Out of scientific curiosity, I am interested in Lagrangian dynamical systems that can be expressed in a "linear" manner. By this, I mean that their Lagrangian can be expressed, linearly, as
L = 1/2 (dq/dt)^T.M.(dq/dt) - 1/2 q^T.K.q^T,
where 'q' is a vector in R^n; 'M' and 'K' are symmetric, positive-definite, n-dimensional square matrices with constant coefficients.
One example of such a system is the "linear" double pendulum which is simply a double pendulum where one assumes small angles during motion (see for example https://math24.net/double-pendulum.html).
Usually, it is easy to obtain the analytic solutions for this kind of system (as for the linear double pendulum). So my question is if there exist any other examples such as this one, with true technical/scientific interest? And particularly, is there a similar example where the analytic solution is not obvious at all (requiring the use of numerical methods)?
The Euler-Lagrange equations for this are simply (d/dt)(M.dq/dt) = -K.q. to find exact solutions all you need are the generalized eigenvalues of (M,K) i.e., the values a where M.v = a.K.v. with v non-zero. If n >= 5 there are symmetric positive definite matrices for which there is no formula for these eigenvalues. But that's the only thing you need numerical methods for.
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