Classical Mechanics Point Particles and Relativity

Classical mechanics : point particles and relativity
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Volume Space Integrals Pages Basic Concepts of Mechanics Pages The General Linear Motion Pages The Free Fall Pages Friction Pages The Harmonic Oscillator Pages The Damped Harmonic Oscillator Pages The Pendulum Pages Mathematical Interlude: Differential Equations Pages Planetary Motions Pages Special Problems in Central Fields Pages The Earth and our Solar System Pages The Lorentz Transformation Pages Properties of the Lorentz transformation Pages Addition Theorem of the Velocities Pages Applications of the Special Theory of Relativity Pages Some authors use m for relativistic mass and m 0 for rest mass, [4] others simply use m for rest mass.

Relativistic space-time

This article uses the former convention for clarity. The energy and momentum of an object with invariant mass m 0 are related by the formulas. The first is referred to as the relativistic energy—momentum relation.

Now, gamma can be replaced in the expression of energy. Rest mass is not a conserved quantity in special relativity unlike the situation in Newtonian physics. However, even if an object is changing internally, so long as it does not exchange energy with surroundings, then its rest mass will not change, and can be calculated with the same result in any frame of reference. A particle whose rest mass is zero is called massless. Photons and gravitons are thought to be massless; and neutrinos are nearly so. In this case:.

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Notice that the rest mass of a composite system will generally be slightly different from the sum of the rest masses of its parts since, in its rest frame, their kinetic energy will increase its mass and their negative binding energy will decrease its mass. In particular, a hypothetical "box of light" would have rest mass even though made of particles which do not since their momenta would cancel. The corresponding energy, which is also the total energy when a single particle is at rest, is referred to as "rest energy".

In systems of particles which are seen from a moving inertial frame, total energy increases and so does momentum. However, for single particles the rest mass remains constant, and for systems of particles the invariant mass remain constant, because in both cases, the energy and momentum increases subtract from each other, and cancel. Thus, the invariant mass of systems of particles is a calculated constant for all observers, as is the rest mass of single particles. For systems of particles, the energy—momentum equation requires summing the momentum vectors of the particles:.

The inertial frame in which the momenta of all particles sums to zero is called the center of momentum frame. This is the invariant mass of any system which is measured in a frame where it has zero total momentum, such as a bottle of hot gas on a scale. In such a system, the mass which the scale weighs is the invariant mass, and it depends on the total energy of the system. It is thus more than the sum of the rest masses of the molecules, but also includes all the totaled energies in the system as well.

Classical Mechanics - Lecture 10

Like energy and momentum, the invariant mass of isolated systems cannot be changed so long as the system remains totally closed no mass or energy allowed in or out , because the total relativistic energy of the system remains constant so long as nothing can enter or leave it. An increase in the energy of such a system which is caused by translating the system to an inertial frame which is not the center of momentum frame , causes an increase in energy and momentum without an increase in invariant mass. Taking this formula at face value, we see that in relativity, mass is simply energy by another name and measured in different units.

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Classical Mechanics: Point Particles and Relativity (Classical Theoretical Physics ) [Walter Greiner] on *FREE* shipping on qualifying offers. More than a generation of German-speaking students around the world have worked their way to an understanding and appreciation of the power and beauty of.

In Einstein remarked about special relativity, "Under this theory mass is not an unalterable magnitude, but a magnitude dependent on and, indeed, identical with the amount of energy. In a "totally-closed" system i. Einstein's equation shows that such systems must lose mass, in accordance with the above formula, in proportion to the energy they lose to the surroundings. Conversely, if one can measure the differences in mass between a system before it undergoes a reaction which releases heat and light, and the system after the reaction when heat and light have escaped, one can estimate the amount of energy which escapes the system.

In both nuclear and chemical reactions, such energy represents the difference in binding energies of electrons in atoms for chemistry or between nucleons in nuclei in atomic reactions. In both cases, the mass difference between reactants and cooled products measures the mass of heat and light which will escape the reaction, and thus using the equation give the equivalent energy of heat and light which may be emitted if the reaction proceeds.

Classical Mechanics Point Particles Relativity by Walter Greiner

Thus, Einstein's formula becomes important when one has measured the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have stored energy that can be released by certain nuclear reactions , providing important information which was useful in the development of nuclear energy and, consequently, the nuclear bomb. Historically, for example, Lise Meitner was able to use the mass differences in nuclei to estimate that there was enough energy available to make nuclear fission a favorable process.

The implications of this special form of Einstein's formula have thus made it one of the most famous equations in all of science. It has been popularly misunderstood to mean that mass may be converted to energy, after which the mass disappears. However, popular explanations of the equation as applied to systems include open non-isolated systems for which heat and light are allowed to escape, when they otherwise would have contributed to the mass invariant mass of the system. Historically, confusion about mass being "converted" to energy has been aided by confusion between mass and " matter ", where matter is defined as fermion particles.

In such a definition, electromagnetic radiation and kinetic energy or heat are not considered "matter". In some situations, matter may indeed be converted to non-matter forms of energy see above , but in all these situations, the matter and non-matter forms of energy still retain their original mass. For isolated systems closed to all mass and energy exchange , mass never disappears in the center of momentum frame, because energy cannot disappear.


Instead, this equation, in context, means only that when any energy is added to, or escapes from, a system in the center-of-momentum frame, the system will be measured as having gained or lost mass, in proportion to energy added or removed. Thus, in theory, if an atomic bomb were placed in a box strong enough to hold its blast, and detonated upon a scale, the mass of this closed system would not change, and the scale would not move. Only when a transparent "window" was opened in the super-strong plasma-filled box, and light and heat were allowed to escape in a beam, and the bomb components to cool, would the system lose the mass associated with the energy of the blast.

In a 21 kiloton bomb, for example, about a gram of light and heat is created. If this heat and light were allowed to escape, the remains of the bomb would lose a gram of mass, as it cooled. In this thought-experiment, the light and heat carry away the gram of mass, and would therefore deposit this gram of mass in the objects that absorb them. This tensor is additive: the total angular momentum of a system is the sum of the angular momentum tensors for each constituent of the system. It is frequently useful to represent physical processes by space-time diagrams in which time runs vertically and the spatial coordinates run horizontally.

Of course, since space-time is four-dimensional, at least one of the spatial dimensions in the diagram must be suppressed. Thus, the set of all events t 2 , x 2 satisfying equation with zero on the right-hand side is the light cone of the event t 1 , x 1.

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Because Lorentz transformations leave invariant the space-time interval , all inertial observers agree on what the light cones are. If the right-hand side of equation is strictly positive, in which case one says that the two events are timelike separated, or have a timelike interval, then one can find an inertial frame with respect to which the two events have the same spatial position. The straight world line joining the two events corresponds to the time axis of this inertial frame of reference.

The proper time would be measured by any clock moving along the straight world line between the two events. Just as an ordinary vector like v has three components, v x , v y , and v z , a 4-vector has four components. Geometrically the 4-velocity and 4-acceleration correspond, respectively, to the tangent vector and the curvature vector of the world line see Figure 2. If the particle moves slower than light, the tangent, or velocity, vector at each event on the world line points inside the light cone of that event, and the acceleration, or curvature, vector points outside the light cone.

If the particle moves with the speed of light, then the tangent vector lies on the light cone at each event on the world line. One can, however, define a so-called affine parameter that satisfies equation with zero on the right-hand side. For the time being this discussion will be restricted to particles moving with speeds less than light.

This requirement is invariant under Lorentz transformations of the form of equations and The tangent vector then points inside the future light cone and is said to be future-directed and timelike see Figure 3.

One may if one wishes attach an arrow to the world line to indicate this fact. One says that the particle moves forward in time. It was pointed out by the Swiss physicist Ernest C. It is possible to interpret these world lines in terms of antiparticles, as will be seen when particles moving in a background electromagnetic field are considered.

Classical Mechanics: Point Particles and Relativity

Equations and , which relate the curvature of the world line to the applied forces, are the same in all inertial frames related by Lorentz transformations. It induces an identical rotation on the 4-acceleration and force 4-vectors. To say that both of these 4-vectors experience the same generalized rotation or Lorentz transformation is simply to say that the fundamental laws of motion and are the same in all inertial frames related by Lorentz transformations.

They also have a natural generalization in the general theory of relativity, which incorporates the effects of gravity. The law of motion may also be expressed as:. This idea is a consequence of special relativity alone. It really comes into its own, however, when one considers relativistic quantum mechanics. This work goes into increasing the energy E of the particle.