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where formula_2 is the latitudinal component of the particle's angular momentum, formula_3 is the energy of the particle, formula_4 is the particle's axial angular momentum, formula_5 is the rest mass of the particle, and formula_6 is the spin parameter of the black hole. Because functions of conserved quantities are a... |
in place of formula_7. The quantity formula_11 is useful because it is always non-negative. In general any fourth conserved quantity for motion in the Kerr family of spacetimes may be referred to as "Carter's constant". |
Noether's theorem states that all conserved quantities are related to spacetime symmetries. Carter's constant is related to a higher order symmetry of the Kerr metric generated by a second order Killing tensor field formula_11 (different formula_11 than used above). In component form: |
where formula_15 is the four-velocity of the particle in motion. The components of the Killing tensor in Boyer–Lindquist coordinates are: |
where formula_17 are the components of the metric tensor and formula_18 and formula_19 are the components of the principal null vectors: |
The spherical symmetry of the Schwarzschild metric for non-spinning black holes allows one to reduce the problem of finding the trajectories of particles to three dimensions. In this case one only needs formula_3, formula_4, and formula_5 to determine the motion; however, the symmetry leading to Carter's constant still... |
By a rotation of coordinates we can put any orbit in the formula_27 plane so formula_28. In this case formula_29, the square of the orbital angular momentum. |
The Bohr–Kramers–Slater theory (BKS theory) was perhaps the final attempt at understanding the interaction of matter and electromagnetic radiation on the basis of the so-called old quantum theory, in which quantum phenomena are treated by imposing quantum restrictions on classically describable behaviour. It was advanc... |
One aspect, the idea of modelling atomic behaviour under incident electromagnetic radiation using "virtual oscillators" at the absorption and emission frequencies, rather than the (different) apparent frequencies of the Bohr orbits, significantly led Born, Heisenberg and Kramers to explore mathematics that strongly ins... |
The initial idea of the BKS theory originated with Slater, who proposed to Bohr and Kramers the following elements of a theory of emission and absorption of radiation by atoms, to be developed during his stay in Copenhagen: |
Slater's main intention seems to have been to reconcile the two conflicting models of radiation, viz. the wave and particle models. He may have had good hopes that his idea with respect to oscillators vibrating at the "differences" of the frequencies of electron rotations (rather than at the rotation frequencies themse... |
As Max Jammer puts it, this refocussed the theory "to harmonize the physical picture of the continuous electromagnetic field with the physical picture, not as Slater had proposed of light quanta, but of the discontinuous quantum transitions in the atom." Bohr and Kramers hoped to be able to evade the photon hypothesis ... |
In particle physics, flavour or flavor refers to the "species" of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with "flavour quantum numbers" that are assigned to all subatomic particles. They can also be described by some of... |
In classical mechanics, a force acting on a point-like particle can only alter the particle's dynamical state, i.e., its momentum, angular momentum, etc. Quantum field theory, however, allows interactions that can alter other facets of a particle's nature described by non dynamical, discrete quantum numbers. In particu... |
In atomic physics the principal quantum number of an electron specifies the electron shell in which it resides, which determines the energy level of the whole atom. Analogously, the five flavour quantum numbers (isospin, strangeness, charm, bottomness or topness) can characterize the quantum state of quarks, by the deg... |
Composite particles can be created from multiple quarks, forming hadrons, such as mesons and baryons, each possessing unique aggregate characteristics, such as different masses, electric charges, and decay modes. A hadron's overall flavour quantum numbers depend on the numbers of constituent quarks of each particular f... |
All of the various charges discussed above are conserved by the fact that the corresponding charge operators can be understood as "generators of symmetries" that commute with the Hamiltonian. Thus, the eigenvalues of the various charge operators are conserved. |
Absolutely conserved flavour quantum numbers in the Standard Model are: |
In some theories, such as the grand unified theory, the individual baryon and lepton number conservation can be violated, if the difference between them () is conserved (see chiral anomaly). |
Strong interactions conserve all flavours, but all flavour quantum numbers (other than and ) are violated (changed, non-conserved) by electroweak interactions. |
If there are two or more particles which have identical interactions, then they may be interchanged without affecting the physics. Any (complex) linear combination of these two particles give the same physics, as long as the combinations are orthogonal, or perpendicular, to each other. |
In other words, the theory possesses symmetry transformations such as formula_1, where and are the two fields (representing the various "generations" of leptons and quarks, see below), and is any unitary matrix with a unit determinant. Such matrices form a Lie group called SU(2) (see special unitary group). This is an ... |
In quantum chromodynamics, flavour is a conserved global symmetry. In the electroweak theory, on the other hand, this symmetry is broken, and flavour changing processes exist, such as quark decay or neutrino oscillations. |
All leptons carry a lepton number . In addition, leptons carry weak isospin, , which is − for the three charged leptons (i.e. electron, muon and tau) and + for the three associated neutrinos. Each doublet of a charged lepton and a neutrino consisting of opposite are said to constitute one generation of leptons. In addi... |
Leptons may be assigned the six flavour quantum numbers: electron number, muon number, tau number, and corresponding numbers for the neutrinos. These are conserved in strong and electromagnetic interactions, but violated by weak interactions. Therefore, such flavour quantum numbers are not of great use. A separate quan... |
All quarks carry a baryon number and all anti-quarks have They also all carry weak isospin, The positively charged quarks (up, charm, and top quarks) are called "up-type quarks" and have the negatively charged quarks (down, strange, and bottom quarks) are called "down-type quarks" and have Each doublet of up and down t... |
For all the quark flavour quantum numbers listed below, the convention is that the flavour charge and the electric charge of a quark have the same sign. Thus any flavour carried by a charged meson has the same sign as its charge. Quarks have the following flavour quantum numbers: |
These five quantum numbers, together with baryon number (which is not a flavour quantum number), completely specify numbers of all 6 quark flavours separately (as i.e. an antiquark is counted with the minus sign). They are conserved by both the electromagnetic and strong interactions (but not the weak interaction). Fro... |
The terms "strange" and "strangeness" predate the discovery of the quark, but continued to be used after its discovery for the sake of continuity (i.e. the strangeness of each type of hadron remained the same); strangeness of anti-particles being referred to as +1, and particles as −1 as per the original definition. St... |
For first-order weak decays, that is processes involving only one quark decay, these quantum numbers (e.g. charm) can only vary by 1, that is, for a decay involving a charmed quark or antiquark either as the incident particle or as a decay byproduct, likewise, for a decay involving a bottom quark or antiquark Since fir... |
A special mixture of quark flavours is an eigenstate of the weak interaction part of the Hamiltonian, so will interact in a particularly simple way with the W bosons (charged weak interactions violate flavour). On the other hand, a fermion of a fixed mass (an eigenstate of the kinetic and strong interaction parts of th... |
The CKM matrix allows for CP violation if there are at least three generations. |
Flavour quantum numbers are additive. Hence antiparticles have flavour equal in magnitude to the particle but opposite in sign. Hadrons inherit their flavour quantum number from their valence quarks: this is the basis of the classification in the quark model. The relations between the hypercharge, electric charge and o... |
Quantum chromodynamics (QCD) contains six flavours of quarks. However, their masses differ and as a result they are not strictly interchangeable with each other. The up and down flavours are close to having equal masses, and the theory of these two quarks possesses an approximate SU(2) symmetry (isospin symmetry). |
Under some circumstances (for instance when the quark masses are much smaller than the chiral symmetry breaking scale of 250 MeV), the masses of quarks do not meaningfully contribute to the system's behavior, and can be ignored to zeroth approximation. The simplified behavior of flavour transformations can then be succ... |
If all quarks had non-zero but equal masses, then this chiral symmetry is broken to the "vector symmetry" of the "diagonal flavour group" , which applies the same transformation to both helicities of the quarks. This reduction of symmetry is a form of "explicit symmetry breaking". The strength of explicit symmetry brea... |
Even if quarks are massless, chiral flavour symmetry can be spontaneously broken if the vacuum of the theory contains a chiral condensate (as it does in low-energy QCD). This gives rise to an effective mass for the quarks, often identified with the valence quark mass in QCD. |
Analysis of experiments indicate that the current quark masses of the lighter flavours of quarks are much smaller than the QCD scale, ΛQCD, hence chiral flavour symmetry is a good approximation to QCD for the up, down and strange quarks. The success of chiral perturbation theory and the even more naive chiral models sp... |
Some of the historical events that led to the development of flavour symmetry are discussed in the article on isospin, the eightfold way (physics) and chiral symmetry. Chief among these would be the November Revolution (physics) in 1974, when the fourth (charm) quark was found. |
Reciprocity in electrical networks is a property of a circuit that relates voltages and currents at two points. The reciprocity theorem states that the current at one point in a circuit due to a voltage at a second point is the same as the current at the second point due to the same voltage at the first. The reciprocit... |
If a current, formula_1, injected into port A produces a voltage, formula_2, at port B and formula_1 injected into port B produces formula_2 at port A, then the network is said to be reciprocal. Equivalently, reciprocity can be defined by the dual situation; applying voltage, formula_5, at port A producing current form... |
The transfer function of a reciprocal network has the property that it is symmetrical about the main diagonal if expressed in terms of a z-parameter, y-parameter, or s-parameter matrix. A non-symmetrical matrix implies a non-reciprocal network. A symmetric matrix does not imply a symmetric network. |
In some parametisations of networks, the representative matrix is not symmetrical for reciprocal networks. Common examples are h-parameters and ABCD-parameters, but they all have some other condition for reciprocity that can be calculated from the parameters. For h-parameters the condition is formula_9 and for the ABCD... |
An example of reciprocity can be demonstrated using an asymmetrical resistive attenuator. An asymmetrical network is chosen as the example because a symmetrical network is fairly self-evidently reciprocal. |
Injecting six amps into port 1 of this network produces 24 volts at port 2. |
Injecting six amps into port 2 produces 24 volts at port 1. |
Hence, the network is reciprocal. In this example, the port that is not injecting current is left open circuit. This is because a current generator applying zero current is an open circuit. If, on the other hand, one wished to apply voltages and measure the resulting current, then the port to which the voltage is not a... |
Reciprocity of electrical networks is a special case of Lorentz reciprocity, but it can also be proven more directly from network theorems. This proof shows reciprocity for a two-node network in terms of its admittance matrix, and then shows reciprocity for a network with an arbitrary number of nodes by an induction ar... |
If we further require that network is made up of passive, bilateral elements, then |
since the admittance connected between nodes "j" and "k" is the same element as the admittance connected between nodes "k" and "j". The matrix is therefore symmetrical. For the case where formula_17 the matrix reduces to, |
From which it can be seen that, |
which is synonymous with the condition for reciprocity. In words, the ratio of the current at one port to the voltage at another is the same ratio if the ports being driven and measured are interchanged. Thus reciprocity is proven for the case of formula_17. |
For the case of a matrix of arbitrary size, the order of the matrix can be reduced through node elimination. After eliminating the "s"th node, the new admittance matrix will have the form, |
It can be seen that this new matrix is also symmetrical. Nodes can continue to be eliminated in this way until only a 2×2 symmetrical matrix remains involving the two nodes of interest. Since this matrix is symmetrical it is proved that reciprocity applies to a matrix of arbitrary size when one node is driven by a volt... |
The Extra Element Theorem (EET) is an analytic technique developed by R. D. Middlebrook for simplifying the process of deriving driving point and transfer functions for linear electronic circuits. Much like Thévenin's theorem, the extra element theorem breaks down one complicated problem into several simpler ones. |
Driving point and transfer functions can generally be found using Kirchhoff's circuit laws. However several complicated equations may result that offer little insight into the circuit's behavior. Using the extra element theorem, a circuit element (such as a resistor) can be removed from a circuit and the desired drivin... |
The general form of the extra element theorem is called the N-extra element theorem and allows multiple circuit elements to be removed at once. |
The (single) extra element theorem expresses any transfer function as a product of the transfer function with that element removed and a correction factor. The correction factor term consists of the impedance of the extra element and two driving point impedances seen by the extra element: The double null injection driv... |
Where the Laplace-domain transfer functions and impedances in the above expressions are defined as follows: is the transfer function with the extra element present. is the transfer function with the extra element open-circuited. is the transfer function with the extra element short-circuited. is the impedance of the ex... |
The extra element theorem incidentally proves that any electric circuit transfer function can be expressed as no more than a bilinear function of any particular circuit element. |
is found by making the input to the system's transfer function zero (short circuit a voltage source or open circuit a current source) and determining the impedance across the terminals to which the extra element will be connected with the extra element absent. This impedance is same as the Thévenin's equivalent impedan... |
is found by replacing the extra element with a second test signal source (either current source or voltage source as appropriate). Then, is defined as the ratio of voltage across the terminals of this second test source to the current leaving its positive terminal when the output of the system's transfer function is nu... |
In practice, can be found from working backwards from the facts that the output of the transfer function is made zero and that the primary input to the transfer function is unknown. Then using conventional circuit analysis techniques to express both the voltage across the extra element test source's terminals, , and th... |
Special case with transfer function as a self-impedance. |
As a special case, the EET can be used to find the input impedance of a network with the addition of an element designated as "extra". In this case, is same as the impedance of the input test current source signal made zero or equivalently with the input open circuited. Likewise, since the transfer function output sign... |
Computing these three terms may seem like extra effort, but they are often easier to compute than the overall input impedance. |
Consider the problem of finding formula_9 for the circuit in Figure 1 using the EET (note all component values are unity for simplicity). If the capacitor (gray shading) is denoted the extra element then |
Calculating the impedance seen by the capacitor with the input shorted, |
Calculating the impedance seen by the capacitor with the input open, |
This problem was solved by calculating three simple driving point impedances by inspection. |
The EET is also useful for analyzing single and multi-loop feedback amplifiers. In this case the EET can take the form of the asymptotic gain model. |
The star-mesh transform, or star-polygon transform, is a mathematical circuit analysis technique to transform a resistive network into an equivalent network with one less node. The equivalence follows from the Schur complement identity applied to the Kirchhoff matrix of the network. |
The equivalent impedance betweens nodes A and B is given by: |
where formula_2 is the impedance between node A and the central node being removed. |
The transform replaces "N" resistors with formula_3 resistors. For formula_4, the result is an increase in the number of resistors, so the transform has no general inverse without additional constraints. |
It is possible, though not necessarily efficient, to transform an arbitrarily complex two-terminal resistive network into a single equivalent resistor by repeatedly applying the star-mesh transform to eliminate each non-terminal node. |
Source transformation is the process of simplifying a circuit solution, especially with mixed sources, by transforming voltage sources into current sources, and vice versa, using Thévenin's theorem and Norton's theorem respectively. |
Performing a source transformation consists of using Ohm's law to take an existing voltage source in series with a resistance, and replacing it with a current source in parallel with the same resistance, or vice versa. The transformed sources are considered identical and can be substituted for one another in a circuit. |
Source transformations are not limited to resistive circuits. They can be performed on a circuit involving capacitors and inductors as well, by expressing circuit elements as impedances and sources in the frequency domain. In general, the concept of source transformation is an application of Thévenin's theorem to a cur... |
Source transformations are easy to compute using Ohm's law. If there is a voltage source in series with an impedance, it is possible to find the value of the equivalent current source in parallel with the impedance by dividing the value of the voltage source by the value of the impedance. The converse also holds: if a ... |
The transformation can be derived from the uniqueness theorem. In the present context, it implies that a black box with two terminals must have a unique well-defined relation between its voltage and current. It is readily to verify that the above transformation indeed gives the same V-I curve, and therefore the transfo... |
In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization ... |
A Hopf bifurcation is also known as a Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Aleksandr Andronov and Eberhard Hopf. |
The limit cycle is orbitally stable if a specific quantity called the first Lyapunov coefficient is negative, and the bifurcation is supercritical. Otherwise it is unstable and the bifurcation is subcritical. |
The normal form of a Hopf bifurcation is: |
Write: formula_2 The number "α" is called the first Lyapunov coefficient. |
Hopf bifurcations occur in the Lotka–Volterra model of predator–prey interaction (known as paradox of enrichment), the Hodgkin–Huxley model for nerve membrane, the Selkov model of glycolysis, the Belousov–Zhabotinsky reaction, the Lorenz attractor, the Brusselator and Classical electromagnetism. |
The phase portrait illustrating the Hopf bifurcation in the Selkov model is shown on the right. |
In railway vehicle systems, Hopf bifurcation analysis is notably important. Conventionally a railway vehicle's stable motion at low speeds crosses over to unstable at high speeds. One aim of the nonlinear analysis of these systems is to perform an analytical investigation of bifurcation, nonlinear lateral stability and... |
The appearance or the disappearance of a periodic orbit through a local change in the stability properties of a fixed point is known as the Hopf bifurcation. The following theorem works for fixed points with one pair of conjugate nonzero purely imaginary eigenvalues. It tells the conditions under which this bifurcation... |
Theorem (see section 11.2 of ). Let formula_6 be the Jacobian of a continuous parametric dynamical system evaluated at a steady point formula_7. Suppose that all eigenvalues of formula_6 have negative real part except one conjugate nonzero purely imaginary pair formula_9. A "Hopf bifurcation" arises when these two eige... |
Routh–Hurwitz criterion (section I.13 of ) gives necessary conditions so that a Hopf bifurcation occurs. Let us see how one can use concretely this idea. |
Let formula_10 be Sturm series associated to a characteristic polynomial formula_11. They can be written in the form: |
The coefficients formula_13 for formula_14 in formula_15 correspond to what is called Hurwitz determinants. Their definition is related to the associated Hurwitz matrix. |
Proposition 1. If all the Hurwitz determinants formula_13 are positive, apart perhaps formula_17 then the associated Jacobian has no pure imaginary eigenvalues. |
Proposition 2. If all Hurwitz determinants formula_13 (for all formula_14 in formula_20 are positive, formula_21 and formula_22 then all the eigenvalues of the associated Jacobian have negative real parts except a purely imaginary conjugate pair. |
The conditions that we are looking for so that a Hopf bifurcation occurs (see theorem above) for a parametric continuous dynamical system are given by this last proposition. |
Consider the classical Van der Pol oscillator written with ordinary differential equations: |
The Jacobian matrix associated to this system follows: |
The characteristic polynomial (in formula_25) of the linearization at (0,0) is equal to: |
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