Direct Energy, 2018a

11 CALCULUS OF VARIATIONS 259
The energy stored in a capacitor is
E cap = 1 2 Cv2 . (11.48)
The second form of energy in this system is the energy stored in the
magnetic eld of the inductor.The current i L through the inductor, measured
in amperes, is proportional to the magnetic ux Ψ, measured in
webers, around the inductor.Inductance L, measured in henries, is the
constant of proportionality between the current and magnetic ux.
Ψ=Li L (11.49)
The current voltage relationship across this inductor can be found by taking
the derivative with respect to time.
dΨ
dt = v = Ldi L
dt
The energy stored in the inductor is given by
(11.50)
E ind = 1 2 Li2 L. (11.51)
We describe the energy conversion process by keeping track of a the
generalized path Q(t), the charge stored on the capacitor.The variable t
represents the independent variable time in seconds, and Q is the dependent
variable charge in coulombs.The Hamiltonian and Lagrangian, H and L,
will be considered functions of three independent-like variables: t, Q, and
dQ
dt . The Hamiltonian is the sum of the energy in the capacitor and the
energy in the inductor.The Lagrangian is the dierence between these
energies.
H = E total = E cap + E ind (11.52)
L = E cap − E ind (11.53)
Electrical engineers typically describe physical circuits using the most easily
measured quantities: current and voltage.However, here to illustrate the
use of the calculus of variations formalism, we write expressions for both
the total energy and the Lagrangian in terms of the specied variables: t,
Q, and dQ . dt
(
H t, Q, dQ )
= 1
dt 2C Q2 + 1 ( ) 2 dQ
2 L (11.54)
dt