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bcrowell submitted a new PF Insights post
Relativistic Work-Kinetic Energy Theorem
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Relativistic Work-Kinetic Energy Theorem
Continue reading the Original PF Insights Post.
I see the ## way of writing formulae does not work well, so here's my comment in plaintext:Jano L. said:I have trouble seeing what is the problem you're trying to solve. The relation ##W=Fd## is definition of work, not the work-energy theorem. The work-energy theorem says work equals change in kinetic energy of the particle. This follows mathematically from the equation of motion ##md(gamma v)/dt = F## and Einstein's definition of energy ##E=gamma mc^2##.
Jano L. said:I see the ## way of writing formulae does not work well, so here's my comment in plaintext:
I have trouble seeing what is the problem you're trying to solve. The relation W=Fd is definition of work, not the work-energy theorem.
Jano L. said:The work-energy theorem says work equals change in kinetic energy of the particle. This follows mathematically from the equation of motion md(γv)/dt=F and Einstein's definition of energy E=γmc^2.
bcrowell said:There are two disadvantages to your method. (1) In the 1905 paper, Einstein uses ##\Delta E=Fd## to prove ##E=m\gamma c^2##, so he can't use the latter to prove the former. (2) Your method doesn't work for massless particles.
pervect said:I'm not too sure my thoughts will be helpful for the intended purpose (which I assume ultimately involves a simple presentation of special relativity), but perhaps they'll provide some insight. And they can be expressed pretty briefly. As long as special relativity has a non-trivial Hamiltonian, we can write Hamilton's equations:
[tex]\frac{\partial H}{\partial q} = \dot {p}[/tex]
Now, if we can also identify H with energy, q with position, and ##\dot{p}## with force, then we have the work-energy theorem, the rate of change of the energy with position must be equal to the force.
Jano L. said:I do not know what Einstein was getting at there. I am not sure proving the relation ##E=\gamma mc^2## makes sense. I think this is just a definition of energy, same as ##\frac{1}{2}mv^2## is definition of kinetic energy in classical mechanics.
vanhees71 said:"massless particles" (which don't make too much sense in classical (i.e., non-quantum) relativistic physics anyway)
vanhees71 said:Can you give an example where one does this? Anyway, there's of course nothing wrong with it from a theoretical point of view. You can have perfectly valid equations of motion for massless particles.
vanhees71 said:In which sense has gravitational lensing to do with the motion of massless particles? Well, you might interpret ray optics (i.e., the eikonal approximation for classical electromagnetic waves) as massless-particle motion.
vanhees71 said:Ok, so that's well understood by classical electromagnetism. No naive-photon picture is needed.
The Relativistic Work-kinetic Energy Theorem is a fundamental concept in physics that relates the work done on an object to its change in kinetic energy. It takes into account the effects of relativity, specifically the increase in mass as an object approaches the speed of light.
The theorem is derived from the laws of motion and the principles of relativity. It involves integrating the force acting on an object over a distance to find the work done, and then using the relativistic expression for kinetic energy to relate the two quantities.
The theorem is significant because it allows us to accurately calculate the work done on an object and its resulting change in kinetic energy, even at high speeds where relativistic effects become important. It also helps us understand the relationship between energy and mass.
The Classical Work-kinetic Energy Theorem does not take into account the effects of relativity, and therefore is only accurate at low speeds. The Relativistic Work-kinetic Energy Theorem, on the other hand, incorporates relativity and is valid at all speeds, including those approaching the speed of light.
The theorem has numerous applications in fields such as astrophysics, particle physics, and nuclear energy. It is used to calculate the energy released in nuclear reactions, the behavior of particles in particle accelerators, and the motion of objects in space at high speeds.