Direct Energy, 2018a

11 CALCULUS OF VARIATIONS 261
Energy
storage
device
Generalized
Path
Capacitor
Charge Q
in C
Linear
Spring
Displacement
−→ x in m
Generalized
Potential
Generalized
Capacity
Constitutive
relationship
Energy
Law for
potential
Voltage v in
J
C = V
Capacitance
C in
F = C2
J
Q = Cv
−→ F Force in
J
m = N
1
K
in
m2
J
−→ x =
1
K
−→ F
1
2 Cv2 1K|−→ x | 2 =
2
1 1
|−→ F | 2
2 K
KVL
Newton's
Second Law
−→ F = m
−→ a
Table 11.2: Summary of the capacitor inductor system in the language of
calculus of variations.
Furthermore, we can show that energy is conserved in this energy conversion
process because the partialderivative of both the totalenergy and
the Lagrangian with respect to time are zero.
∂L
∂t = ∂H
∂t
dL
dt = dH dt
=0 (11.62)
=0 (11.63)
Table 11.2 summarizes this example. It also illustrates the relationship
between parameters of this example and parameters of the mass spring
example.