Abstract
In this chapter, we will apply the Lagrangian formalism to the general case of coupled N-body systems perturbed from equilibrium. We will first look at the simple one-dimensional oscillator as a refresher, and then go on to develop the general formulation of the problem, applying it to solve a few simple example problems.
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Notes
- 1.
If M is not already independent of q, we can always change variables to a new generalized coordinate \(q'\equiv \int \text {d}q\>\sqrt{M(q)/M_0}\), in terms of which \(M(q)\dot{q}^2 = M_0 \dot{q}'{}^2\), with \(M_0\) independent of q.
- 2.
There is no loss of generality in assuming a sinusoidal driving force as we have done here, since most real driving forces can be expressed as a Fourier series or a Fourier transform, which involve sums of such oscillatory terms.
- 3.
If there were more than one redundant equation, then we can simply repeat this process until we arrive at an invertible matrix with a non-zero determinant.
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Benacquista, M.J., Romano, J.D. (2018). Small Oscillations. In: Classical Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68780-3_8
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DOI: https://doi.org/10.1007/978-3-319-68780-3_8
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