Abstract
This chapter will be devoted to the important phenomenon of oscillations, which has numerous applications and generalizations in several areas of Physics. The small oscillations without friction manifest a property of universality, such that they can be treated as harmonic ones. More complicated cases include harmonic oscillations in the presence of damping force and external source. We shall present a few simple examples illustrating the treatment of multidimensional oscillators by means of normal modes.
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Notes
- 1.
It is worth noting that this restriction is only valid in classical mechanics. In quantum physics, the passage through a potential barrier is possible, in a phenomenon called “quantum tunneling”.
- 2.
The real meaning of this is that the repetition of cycles will not end in a finite time. In practice, there will always be dissipative forces, and there will be a moment at which the particle stops at some point near the equilibrium position x 0.
- 3.
Two functions q 1 (t) and q 2 (t) are linearly independent if \(B_{1}\,q_{1}(t) + B_{2}\,q_{2}(t) \equiv 0\) implies \(B_{1} = B_{2} = 0\) . We leave it as an exercise to show that if this criterion is not satisfied, the Wronskian W(t) is identically zero. The demonstration of the sufficiency of W(t)≠0 for linear independence requires greater efforts, but can be found in many courses of differential equations (see, e.g., [4, 7] ).
- 4.
It is easy to see that the difference between two solutions of the non-homogeneous equation is a solution of the homogeneous equation. In other words, the difference between two such solutions can always be compensated by adjusting the constants of integration and eventually disappear at the moment when we implement initial conditions.
- 5.
There is a mathematically rigorous way to prove this statement in Lagrangian (Analytical) Mechanics. Here we rely on our intuition.
References
V.I. Arnold, Ordinary Differential Equations, 3rd edn. (Springer, 1997)
M. Braun, Differential Equations and Their Applications: An Introduction to Applied Mathematics. Texts in Applied Mathematics, vol. 11, 4th edn. (Springer, New York, 1992)
H. Goldstein, C.P. Poole, J.L. Safko, Classical Mechanics, 3rd edn. (Addison Wesley, San Francisco, 2001)
I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 7th edn. (Elsevier, Burlington, 2007)
L.D. Landau, E.M. Lifshits, Course of Theoretical Physics: Mechanics, 3rd edn. (Butterworth-Heinemann, 1982)
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Shapiro, I.L., de Berredo-Peixoto, G. (2013). Movement in a Potential Field: Oscillations. In: Lecture Notes on Newtonian Mechanics. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7825-6_7
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DOI: https://doi.org/10.1007/978-1-4614-7825-6_7
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