Abstract
This chapter and the next one illustrate the basic principles of Analytical Mechanics. Their purpose is to introduce a number of concepts that are not only useful per se, but also constitute a basis for the concepts of Quantum Mechanics that are introduced in later chapters. The Lagrangian function and the Lagrange equations are derived as a consequence of the variational calculus, followed by the derivation of the Hamiltonian function and Hamilton equations. Next, the Hamilton-Jacobi equation is derived after discussing the time-energy conjugacy. The chapter continues with the definition of the Poisson brackets and the derivation of some properties of theirs, and concludes with the description of the phase space and state space.
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Notes
- 1.
The units in (1.9) are: [m] = kg, [r] = m, \([\dot{\mathbf{r}}] =\mathrm{ m}\ \mathrm{s}^{-1}\), \([\ddot{x_{i}}] =\mathrm{ m}\ \mathrm{s}^{-2}\), [F i ] = N, where “N” stands for Newton.
- 2.
The units in (1.13) are: [F] = N, [e] = C, [E] = V m−1, [u] = m s−1, [B] = V s m−2 = Wb m−2 = T, where “N,” “C,” “V,” “Wb,” and “T” stand for Newton, Coulomb, Volt, Weber, and Tesla, respectively. The coefficients in (1.13) differ from those of [10] because of the different units adopted there. In turn, the units in (1.14) are: [φ] = V, [A] = V s m−1 = Wb m−1.
- 3.
- 4.
The letter “μ” stands for “molecule,” whereas the letter “γ” in the term “γ-space” stands for “gas.”
References
R. Becker, Electromagnetic Fields and Interactions (Dover, New York, 1982)
H. Goldstein, C. Poole, J. Safko, Classical Mechanics, 3rd edn. (Addison Wesley, San Francisco, 2002)
C. Lanczos, The Variational Principles in Mechanics, 4th edn. (Dover, New York, 1970)
L. Landau, E. Lifchitz, Mécanique (mir, Moscou, 1969) (in French)
E. Schrödinger, Quantisierung als eigenwertproblem (erste mitteilung). Annalen der Physik 384(4), 361–376 (1926) (in German)
R.C. Tolman, Statistical Mechanics (Dover, NewYork, 1979)
R. Weinstock, Calculus of Variations (Dover, New York, 1974)
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Rudan, M. (2018). Analytical Mechanics. In: Physics of Semiconductor Devices. Springer, Cham. https://doi.org/10.1007/978-3-319-63154-7_1
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DOI: https://doi.org/10.1007/978-3-319-63154-7_1
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