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Is this true?

The Hamiltonian always equals the total energy

No — it is false.

The counterexample

A bead on a wire rotating at forced angular velocity

The constraint is time-dependent, so H is conserved but differs from T + V.

The kind of mistake this is

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this
mechanics

Others that fail the same way

From Classical Mechanics › Hamiltonian formalism and conservation laws

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