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

• In the 2nd century BCE Hero of Alexandria noted that the laws of the reflection of light\ can be derived from the principle that light always takes the shortest path. This principle does mot explain rhe the laws of the refraction of light.
• In 1657 Fermat formulated the Principle of Least Time; i.e., that light travels from its initial point to its final point through reflection and refraction through media in which light travels at different speeds by folowing the path requiring the least time.
• In 1747 Maupertuis asseted the Principle of Least Action. Action is defined as the a variable having the same dimensions as length times momentum or energy times time. Maupertuis asserted that the motion of a physical system takes place in a way to minimize action. Maupertuis did not give any basis for this principle other than the theological one that it is God's Wisdom. han a theological one of God's wisdom.
• In 1760: Lagrange gave a mathematically rational basis for the Principle of Least Action.
• In 1828: Gauss formulated the Principle of Least Constraint. Gauss tried to incorporate the matter of constraints on motion into the principles of physical dynamics.
• In 1834-35 Hamiltonian formulated the principle that the path taken by a physical system is the one which minimizes the integral of the difference between kinetic and potential energy. The difference between the kinetic and potential energy came to be known as the Lagragian of a system.

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 {q_i; i=1,2,…,n} and their time derivatives {dq_i/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/∂v_i)/dt − (∂L/∂q_i) = 0

where v_i=dq_i/dt.

The expression (∂L/∂v_i) is the momentum p_i with respect to the variable q_i. If (∂L/∂q_i) = 0 then

(dp_i/dt) = 0

and thus p_i 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
−½(m₀c²/(1−β²)^3/2)(−2β(1/c))
which reduces to
(m₀c²/(1−β²)^3/2)(v/c)(1/c)
or, equivalently
(m₀/(1−β²)^3/2)v
which can be expressed as
mv/(1−β²)

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.

Precedents

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.,

m_l = m₀/(1−β²)^3/2
m_t = 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 = m_lv

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).

Conclusions

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.

APPENDIX
Extrema and Variational Conditions

Pontryagin's Maximum Principle

Let X₁, X₂, ..., X_n be the state variables for an extremum problem and u₁, u₂, ..., u_m 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