Partial differential notation

Given

$\displaystyle 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):

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

or I can write the three partial derivatives

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

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

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

Re: Partial differential notation

I think I have worked it out!

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

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

Does that make sense?

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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Re: Partial differential notation

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

But I worry that I can write:

$\displaystyle \partial_\mu (\delta^\mu_\rho \mathcal{L})=(\partial_\mu \delta^\mu_\rho) \mathcal{L}+\delta^\mu_\rho (\partial_\mu \mathcal{L})=\delta^\mu_\rho (\partial_\mu \mathcal{L})$ 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 $\displaystyle \mathcal{H}^{.\mu}_\rho=(\phi_\rho p^\mu - \delta^\mu_\rho \mathcal{L})$

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.