In classical mechanics, kinetic energy is KE = ½mv², where m is mass and v is velocity. So mass multiplied by the square of the speed is an energy. The concept of energy plays a fundamental role in understanding the behaviour of objects in motion. One of the key forms of energy is kinetic energy, which is intimately linked to an object's mass and velocity. Additionally, in the realm of relativity, Einstein's famous equation E = mc² introduces a profound understanding of energy in terms of mass and the speed of light. This discussion aims to delve into the classical expression for kinetic energy KE = ½mv² and its connection to relativistic energy (mc²).
No. The distinction between kinetic and potential energy is a non-relstivistic artifact. Incidentally, relstivistic and non-relstivistic mechanics are both classical mechanics; non-relativistic mechanics is the low-velocity approximation, in an expansion in powers of v/c, of relstivistic mechanics.
Energy and momentum are components of a 4-vector and satisfy the relation E^2-|p|^2c^2=m^2c^4. This implies that E = mc^2 coshφ, where tanhφ = v/c. Kinetic energy can thus be defined as E-mc^2 = mc^2(coshφ -1). Expanding in powers of v/c, one finds E-mc^2=mv^2/2 + O((v/c)^4). All this is now standard material in any introductory course in undergraduate physics... There's no point pretending to live 100 years ago...
Dear Mr. Robert A. Phillips
The main question is, "Is relativistic 'energy' the same representation of kinetic energy as 'energy' in classical mechanics, which is questioned on the basis that the main point of discussion is, "mass multiplied by the square of speed is 'energy'.” The question is between relativistic and mechanical 'energy', it doesn't ask the differences between relativistic and mechanical processes.The relativistic energy, as represented by the mass-energy equivalence principle (E = mc²), can be seen as a similar representation of energy as kinetic energy in classical mechanics. Both concepts involve the multiplication of mass by the square of speed to yield energy.
In classical mechanics, kinetic energy (KE = ½mv²) quantifies the energy associated with the motion of an object. This expression illustrates that there's an energy associated with the mass of an object and its velocity squared.
Similarly, in relativistic physics, Einstein's equation E = mc² reveals that mass and energy are interchangeable, with the speed of light squared acting as a conversion factor. In this context, mass multiplied by the square of the speed of light represents energy.
So, while there are conceptual differences between classical and relativistic frameworks, both expressions involve mass multiplied by the square of speed to quantify energy, suggesting a fundamental connection between mass, velocity, and energy in both contexts.
Best regards,
Soumendra Nath Thakur
Dear Dr. Stam Nicolis The main question is, "Is relativistic 'energy' the same representation of kinetic energy as 'energy' in classical mechanics, which is questioned on the basis that the main point of discussion is, "mass multiplied by the square of speed is 'energy'.” The question is between relativistic and mechanical 'energy', it doesn't ask the differences between relativistic and mechanical processes. The relativistic energy, as represented by the mass-energy equivalence principle (E = mc²), can be seen as a similar representation of energy as kinetic energy in classical mechanics. Both concepts involve the multiplication of mass by the square of speed to yield energy. In classical mechanics, kinetic energy (KE = ½mv²) quantifies the energy associated with the motion of an object. This expression illustrates that there's an energy associated with the mass of an object and its velocity squared.
So, while there are conceptual differences between classical and relativistic frameworks, both expressions involve mass multiplied by the square of speed to quantify energy, suggesting a fundamental connection between mass, velocity, and energy in both contexts.
Regards
Soumendra Nath Thakur
Once more, the statement that relativistic energy is mc^2 is WRONG. The statement that relativistic mechanics isn't classical mechanics is, also, wrong.
Relativistic energy is given by the expression mc^2 coshφ, where tanhφ=v/c.
That is deduced by writing the Lagrangian for the massive relativistic particle and deducing the corresponding Hamiltonian. The upshot is that
E = mc2 coshφ = mc2/(1-(v/c)2)1/2
upon expressing coshφ in terms of tanhφ.
This implies that the kinetic energy in relativistic mechanics is given by the expression
Ecin=E-mc^2=mc2/(1-(v/c)2)1/2 - mc2 (1)
as a function of velocity, This is another standard exercise. Expanding this expression in powers of v/c gives
Ecin=(1/2)mv2 + corrections in powers of v/c. (2)
This isn't a research topic anymore! It's in any textbook of undergraduate physics the last 100 years.
That energy is equal to the product of mass and the square of the velocity is, just, dimensional analysis. Both expressions (1) and (2) are, obviously, consistent with this statement. Just that, in the non-relativistic limit, the expression for the kinetic energy is independent of c, since this limit is defined by v/c->0.
Energy in relativistic mechanics is given by, dimensional analysis, by the expression
E = mc2f((v/c)2)
and, of course, it is equal to the product of mass and the square of the velocity. f is dimensionless, so it can only depend on a dimensionless quantity.
The only question is what the function f((v/c)2) can be. The transformation properties of energy and momentum under Lorentz transformations imply that
f((v/c)2)=1/(1-(v/c)2)1/2.
The reason why it wouldn't be correct to write
E = mv^2 F( (v/c)^2)
which is, also, dimensionally correct, is because v isn't invariant under Lorentz trsnsformations, so it can't define a unit of velocity-which can't depend on the frame.
The equations express and follow from the symmetries, in this case from invariance under global Lorentz transformations-which can be shown to be equivalent to the freedom to choose the parameter of the trajectory in spacetime. This is another standard exercise, that's been understood since 1918. All this is part of what is taught in any undergraduate physics curriculum (and reviewed as background for more advanced courses). The first ten pages of these notes: https://www.liverpool.ac.uk/~mohaupt/Strings07.pdf suffice.
How a subject developed historically is part of history, not of its meaning; but only the meaning matters, not the history.
The ``instructions'' involve considerable guesswork: https://www.youtube.com/watch?v=ysYEAC4z66c from the fragmentary knowledge at any one moment to try to understand how the pieces are part of a bigger picture. For the relativistic particle now everything is competely understood so it suffices to learn it-which amounts to learning the content of the ten pages of the lecture notes in the link. The rest is just exercises.
Newtonian mechanics is a limiting case of relativistic mechanics, not the other way around. And the reason to get interested in relativistic mechanics at all is because particles can carry electric charge and thus be sources of the electromagnetic field, which is described by Maxwell's equations-and these aren't invariant under the transformations of coordinates, that leave invariant the equations of Newtonian mechanics, but under global Lorentz transformations (and gauge transformations). This took some time to be understood, but now this is the case, so it doesn't make sense pretending to live 100 years ago.
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