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Math Help - Partial differential notation

  1. #1
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    Partial differential notation

    Given

     L=L(x,y,z,\dot x,\dot y,\dot z)

    then I can write the six partial derivatives (as I would when forming Euler-Lagrange equations):

    \frac{\partial L}{\partial x^\mu}, \frac{\partial L}{\partial \dot x^\mu}

    or I can write the three partial derivatives

    \partial_\mu L=\frac{\partial L}{\partial x^\mu}=\frac{\partial L}{\partial x^\mu}+\frac{\partial L}{\partial \dot x^\mu}\frac{\partial \dot x^\mu}{\partial x^\mu}

    These two options use the same notation but give different things. How am I to know which is required? Does the \partial_\mu notation mean that the derivatives are of the second kind?
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  2. #2
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    Re: Partial differential notation

    Hey Kiwi_Dave.

    As an educated guess, I would say to try and relate them to grad and div. These have very specific meanings:

    Vector calculus identities - Wikipedia, the free encyclopedia
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  3. #3
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    Re: Partial differential notation

    I think I have worked it out!

    \delta^\mu_\rho (\partial_\mu \mathcal{L}) is the first type because of the way the Lagrangian is defined.

    But \partial_\mu (\delta^\mu_\rho \mathcal{L}) is the second kind (Divergence) because the partial is being applied to a function that is not a Lagrangian.

    Does that make sense?
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  4. #4
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    Re: Partial differential notation

    It does based on the sub-scripts and super-scripts but how do you define contraction and expansion of tensors?
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  5. #5
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    Re: Partial differential notation

    Thanks Chiro, I am not entirely sure what you are driving at.

    But I worry that I can write:

    \partial_\mu (\delta^\mu_\rho \mathcal{L})=(\partial_\mu \delta^\mu_\rho) \mathcal{L}+\delta^\mu_\rho (\partial_\mu \mathcal{L})=\delta^\mu_\rho (\partial_\mu \mathcal{L}) which turns my divergence back into a gradient problem, making my type two back into type 1.

    Where this is coming from is that I am trying to calculate the divergence of the Hamiltonian tensor \mathcal{H}^{.\mu}_\rho=(\phi_\rho p^\mu - \delta^\mu_\rho \mathcal{L})

    Now I have a text book that shows me step by step how to do this. But in the first step they expand the partial of L like my type 2, they then cancel some terms but ultimately they seem to treat the partial of L like type 1 again. I have attached an image of my textbook page.
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