What Makes a Circuit Element an Energy Storage Element?

The constitutive equation of a circuit element is a relationship imposed by the element on the solution of the circuit.

If the constitutive equation of an element does not depend on the state of the element at previous time points, then this element is a memoryless element. For example the constitutive equation of a resistor at time t is v(t) = i(t) * R. This is clearly independent of the voltage across and the current through the resistor at any time before t. This makes the capacitor a memoryless element.

Conversely, if the constitutive equation of an element does depend on the state of he element at a previous time point, then the element is an energy storage element and does have memory. Equations (1), (2), (3) and (4) in the previous post show that the inductor, the capacitor, and the transformer are energy storage elements because their constitutive equations at time t + 1 depend on their state at time t.