics

  1. Oct 30, 2011 #1
    In ordinary multivariable calculus the following

    situation is common:

    We have some letters, say, x, y, z and call them variables.

    We have some relations, say, there is only one f(x,y,z)=0.

    Then you have to choose your dependent variable and two (three

    variables minus one relation) independent variables. Ordinary

    taylor series up to first order for infinitesimal changes in all

    variables gives f1*dx+f2*dy+f3*dz=0, where f1, f2, f3 are derivatives

    of f which are taken "explicitly", right from f(x,y,z)=0. Now the

    following questions can be answered:
    1) If y is the dependent variable, what is its partial derivative with
    respect to x? the answer is -f1/f2, because you hold z: dz=0, calculate
    dy/dx and formally take the limit.

    2) if z is the dependent variable, what is its partial derivative with
    respect to y? the answer is -f2/f3, procedure being the same.

    Everything works just perfect for calculus and thermodynamics.

    But in mechanics we often take derivatives with respect to derivatives.

    For example, I have a set of variables q_1,...q_N and time t.

    Supposedly I have to add to these their time derivatives (which I denote by upper case)

    Q_1,...Q_N. Now I have a set of 2*N+1 variables and no relations (the lagrangian would

    be a relation, but then the number of variables would be 2*N+2)

    I am confused how to interpret partial derivatives:

    1) q_i with respect to q_j
    2) Q_i with respect to Q_j
    3) q_i with respect to Q_j and Q_j with respect to q_i

    Thank you in advance for your comments
  2. jcsd
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