- #1
vin300
- 603
- 4
(gamma)ma=F-[F.(beta)]beta
How derive this?
How derive this?
HallsofIvy said:Would you mind explaining what it is? Until you explain what "gamma", "m", "a", "F", and "beta" mean, it's just a string of symbols.
Just add tex tags.clem said:I couldn't find my latex errors. I hope you can read it as is.
clem said:I think the equation you are looking for is
[tex]{\bf F}=\frac{d{\bf p}}{dt}[/tex]
[tex]\frac{d}{dt}\frac{m{\bf v}}{\sqrt{1-v^2}}
=m{\bf a}\gamma+m{\bf v}({\bf v}\cdot{\bf a})\gamma^3[/tex].
This is the usual form.
To get your form, use the above equation to show
[tex]{\bf v}\cdot{\bf F}=m\gamma^3{\bf v}\cdot{\bf a}[/tex]
in a few steps.
I use units with c=1.
The equation (gamma)ma=F-[F.(beta)]beta is the relativistic equation of motion, also known as the Lorentz force law. It describes the relationship between the mass of an object (m), its acceleration (a), and the applied force (F). It is derived by applying the principles of special relativity to the classical Newton's second law of motion.
(gamma) represents the Lorentz factor, which takes into account the effects of time dilation and length contraction due to an object's velocity. It is defined as (1 - (v^2/c^2))^(-1/2), where v is the velocity of the object and c is the speed of light.
The Lorentz force law takes into account the effects of special relativity, such as time dilation and length contraction, which do not exist in classical mechanics. It also introduces the concept of a relativistic mass, which is dependent on an object's velocity and increases as the object approaches the speed of light.
Yes, the Lorentz force law can be applied to all objects, as long as they are not accelerating at a significant fraction of the speed of light. At very high velocities, the effects of special relativity become more pronounced, and a more advanced version of the equation, known as the relativistic Newton's second law, must be used.
The Lorentz force law is used in many fields, including particle physics, astrophysics, and engineering. It is essential for understanding the behavior of charged particles in electric and magnetic fields, which is crucial in particle accelerators and plasma physics. It is also used in the design of particle detectors, such as those used in medical imaging and nuclear physics experiments.