# Conserved charge

Is there an inconsistency between the definition of conserved charge and conserved current in Hamiltonian and Lagrangian formulation?

For example, $$H = \int T^{00} d^3x$$ is a conserved charge,
$$\frac{dH}{dt} = \{H, H\} = 0$$

But we have $$\partial_\mu T^{\mu\nu} = 0$$ implies
$$\int (\partial_\mu T^{\mu 0}) d^3x = \int (\partial_0 T^{00} + \partial_i T^{i0}) d^3x = 0$$ so it seems
$$\frac{d}{dt}\int T^{00}d^3x = - \int \partial_i T^{i0} d^3x \neq 0$$

I'm very puzzled.

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Why is the last integral non-zero? If the charge is constant, shouldn't the current be zero?

So there's a contradiction. In general if there is a boundary, the last integral is not zero.

dextercioby
$$T^{i0} = 0$$ at boundary?