A problem about decomposing a Lagrangian

[Haorong Wu]

Hi, there. Suppose I write down a Langrangian as $L(t, x, y, z, f)$ where $t$, $x$, $y$ and $z$ are identified as variables, and $f=f(t, x, y, z)$. Now the Euler-Lagrange equations$$\sum_{\mu=0} ^3 \frac d {dx^{\mu}}\left (\frac {\partial L }{\partial (\partial_\mu f)} \right ) -\frac {\partial L}{\partial f} =0$$ give the right equations of motion. The corresponding action is then $S=\int dx^4 L(t, x, y, z, f)$. Let us suppose that $t$ is coupled to other variables in $L(t, x, y, z, f)$.

Next, we will utilize the 3+1 decomposition on the action to separate the time variable $t$, yielding $$S=\int dt \int dx^3 L(t, x, y, z, f).$$

My problem is that could we identify $\int dx^3 L(t, x, y, z, f)$ as a new Lagrangian $L'(t, x, y, z, f)$ such that only $t$ is identified as the variable and $x$, $y$, $z$, $f$ are functions on $t$. If this is valid, then we can write the canonical momentum conjugate to $f$ as $\pi_f=\frac {\partial L'}{\partial (\partial_t f)}$.

cf. A master equation for gravitational decoherence: probing the textures of spacetime section 2.1 to 2.2.

Many thanks.

 

[eljose79]

Hello,

Thank you for your interesting question. It seems like you have a good grasp on the Euler-Lagrange equations and the 3+1 decomposition of the action. To answer your question, yes, it is possible to identify $\int dx^3 L(t, x, y, z, f)$ as a new Lagrangian $L'(t, x, y, z, f)$. This is known as the Lagrangian density, and it is often used in field theories to describe the dynamics of a system.

In this case, the Lagrangian density is a function of all the variables $t$, $x$, $y$, $z$, and $f$, but only the time variable $t$ is considered as a variable in the Lagrangian density. The other variables are considered as functions of $t$. So, in a sense, the Lagrangian density encapsulates all the information about the system's dynamics at a given time $t$.

As for your question about the canonical momentum conjugate to $f$, yes, it can be defined as $\pi_f = \frac{\partial L'}{\partial (\partial_t f)}$. This is known as the canonical momentum density, and it is the momentum conjugate to the field $f$ at a given point in space and time.

I hope this clarifies your doubts. Feel free to ask any further questions if needed. Good luck with your research!