# Signs in the Field-Theoretic Euler-Lagrange Equation

• Xezlec
In summary, the conversation discusses two different situations involving Lagrangian densities and concludes that the time-varying and space-varying terms have opposite signs. The conversation also mentions the concept of covariant and contravariant vectors and how they can affect the sign of the components in the four-dimensional divergence equation. The speaker realizes they need to refresh their knowledge on these concepts before continuing.
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?

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 $x^0=t$.

1 person
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.

## 1. What is the Field-Theoretic Euler-Lagrange Equation?

The Field-Theoretic Euler-Lagrange Equation is a mathematical equation used in the field of physics to describe the behavior of a system of particles in terms of a continuous field. It is derived from the Lagrangian formalism and is used to find the equations of motion for a system.

## 2. How is the Field-Theoretic Euler-Lagrange Equation different from the standard Euler-Lagrange Equation?

The main difference between the two equations is that the Field-Theoretic Euler-Lagrange Equation takes into account the interactions between particles in a continuous field, while the standard Euler-Lagrange Equation only considers the motion of individual particles.

## 3. What types of systems can the Field-Theoretic Euler-Lagrange Equation be applied to?

The Field-Theoretic Euler-Lagrange Equation can be applied to a wide range of systems, including classical mechanics, quantum mechanics, and field theories such as electromagnetism and general relativity.

## 4. What are the applications of the Field-Theoretic Euler-Lagrange Equation?

The Field-Theoretic Euler-Lagrange Equation has various applications in physics, including predicting the behavior of particles in a field, studying the dynamics of complex systems, and understanding the behavior of physical systems at a fundamental level.

## 5. Are there any limitations to the use of the Field-Theoretic Euler-Lagrange Equation?

Like any mathematical model, the Field-Theoretic Euler-Lagrange Equation has its limitations. It may not accurately describe certain systems with extreme conditions or interactions, and it does not take into account quantum effects. It is also limited by the assumptions and simplifications made in its derivation.

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