Mathematics of Physics and Engineering
86 The Lagrange-Hamilton Method (2.3.7) Similarly, d / dx \ dx d ( dy\ _ dy dt\dqi) dqi' dt \dqiJ dqi' d ( dx \ _ dx d f dy\ _ d ...
Lagrange's Equations^87 Recall that the work W done by the force F is the line integral JcF • dr. Hence, the differential is dW ...
88 The Lagrange-Hamilton Method and the generalized coordinates are (9i,92,93,.-.,9fc,9i,---,9fc)> two for each degree of fre ...
Lagrange's Equations^89 function: F = — VV; the negative sign is used by convention. The func- tion V is called the potential of ...
90 The Lagrange-Hamilton Method corresponding time derivatives, so that the vector (q, q) uniquely deter- mines the state of the ...
An Example of Lagrange's Method 91 see also Exercise 2.3.4 below. A bead of mass m placed on the wire will slide down under the ...
(^92) The Lagrange-Hamilton Method Ptan = P • — + P • "J-J Q- definition of Q agrees with (2.3.22) on page 88. Differentiating p ...
Hamilton's Equations 93 where Q is the generalized force from (2.3.16), that is, Q = F • dr/dq = -mg k • dr/dq. (2.3.41) We ther ...
94 The Lagrange-Hamilton Method To obtain another k equation, define the Hamiltonian k H(p, q) = Y<PM ~ L(l> <?)> (^ ...
Hamilton's Equations 95 Substituting (2.3.50) into 1 ™ results in 3fe £K = Yl aMiQi = QTAq, (2.3.51) where A = (a,j:e, j,t= 1,.. ...
96 Theory of Relativity close to the speed of light the description is provided by the theory of relativity. Special relativity ...
Historical Background 97 eral relativity. An interested reader can follow up on some of the topics in Problem 2.2, page 414, and ...
98 Theory of Relativity aether existed, then the Earth's motion around the Sun, at the speed of about 30km/sec, would affect the ...
The Lorentz Transformation 99 2.4.2 The Lorentz Transformation and Special Relativity Consider two frames, the fixed frame O and ...
100 Theory of Relativity the frame 0\, then A + y\ + *\ = cH\ (2.4.4) By Postulate (ii), and because of relations (2.4.1), (2.4. ...
The Lorentz Transformation 101 Equations (2.4.9), (2.4.10), and (2.4.11) are the Lorentz transformation equations that govern sp ...
(^102) Theory of Relativity total energy £ of the same object moving with velocity v are m m 0 Vl-(«^2 /c^2 ) £ = mc^2. (2.4.13) ...
The Lorentz Transformation 103 EXERCISE 2.4.2. (a)c Verify that ||p||^2 + mlc^2 = m^2 c^2. (2.4.16) Hint: direct computation usi ...
104 Theory of Relativity world line would be £,,,,„„ = f£ ^\x'(s)\^2 + |y'(s)|^2 + |z'(s)|^2 + \t'(s)\^2 ds. Even if we disregar ...
Einstein's Field Equations 105 called the Minkowski space, after the German mathematician HERMANN MINKOWSKI (1864-1909), who int ...
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