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Lagrangian Analysis and
its Conservation Principles

The Timeline History of the Developmeny of
Minimum Principles of Physical Dynamics

Lagrangian Dynamics

Let K denote the kinetic energy of a physical system and V its potential energy. Then the Lagrangian L of the system is defined as

L = K − V

If the Lagrangian of a system is a function of a set of variables {qi; i=1,2,…,n} and their time derivatives {dqi/dt; i=1,2,…,n} and the system is not subject to external forces then the dynamics of the system is given by the set of equations

d(∂L/∂vi)/dt − (∂L/∂qi) = 0

where vi=dqi/dt.

The expression (∂L/∂vi) is the momentum pi with respect to the variable qi. If (∂L/∂qi) = 0 then

(dpi/dt) = 0

and thus pi is conserved.

The Newtonian Case

In Newtonian mechanics the kinetic energy of a body is ½mv² and v=dx/dt for some x. If the potential energy function does not depend upon x or v then

L = ½mv² − V
and therefore
L/∂v = mv
and since ∂L/∂x = 0
d(mv)/dt = 0
and thus linear momentum
is constant over time

The Relativistic Case

Now consider the relativistic case. The kinetic energy is then (mc²−m0c²) where m=m0/(1−β²)½ and β=v/c.

The expression ½mv² is not the full kinetic energy; it is just the first order approximation of the kinetic energy for low velocities.

The Lagrangian for a particle moving in a potential field V is

L = (mc²−m0c²) − V
and the partial derivative of L
with respect to v is
which reduces to
or, equivalently
which can be expressed as

In other words, relativistic linear momentum requires not only the relativistic adjustment of mass but also division by a factor of (1−β²).

Thus if the kinetic and potential energies are independent of the spatial variable that defines velocity then it is mv/(1−β²) which is constant over time.

That is to say,

d(mv/(1−β²))/dt = 0

The expression m0v/(1−β²)3/2 asymptotically approaches m0v as β→0 just as m0v/(1−β²)½ does.



Herbert Goldstein's works on mechanics have long been considered the definitive source on the topic of classical mechanics. In the two editions of his Classical Mechanics, he refers to the longitudinal and the transverse masses for a body; i.e.,

ml = m0/(1−β²)3/2
mt = m0/(1−β²)1/2

where longitudinal means in the direction of the motion and tranverse is perpendicular to the direction of motion. Transverse mass is just the relativistic mass but longitudinal mass is just the relativistic mass divided by (1−β²). This is the same results that came out of the analysis above. The formula for longitudinal mass was apparently established by Lord Kelvin (William Thomson) empirically around 1850. Goldstein says however that the use of these concepts of mass has been decreasing because they obscure the physics. This may be true that there is no reason for defining mass in this way, but that is no reason not to take into account the (1−β²) term in the computation of momentum. The other interpretation of the result of Lagrangian analysis is that the correct formula for relativistic momentum is

p = mlv

Goldstein adopts a different approach to the issue. He argues that the true equation of motion is just

d[m0v/(1−β²)½]/dt = F

and to get that equation of motion he changes the nature of the Lagrangian so that it is not the difference of kinetic and potential energies. This appears to be an unjustified distortion of the universal principle of Lagrangian Dynamics. In the 1950's edition of his book Golstein takes the definition, of the Lagrangian to be whatever formula gives relativistic momentum to be mv under Lagrangian analysis. Other authors of books on classical mechanics, such as Jerry Marion and Stephen Thornton, adopt the same approach as Goldstein. For a particle in a potential field V that is

L* = −m0c²(1−β²)½ −V
(page 206 equation 6-49)

Intellectually this is invalid. There is no support for the mv formula for relativistic momentum provided by showing that there is a pseudo-Lagrangian from which it can be derived by Lagrangian analysis. In the 1981 edition Goldstein repeats this invalid exercise (page 321, equation 7-136) but gives the formulas for longitudinal and transverse only in a suggested exercise.

Here is a graph of the supposed kinetic energy from Goldstein's pseudo-Lagrangian as a function of relative velocity β. expressed in terms of its ratio to m0c².

Note that at β=0 this supposed kinetic energy is −m0c² and at at β=1 this supposed kinetic energy is zero. Negative kinetic energy is of course complete nonsense. Apparentl conventional physicists are so set on the formula mv for relativistic momentum that they are willing to accept a derivation of it from nonsense. The negativity and the nonzero value for the supposed kinetic energy at β=0 can be remedied by added the term m0c² to it, as does S.W. McCuskey in his An Introduction to Advaand anced Dynamics. But that does not take care of the finiteness of the value at β=1 and that finiteness is also nonsense.

The standard formula for kinetic energy is

K = mc² − m0c² = m0c²(1/(1−β²)½ −1)

Here is a graph of K/(m0c²).


Either the conventional formula for relativistic momentum as relatisic mass times velocity is wrong or Lagrangian analysis is wrong. Apparently the conventional formula is wrong and has been based only upon intuition.

Extrema and Variational Conditions

Pontryagin's Maximum Principle

Let X1, X2, ..., Xn be the state variables for an extremum problem and u1, u2, ..., um be the control variables. Let the state variables b e represented by the column vector X and the control variables by U. Both X and U are functions of time which ranges from 0 to T. The dynamics of the problem are then given by

(∂X/∂t) = F(X, U)

where F is a vector of functions.

The objecive is to choose U(t) such that C·X(T) reaches a maximum where C is a row vector of constants. Lev Pontryagin was able to show that there exists a function Ψ(X, U, t) such that if U(t)is chosen to maximize Ψ at each instant C·X(T) will achieve a maximum.

The Lagrange Principle

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