The math is wondrous not in its complexity but in its simplicity. The simplest E=mc^2. Mass and Energy are the same thing.
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-----------------------------1938 - Favorite Equations
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- My 10 favorite equations and the physicists who invented them.
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------- (1) Pythagoras’ equation for hypotenuse of a right triangle, ...... a^2+b^2 = c^2
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------- (2) Newton’s equation for motion, ............. F = m*a
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------- (3) Newton’s equation for gravity,.......... F = G*M*m/r^2
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------- (4) Euler’s equations for natural logarithm, “e“, and imaginary numbers, “square root of -1“.
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------- (5) Clausis’ equation for entropy
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------- (6) Maxwell’s equations for electromagnetism
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------- (7) Einstein’s equation for energy,............ E = m*c^2
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------- (8) Einstein’s equation for gravity……….. G = k(R - 1/2 g R)
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------- (9) Schrodinger’s equation for wave mechanics.
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------- (10) Heisenberg’s equation for uncertainty , .......... dp*dx > h/2*pi
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- No one knows how mathematics can so well describing the natural Universe. Were these equations invented or discovered? I think they were invented by great men and women. Nature has symmetry and invariance that is a quality of math. So the two work together well to make new discoveries:
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------- (1) a^2+b^2 = c^2
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------- (2) F = m*a
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------- (3) F = G*M*m/r^2
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------- (4) e^i*pi + 1 = 0
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------- (5) S1 - S2 > 0
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------- (6) Integral E* dA = Q / Eo, and Integral B*dA = 0
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------- (7) E = m*c^2
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------- (8) G = K ( T -1/2 gT)
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------- (9) d^2U/dr^2 + 2(a+1) / r* dU/dr + 2*m*(E + e^2/r)*U/K^2 = 0
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------- (10) dp*dx > h/2*pi
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- Here are the famous physicists that invented these equations. Note all from Europe, none from Asia or the United States.
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------- (1) Pythagoras (560 B.C. - 480 B.C.) Age 80, Greece
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------- (2) Sir Isaac Newton (1642 - 1727) Age 85, Lincolnshire, England.
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------- (3) ) Sir Isaac Newton (1642 - 1727) Age 85, Lincolnshire, England.
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------- (4) Leonhard Euler (1707 - 1783) Age76, Basel, Switzerland
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------- (5) Rudolf Julius Emmanuel Clausis ( 1822 - 1888) Age 66, Poland
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------- (6) James Clerk Maxwell (1831 - 1879) Age 48, Edinburgh, Scotland
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------- (7) Albert Einstein (1879- 1955), Age 76, Ulm, Germany
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------- (8) Albert Einstein (1879- 1955), Age 76, Ulm, Germany
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------- (9) Erwin Schrodinger (1867 - 1961) Age 85, Vienna, Austria
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------- (10) Werner Karl Messenger (1901 - 1976) Age 75, Wurzburg, Germany
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------- (1) Pythagoras - the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides, ................... a^2+b^2 = c^2.
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- Pythagoras founded a cult of secrecy, asceticism, and mysticism on a Greek island. He moved to Italy at age 31. In his study of sound he discovered that a string of twice the length would emit a sound an octave lower. He showed that the square root of 2 could not be equal to any conceivable fraction. He thus discovered “irrational numbers“. He discovered that the Evening Star and the Morning Star were the same star, the planet Venus. There are over 50 proofs for the Pythagorean Theorem. Below is one of those proofs. The general equation is a^2 + b^2 - 2*a*b* cosine theta = c^2. When the angle theta is 90 degrees the cosine is zero and the equation for a right triangle becomes the simpler: ......................... a^2 + b^2 = c^2.
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- Start with a large square with equal sides. Put a smaller square inside at a diagonal creating 4 right triangles around the smaller square and inside the larger square. Label the hypotenuse “c” and the longer side “a” and the shorter side ”b” for each triangle. The area of the smaller square on the inside is c^2. The area of each right triangle is 1/2 a*b. The total area of the four triangle is 2*a*b. The area of the larger outside square is (a+b)^2.
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- The area of the small square plus the area of the 4 triangles = the area of the large square, Therefore:
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------------- c^2 + 2*a*b = (a+b)^2
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------------ c^2 + 2*a*b = a^2 + 2*a*b + b^2
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------------- c^2 = a^2 + b^2
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------- (2) Newton’s equation for motion, F = m*a, Force = mass * acceleration, was applying math to the theories of Galileo. Sir Isaac Newton (1642 - 1727) was born in Lincolnshire, England, as a premature baby. His mother remarried and left him with his grandparents to raise him. They lived on a farm. In school he was always building mechanical things, kites, sundials, water clocks. In 1650 he left school at age 8 to work on the farm. His uncle recognized his talent and sent him to Cambridge in 1660 at age 18. In 1665 he graduated and invented the “Binominal Theorem“, A simple version is:
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---------------- (x+y)^2 = x^2 + 2xy + y^2
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--------------- but, it gets much more complicated with larger exponents of a binomial and he invented this when he was 18:
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----------- (x+y)^n = (n:0)x^n + (n:1)*x^(n-1)* y + (n:2)*x^(n-2)* y^2 + (n:3)*x^(n-3)* y^3 + ................. (n:k)*x^(n-k)*y^k .......................(n:n)y^n
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----------- where : (n:k) = n ! / k! * (n-k)!
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----------- where : For example:
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----------- The exponent is 6, n+6 and, 5! = 5 factorial = 5*4*3*2*1 = 120
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------------ (6:5) = 720 / 120 * (6-5) = 6
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----------- (x+y)^6 = (6:0)x^6 + (6:1)*x^(6-1)* y + (6:2)*x^(6-2)* y^2 + (6:3)*x^(6-3)*y^3 + (6:4)*x^(6-4)*y^4 + (6:5)*x^(6-5)*y^5 + (6:6)y^6
----------- (x+y)^6 = x^6 + 6x^5 y + 15*x^4 y^2 + 20*x^3 y^3 + 15*x^2 y^4 + 6*x^6 y^5 + y^6
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------- (3) Newton’s equation for gravity, F = G*M*m/r^2
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- At age 27 Newton became a professor at Cambridge. He developed the equations for Galileo’s laws of motion. He recognized that the force of gravity fell off according to the square of the distance from the center of Earth. He used this to determine the equation for the orbit of the Moon inventing Calculus in the process. He used prisms to prove that white light was a composition of colors of the rainbow. In 1668 he invented the reflecting telescope using a parabolic mirror. In 1687 he published “ Mathematical Principles of Natural Philosophy”, and was elected to Parliament that same year. 1727 he was buried in Westminster Abbey at the age of 85
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- Centripetal force is not a force at all it is simply a body resisting its natural inertia to move at the same speed in a straight line. It does not want to go in a circle. To make it you need a force to constantly accelerate it into changing directions. A change in motion is proportional to the force applied. The force causes a change in momentum which is mass * velocity.
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-------------------- Force = d ( m*v) = m*dv = m*a
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-------------------- F = ma
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--------------------- where: “m” is mass and acceleration “a” is a change in velocity, “dv”.
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------------------- For any action there is an equal and opposite reaction. Action is mass * (length)^2 / time. Action is Energy * time.
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------------------- The force of gravity is proportional to the masses and inversely proportional to the square of the distance between them. “G” is the constant of proportionality.
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-------------------- F = G*m*M / r^2
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- (4) Euler’s equations for natural logarithm, “e“, and imaginary numbers, “square root of -1“. Leonhard Euler (1707 - 1783) Age76, Basel, Switzerland. He got his masters degree at University of Basel at age 16. He went to the University of St. Petersburg, Russia, and became a math professor in 1733 at age 26. In 1735 he lost his right eye trying to look at the Sun and invent a new method for time determination. He published over 800 papers as a prolific mathematician.
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- The natural logarithm, “e” was invented from this math called the Taylor series”
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----------- e = 2.718 ..... = 1 + 1 / 1! + 1/2! + 1/3! + 1/4! + ...........................
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- The natural logarithm, “e”, raised to a power, an exponent “x”, 3! = 3*2*1 = 6:
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----------- e^x = 1+x + x^2/2! + x^3/3! + x^4/4! + .........................
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----------- sin x = x - x3/3! + x^5/5! - x^7/7! + .........................
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----------- cos x = 1 - x^2/2! + x4/4! - x^6/6! + .........................
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------------ e^ix = cos x + i sinx
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------------ e^i*pi = 1 + i*pi + (i*pi)^2 / 2! + (i*pi)^3 / 3! + (i*pi)^4 / 4! + .............
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------------ where “i” = the square root of -1 and “x” = pi, cos pi =0 sin pi = -1
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------------ e^ i*pi = -1
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- “e^ i*pi + 1 = 0” is considered the most famous equation in all of mathematics. It has a natural logarithm, “e“, pi, an angle of 180 degrees, the imaginary number “i”, the square root of -1 and +1 and a zero. There is nothing intuitive about this equation.
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- (5) Clausis’ equation for entropy. Rudolf Julius Emmanuel Clausis ( 1822 - 1888) Age 66, Poland. The energy in the Universe is a constant number. It can never increase or decrease. However, the entropy in the Universe constantly increasing, it can never decrease. Entropy constantly strives toward a maximum.
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------------------ (Entropy) - (Entropy at a later time) > 0
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- At absolute zero it is not to the energy that becomes zero, but, the property entropy becomes zero. Atoms are still moving in quantum at absolute zero, but, entropy stops. The capacity of any closed system to do work is called energy. That portion of energy that is lost in the system is called entropy, lost energy. There is always lost energy, because no system is perfectly efficient. The higher the entropy the less energy available to do work. Entropy is maximum when the Universe is in temperature equilibrium. Entropy is maximum randomness, maximum disorder.
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- (6) Maxwell’s equations for electromagnetism. James Clerk Maxwell (1831 - 1879) Age 48, Edinburgh, Scotland. James was nicknamed “Daffy” in school because of his unusual mathematical talent. He attended Cambridge in 1850, became a professor at Aberdeen in 1856. In 1857 he proved mathematically that the rings of Saturn were composed of small particles. He teamed up with Boltzman and proved that temperature was the average kinetic energy of atoms and that “P*V /T” for gases was a constant, pressure times volume / temperature was a constant, 8.314 per mole. From his equations he worked out Avogadro’s number 6.023*10^23 molecules in every mole of material. In 1864 he applied his math talent to Faraday’s magnetic lines of force and invented 4 equations that completely defined electromagnetic theory.
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------------ Integral E* dA = Q / Eo, and, Integral B*dA = 0
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------------- The 1st equation is Gauss’ Law for an electric field. Calculus applied to an electric flux though a closed surface with an electric charge inside. The 2nd equation is the same thing only applied to a magnetic monopole. This equation proved that a monopole cannot exist. The 3rd equation applied Amperes law relating a magnetic field to the current that creates it. The 4th equation applied Faraday’s law relating an induced electric field generated by changing magnetic field. Together these equations coupled electric and magnetic field that produced traveling waves of electromagnetic energy. Only accelerating charges can produce propagating electromagnetic fields. A changing electric field always produces a magnetic field which in turn induces another electric field. The fields traveled at right angles to each other at the speed of light. He proved light was actually electromagnetic energy.
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- (7) Einstein’s equation for energy, E = m*c^2. Albert Einstein (1879- 1955), Age 76, Ulm, Germany. Einstein was the son of a chemical engineer. He went to college in Switzerland , but failed to get a job as a professor upon graduation. So, at age 22 he became a clerk in the patent office in Berne. In 1905 at age 26 he produced 5 papers that would turn physics on its head . He did this with no money , no laboratory, using only his mind and pencil & paper. The Special Theory of Relativity had his famous E=mc^2. Nothing in the Universe could be viewed at absolute rest. No motion could be viewed as absolute motion, all motion is Relative. The speed of light was constant regardless of motion. His paper on the photoelectric effect showed that light was “quanta” of energy and a function of wavelength. The shorter the wavelength the more the energy. He showed statistically that Brownian Motion was atoms in vibration using Maxwell and Boltzmans’ kinetic energy of molecules.
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- At age 34 he finally became a professor in Berlin in 1913. In 1915 he came to Princeton, New Jersey, because the Nazi’s were coming to power and he was a Jew. The same year he published his theory for gravity in General Relativity. In 1921 he won the Nobel Prize for his photoelectric paper. In 1940 he became a U.S. citizen.
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- (8) Einstein’s equation for gravity. Time and energy could not be measured simultaneously. In 1895 George Francis Fitzgerald ( 1851 - 1901) proposed that distance would contract with the velocity of motion enough to keep the speed of light constant. The contraction of length with motion. In 1902 Hendrik Antoon Lorentz (1853 - 1928) proposed that mass increases with velocity. In 1905 Einstein pulled it all together. He proposed that physics was the same everywhere regardless of Relative motion. There was symmetry in the Universe whereby the laws of space-time were invariant. If force = mass * acceleration. Then, acceleration = force / mass. If the force was constant and the mass was constant then acceleration was constant. According to Newton’s equations then acceleration would increase forever. Einstein proposed that acceleration which was the rate of change of velocity was limited because velocity could not exceed the speed of light. A force causes motion and faster motion is a gain in Kinetic Energy and both mass and velocity must increase.
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- Einstein used Euler’s Binominal Theorem on Lorentz factor for this increase in mass with relativity.
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-------------- mass at rest = mass in motion * Lorentz Factor.
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Lorentz Factor = 1 / square root ( 1 - v^2/c^2). This could be written as:
( 1 - v^2/c^2)-.5. And, this has the binomial form of the Binomial Theorem :
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----------- (x+y)^n = (n:0)x^n + (n:1)*x^(n-1)* y + (n:2)*x^(n-2)* y^2 + (n:3)*x^(n-3)* y^3 + (n:2)*x^(n-4)* y^4 +……..(n:k)*x^(n-k)*y^k ......................... (n:n)y^n
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---------- where for: ( 1 - v^2/c^2)-0.5 , x = 1, y = (-v^2/c^2), n = ( -½ )
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----------- (1-v^2/c^2)^(-½)= 1^-½ + 3/4*1^(-½-1)* v^2/c^2 + 5/8*1^(-½-2)* (-
v^2/c^2),^2 + 7/24*1^(-½-3)* (-v^2/c^2)^3 + 9/96*1^(-½-4)* (-v^2/c^2)^4
+……..(-½:k)*1^(-½-k)*(-v^2/c^2), ^k ......................... (-½:-½)(-v^2/c^2), ^-½
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----------- where : (n:k) = n ! / k! * (n-k)!
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----------- where : (-½:0) = -½! / 0! * (-½-0)! = 1
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----------- where : (-½:1) = -½! / 1! * (-½-1)! = ¾
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----------- where : (-½:2) = -½! / 2! * (-½-2)! = 5/8
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----------- where : (-½:3) = -½! / 3! * (-½-3)! = 7/24
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----------- where : (-½:4) = -½! / 4! * (-½-4)! = 9/96
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--------- (1-v^2/c^2)^(-½)= 1 + 3/4* v^2/c^2 + 5/8* (v^2/c^2)^2 + 7/24*
(v^2/c^2)^3 + 9/96* (v^2/c^2)^4
+……..(-½:k)*1^(-½-k)*(v^2/c^2)^k ......................... (-½:-½)(v^2/c^2)^-½
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1 + v^2 / 2c^2 + 3v^4 /8c^4 + .................
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------------ m = m*( 1 - v^2/c^2)-.5
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------------ where: E = mc^2 , mass at rest = E/c^2
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----------- where E/c^2 = m * (1 + v^2 / 2c^2 + 3v^4 /8c^4 + .................)
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----------- where E/c^2 = m + m*v^2 / 2c^2 + 3m*v^4 /8c^4 + m*.................
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----------- where E = m c^2 + mv^2 / 2+ 3m*v^4 /8c^2 + mc^2*.................
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----------- where: Kinetic Energy = 1/2 mv^2
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----------- therefore: E = m c^2 + Kinetic Energy + smaller terms that are usually ignored .................)
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To get total Energy you must add Energy with mass at rest according to mc^2 plus the kinetic energy due to relative motion. Einstein’s equation for General Relativity G = k(R - 1/2 g R) where R is the geometry of space. k(R -1/2 gR) is the distribution of energy and momentum. Space is telling mass how to move and mass is telling space-time how to curve.
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- (9) Schrodinger’s equation for wave mechanics. Erwin Schrodinger (1867 - 1961) Age 85, Vienna, Austria. Here is his equation to define the structure of an atom in particle-waves.
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d^2U/dr^2 + 2(a+1) / r* dU/dr + 2*m*(E + e^2/r)*U/K^2 = 0
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His picture of the atom has the electron orbiting around the nucleus is a standing wave that extends an exact number of electron wavelengths. As long as the electron was a standing wave it did not accelerated in its orbit and did not emit electromagnetic radiation. A charged particle under acceleration always emits electromagnetic energy. A standing wave is not accelerating.
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- Schrodinger fled Germany under Hitler for England in 1940. He became a professor in Dublin, Ireland and did not return to Vienna until 1956. He died in 1961 at the age of 94.
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- (10) Heisenberg’s equation for uncertainty , dp*dx > h/2*pi. Werner Karl Messenger (1901 - 1976) Age 75, Wurzburg, Germany. Heisenberg got his Phd form the University of Munich in 1923 at the age of 22. He worked for Bohr, Sommerfeld, De Broglie and Schrodinger in trying to picture the atom. He began using Schrodinger’s wave mechanics to explain the wavelengths of spectral lines. Then Heisenberg invented matrix mechanics, the forerunner to Quantum Mechanics to get the math he wanted. In 1927 at the age of 26 he proposed that it was impossible to simultaneously determine both position and momentum of any body. This law at the time completely destroyed the deterministic philosophy of the physics community. He suggested the proton-neutron model for the atomic nucleus. With the number of neutrons being the Atomic Weight - the Atomic Number. During World War II Heisenberg was in charge of the Nazi program to develop the atomic bomb After the war he became director of the Max Planck Institute in Gottingen. He got the Nobel Prize in 1932.
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------------------ dp*dx > h/2*pi
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------------------ where: h/2*pi is a constant = 6.58*10^-16 electron volt * seconds.
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------------------ The probability of momentum * probability of position is always greater than Planck’s Constant.
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----------------- dp * dx > 10^-16 eV*seconds.
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