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for a Particle in a Potential Field |
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Let m_{0} be the rest mass for a particle. If its velocity is v then its relativistic mass m is
where β is velocity relative to the speed of light; i.e., v/c.
The total energy of the particle is its kinetic energy K(v) plus its potential energy V(x). The kinetic energy component of the Hamiltonian function is expressed as a function of its momentum. Relativistic kinetic energy is K(v) = (m − m_{0})c²
So far everything is as expected. The problem is that relativistic momentum in the direction of travel is not simply mv. Instead it is
For the details on this matter see Relativistic Momentum
The solution for β proceeds as follows.
This is an equation in β². Let
The equation to be solved is then the cubic
In standard form this is
Such an equation is called a depressed cubic equation. The quantity (m_{0}c/p) is intriguingly interesting. It is a dimensionless number in the nature of a ratio of momenta.
One real solution to
The equation for γ is such that b=−a. The above solution reduces to
With a equal to (m_{0}c/p)²
The kinetic energy of the particle is
The relativistic Hamiltonian function H(p, z) for a particle in a potential field V(x) is then given by
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