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

For example, [tex]H = \int T^{00} d^3x[/tex] is a conserved charge,

[tex]\frac{dH}{dt} = \{H, H\} = 0[/tex]

But we have [tex]\partial_\mu T^{\mu\nu} = 0[/tex] implies

[tex]\int (\partial_\mu T^{\mu 0}) d^3x = \int (\partial_0 T^{00} + \partial_i T^{i0}) d^3x = 0[/tex] so it seems

[tex]\frac{d}{dt}\int T^{00}d^3x = - \int \partial_i T^{i0} d^3x \neq 0[/tex]

I'm very puzzled.

For example, [tex]H = \int T^{00} d^3x[/tex] is a conserved charge,

[tex]\frac{dH}{dt} = \{H, H\} = 0[/tex]

But we have [tex]\partial_\mu T^{\mu\nu} = 0[/tex] implies

[tex]\int (\partial_\mu T^{\mu 0}) d^3x = \int (\partial_0 T^{00} + \partial_i T^{i0}) d^3x = 0[/tex] so it seems

[tex]\frac{d}{dt}\int T^{00}d^3x = - \int \partial_i T^{i0} d^3x \neq 0[/tex]

I'm very puzzled.

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