2.2.1 Principle of Least Action: Nonlinear Systems
2.2.1.1 Electrically Uncharged Particle in Relativistic Mechanics
Deriving from the general postulates associated with relativistic mechanics and in particular from the postulate that action S of a mechanical system may not be relative to the selection of any inertial reference system and, hence, it has to be an invariant of Lorentz transformation [13,16], Lagrange’s function for a particle can be stated in the following form
2
2 1 ⎟
⎠
⎜ ⎞
⎝
−⎛
−
= c
mc v
L (2.168) Concurrently, the action on the particle is
( )v c dt
mc dt L S
t
t t
t ∫
∫ =− −
=
2
1 2
1
2 1 / 2 (2.169)
where:
m - is the rest mass of a particle c - is the velocity of light in vacuum
Since always the velocity of a particle v is considerably lower than the speed of light, it is possible to expand Lagrange’s function (2.168) so that it takes the form of Taylor’s power series of a small quantity (ν/c2). As a result of such expansion we obtain:
+…
⎟⎠
⎜ ⎞
⎝ + ⎛
⎟⎠
⎜ ⎞
⎝ + ⎛ +
−
= 2 21 2 81 2 2 161 2 4 c mv v c
mv v mv mc
L (2.170)
For small velocities ν << c only the first two terms of the power expansion of La- grange’s function are relevant in comparison to the following ones; however, the first term in them, as a constant value, does not contribute anything due to the in- definite form of Lagrange’s function (2.143). Hence the result takes the form of familiar classical result well known in mechanics L = ẵ mν 2, which denotes
kinetic energy of a particle. The calculation of particle’s momentum according to the general definition (2.137) results in:
)2
/ (
1 v c
m L
L
= −
∂
=∂
∂
=∂ v
v
p q (2.171) which when expanded into power series with respect to (ν /c2), yields that
+…
⎟⎠
⎜ ⎞
⎝ + ⎛
⎟⎠
⎜ ⎞
⎝ + ⎛
=
4 2
8 3 2
1
c m v c
m v
mv v v
p (2.172)
For small velocities ν << c this results in the classical shape v
p=m (2.173) The calculation of the total energy of a particle in accordance with relation (2.139) gives the result, which is relevant to our considerations
2 2
) / ( 1 v c L mc
L
E=qp− =vp− = − (2.174) By expansion into power series (1.174), this gives
+…
⎟⎠
⎜ ⎞
⎝ + ⎛ +
= 2 2 2 2
8 3 2
1
c mv v mv
mc
E (2.175)
This energy consists of rest mass energy E0 = mc2 and the energy associated with the velocity of particle’s motion. The relevance of this result is associated with the fact that the comparison between expressions in (2.170) and (2.175) offers a con- clusion that Lagrange’s function for a particle in motion is equal to its kinetic energy only for the case of classical mechanics. In the relativistic mechanics the expressions in (2.168) and (2.174) are completely different. The formal reason for this is related to nonlinearity of parameters, or more precisely, nonlinearity of mass, which increases along with speed as illustrated by the formula for the parti- cle’s momentum. An illustration of this is found in Fig. 2.23.
From Fig. 2.23 one can conclude that surface areas that illustrate kinetic energy and the supplementary term denoted as co-energy overlap only for small velocities of a particle in motion. This is the case when the function of kinetic energy is a homogenous function of velocity (2.140). Accordingly, from formula (2.175) one can conclude that this is so only when the expansion into power series may omit the terms containing (ν /c)2k. In practice this means that equality between kinetic energy and kinetic term in Lagrange’s function takes place only in case when the mass of the particle is constant, which is represented by a linear system in the sense of involvement of constant parameters. In systems with non-linear pa- rameters relative to velocity q , Lagrange’s function does not account for kinetic
Fig. 2.23 Kinetic energy and co-energy of a particle in motion
energy; however, in accordance with (2.171) the integral of system’s momentum takes the form
∫
∫ =
=
′=L pdv pdq
T (2.176) This integral denotes the kinetic term of Lagrange’s function written as T' and is called kinetic complementary energy: co-energy. Its graphical representation (Fig. 2.23) takes the form of the surface area under the curve denoting the relation between the momentum of a system and velocity. From both the formal point of view and its graphical illustration it is clear that kinetic energy T is equal to kinetic co-energy T' only for a system with constant parameters.
const m T
T′= = (2.177) Concurrently, the total energy of particle is equal to
∫
∫ =
= vdp qdp
T (2.178) which can be integrated and the result takes the form (2.174), and the one in (2.175) after power expansion. Both functions of co-energy (2.176) and kinetic energy (2.178) add up to form a rectangle, which results from formula (2.174) and has also a graphical representation in Fig. 2.23. This also justifies the definition at- tributed to co-energy, which states that it completes the function of energy to the bound of a rectangle.
q
=p
′+T
T (2.179) The result of these considerations conducted for motion of a particle in relativistic mechanics shows that kinetic co-energy T' (2.176) is involved in the Lagrange’s function for a system whereas kinetic energy T is not. The distinction between ki- netic co-energy and kinetic energy is only necessary in non-linear systems whose parameters are relative to generalized velocities. In the examined case of a particle
in motion mass m relative to the velocity forms the only variable parameter. In en- gineering practice associated with issues of electric drives we have to do with such low velocities that the relativistic variation in mass is insignificant and, as a result, we assume that T' = T for mechanical variables. The studies conducted here have an even more comprehensive application besides theoretical considerations, as in electromagnetism we have to do with systems with non-linear parameters, in particular with inductance of the windings containing ferromagnetic core, which is relative to the current applied to the windings.