Signs in the Field-Theoretic Euler-Lagrange Equation

[Xezlec]

So I have this book that considers the problem of a flexible vibrating string, taking \phi(x,t) as the string's displacement from equilibrium. It then writes a Lagrangian density in terms of this \phi, takes \delta \mathcal{S} = 0, and eventually concludes that \frac{\partial}{\partial t}(\frac{\partial \mathcal{L}}{\partial \dot{\phi}}) + \frac{\partial}{\partial x}(\frac{\partial \mathcal{L}}{\partial \phi'}) = 0. Notice that the time-varying and space-varying terms have the same sign.

Two pages later, it considers a scalar field \phi(x^0,\mathbf{x}) with a Lagrangian density \mathcal{L}=\mathcal{L}(\phi,\partial_\mu\phi), and concludes that \frac{\partial\mathcal{L}}{\partial\phi}-\partial_\mu(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)})=0. Now, unless I am having some massive brain fart on how covariant and contravariant work, the time-varying and space-varying terms have opposite signs. Right?

What gives? Why are the signs different between these two situations?

 

[vanhees71]

In
\partial_{\mu} \frac{\partial \mathcal{L}}{\partial(\partial_{\mu} \phi)}
the temporal and spatial components have the same sign. How do you come to the conclusion that might be not so?

The same is true for the four-dimensional divergence.
\partial_{\mu} A^{\mu}=\frac{\partial A^{\mu}}{\partial x^{\mu}}=\frac{\partial A^0}{\partial t} + \vec{\nabla} \cdot \vec{A},
where I've used natural units, c=1, and x^0=t.

 

[Xezlec]

Thanks. It's clearly been too long since I've done anything with covariant and contravariant vectors. I need to go back and refresh before jumping back into this stuff.