dHdt

dHdt is the time derivative of a quantity H, written without parentheses as dH/dt. It represents the instantaneous rate at which H changes with time along a process or trajectory. The meaning of H depends on the context, so dH/dt is interpreted differently in various fields.

In thermodynamics, H commonly denotes enthalpy, defined as H = U + pV, where U is internal energy, p is pressure, and V is volume. The rate dH/dt describes how a system’s enthalpy evolves due to changes in internal energy and pressure–volume work. In practical analyses of flowing substances, relationships such as dh = c_p dT describe how enthalpy changes with temperature for a given substance, under appropriate conditions.

In classical mechanics and related disciplines, H often denotes the Hamiltonian, a function H(q,p,t) of generalized coordinates q, momenta p, and possibly time t, representing the system’s total energy. The total time derivative along a trajectory satisfies dH/dt = ∂H/∂t + {H, H}. Since the Poisson bracket {H, H} is zero, this reduces to dH/dt = ∂H/∂t. Consequently, a time-independent Hamiltonian yields dH/dt = 0, reflecting energy conservation, while a explicitly time-dependent Hamiltonian indicates an exchange of energy with external forces or fields.

In summary, dH/dt is a general notation for the rate of change of H with respect to time, with common interpretations in enthalpy and in the Hamiltonian framework.