# Evaluating conjugate momentum from a given Lagrangian density

I have the following Lagrangian density $$\mathcal{L}$$ where

$$\mathcal{L}=\frac{1}{2}\left(c[\partial_{t}\phi(x,t)]^{2}-\frac{1}{l}[\partial_{x}\phi(x,t)]^{2}+\frac{1}{\omega_{J}^{2}l}[\partial_{x}\partial_{t}\phi(x,t)]^{2}+\gamma[\partial_{x}\phi(x,t)]^{4}\right)\tag{1}$$

where $$c,l,\omega_{J},\gamma$$ are constants. Defining the usual conjugate momenta $$\pi$$ such that $$\pi=\frac{\partial\mathcal{L}}{\partial(\partial_{t}\phi(x,t))}.\tag{2}$$

How should I evaluate the third term $$[\partial_{x}\partial_{t}\phi(x,t)]^{2}$$ where there is also an $$x$$-derivative?

Edit: I found a solution to this. It seems that I cannot use the regular convention for defining conjugate momenta. Rather I have to define it such that $$\pi=\frac{\delta\mathcal{L}}{\delta[\partial_{t}\phi]}=\frac{\partial\mathcal{L}}{\partial[\partial_{t}\phi]}-\partial_{x}\frac{\partial\mathcal{L}}{\partial[\partial_{x}\partial_{t}\phi]} = c\partial_{t}\phi-\frac{1}{\omega_{J}^{2}l}\partial_{x}^{2}\partial_{t}\phi.\tag{3}$$

I do not understand this definition of conjugate momenta. Can someone explain why is it defined like so?

1. To properly define the momentum density $$p(x,t)$$ it is important to be able to distinguish between the dependence of the position field $$q(x,t)$$ and the velocity field $$v(x,t)$$. Therefore we cannot use the action $$S[q]~=~\left. \int\! dt~ L[q(\cdot,t),v(\cdot,t);t]\right|_{v=\dot{q}}, \tag{A}$$ which is only a functional of the position fields $$q(x,t)$$.

2. Instead the fundamental object is the Lagrangian $$L[q(\cdot,t),v(\cdot,t);t], \tag{B}$$ which in field theory is a functional.

3. The momentum density is then defined as the functional/variational derivative $$p(x,t)~:=~\frac{\delta L[q(\cdot,t),v(\cdot,t);t]}{\delta v(x,t)}\tag{C}$$ wrt. the velocity field $$v(x,t)$$.

4. In OP's last eq. (3) there appears the notation of a 'same-spacetime' functional derivative $$\frac{\delta {\cal L}(x,t)}{\delta v(x,t)}\tag{D},$$ which is a somewhat-misleading-although-common notation for above definition (C). In particular (D) is not an actual variational derivative of the Lagrangian density $${\cal L}(x,t)$$: If it were it would be infinite, since the 'numerator' and 'denominator' of eq. (D) are evaluated at the same spacetime point $$(x,t)$$.

It is the usual definition of conjugate momenta. The extra term comes from taking the $$\textbf{variation}$$ of the Lagrangian respect to $$\partial_t\phi$$ while allowing for mixed partial terms $$\mathcal{L}[\partial_t\phi,\partial_x\partial_t\phi]$$.
$$\pi=\frac{\delta\mathcal{L}}{\delta[\partial_{t}\phi]}=\frac{\partial\mathcal{L}}{\partial[\partial_{t}\phi]}-\partial_{x}\frac{\partial\mathcal{L}}{\partial[\partial_{x}\partial_{t}\phi]} = c\partial_{t}\phi-\frac{1}{\omega_{J}^{2}l}\partial_{x}^{2}\partial_{t}\phi$$