Classical Mechanics Joel phần 8 ppt

... is unchanged, it must be independent of q I ,andq I is an ignorable coordinate. 6 .8. ACTION-ANGLE VARIABLES 187 Exercises 6.1 In Exercise 2.6, we discussed the connection between two Lagrangians, L 1 and ... in the 2n + 1 dimensional extended phase space which includes time as one of its coordinates. 188 CHAPTER 6. HAMILTON’S EQUATIONS A 1-form acts on such a vector by dx j (D f )=f j . Sh...

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... 88 CHAPTER 4. RIGID BODY MOTION and we may conclude from the fact that the ˆe j are linearly independent that ... rather than Euclidean space arise often in applications in quantum mechanics and quantum field theory, in ad- dition to the classical problems we will consider such as gyroscopes and tops. 4.1 ... are very well approximated by the assumption that the distances between the atom...

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... 2nd edition, 1969. QA805.L 283 /1976. [7] Jerry B. Marion and Stephen T. Thornton. Classical Dynam- ics. Harcourt Brace Jovanovich, San Diego, 3rd ed edition, 1 988 . QA845.M 38/ 1 988 . [8] R. A. Matzner ... Math. Methods of Classical Mechanics. Springer- Verlag, New York, 1 984 . QA805.A 681 3. [3] R. Creighton Buck. Advanced Calculus. McGraw-Hill, 1956. [4] Herbert Goldstein. Classi...

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... will need to go beyond classical mechanics, we begin with a few words of justification for investing effort in under- standing classical mechanics. First of all, classical mechanics is still very ... 181 6 .8 Action-AngleVariables 185 7 Perturbation Theory 189 7.1 Integrablesystems 189 7.2 CanonicalPerturbationTheory 194 7.2.1 TimeDependentPerturbationTheory 196 7.3 AdiabaticInv...

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... arose in 183 5 in his unification of the language of optics and mechanics. It too had a usefulness far beyond its origin, and the Hamiltonian is now most familiar as the operator in quantum mechanics ... begin to feel the asym- metry of the potential, and the orbits become egg-shaped. 1 -1 x p 1.210 .8 0.6 0.40.2-0.2-0.4 0 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 x U U(x) Figure 1.3. Motion in a cubic...

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... fields  E(r,t)and  B(r,t),  F = q   E + v c ×  B  . (2. 18) If the motion of a charged particle is described by Lagrangian mechanics with a potential U(r,v, t), Lagrange’s equation says 0= d dt ∂L ∂v i − ∂L ∂r i = ... discussed in Chapter 8, but until then we will assume that the phase space (q, p), or cotangent bundle, is equivalent to the tangent bundle, i.e. the space of (q,...

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... generally used in quantum mechanics. Many of the standard classical mechanics texts 10 take the second rotation to be about the x 1 - axis instead of y 1 , but quantum mechanics texts 11 avoid ... add or subtract, and many choices for the Euler angles (φ, ψ) will give the same full rotation. 1 18 CHAPTER 4. RIGID BODY MOTION Then E becomes E = 1 2 I 1 ˙ θ 2 + U  (θ)+ 1 2 I 3 ω 2 3 ,...

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... expression in (4.29). [For this and exercise 4 .8, you might want to use a computer algebra program such as mathematica or maple, if one is available.] 4 .8 Find the expression for ω in terms of φ, ... 1 28 CHAPTER 5. SMALL OSCILLATIONS Note that A is a constant symmetric real matrix. The kinetic energy ... messy.] 4.9 A diamond shaped object is shown in top, front, and side views. It is a...

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... canonical. The Poisson bracket plays such an important role in classical me- chanics, and an even more important role in quantum mechanics, that it is worthwhile to discuss some of its abstract ... R n and totally antisymmetric in its action on the k copies, and is a smooth function of x ∈M.Ata 1 58 CHAPTER 6. HAMILTON’S EQUATIONS where f j are an arbitrary set of functions on phase space...

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... 2 08 CHAPTER 7. PERTURBATION THEORY Consider the evolution in time, un- der the full hamiltonian, of ... magnetic field. It is interesting to note that in Bohr-Sommerfeld quantization in the old quantum mechanics, used be- fore the Schr¨odinger equation clarified such issues, the quantization of bound ... + 1 ω 2 ξ  η(t)= 1 ω ∂U ∂q cos ωt + 1 ω 2 η  and assume ¯q and ¯p obey Hamiltons equat...

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