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What is the definition of the conjugate momentum of the function $f$ if the Lagrangian has more than one independent variable, such as:

$$L=L(t,x,f,f_x,\dot f, g,g_x,\dot g)$$

where $f=f(t,x)$, $g=g(t,x)$?

I know that for a one-independent variable case $$p_i={\partial L\over \partial\dot q_i}$$.

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up vote 1 down vote accepted

The conjugate momenta are still defined by $ \pi_i = \frac{\partial L}{\partial \dot{q_i}}$. For example, look at

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