Chapter 5: Problem 11
Show that if we increase the k?netic energy of a system without decreasing the potential energy (for example, we increase the mass on a given spring), then every characteristic frequency decreases.
Chapter 5: Problem 11
Show that if we increase the k?netic energy of a system without decreasing the potential energy (for example, we increase the mass on a given spring), then every characteristic frequency decreases.
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Get started for freeShow that the number of independent real characteristic oscillations is equal to the dimension of the largest positive definite subspace for the potential energy \(\frac{1}{2}(B \mathbf{q}, \mathbf{q})\).
Find the shape of the region of stability in the \(\varepsilon_{.}(0\)-plane for
the system described by the equations
$$
\begin{gathered}
\dot{x}=-f^{2}(t) x \quad f(t)=\left\\{\begin{array}{ll}
\omega+\pi & 0
Show that under the orthogonal projection of an ellipsoid lying in one subspace of euclidean space onto another subspace, all the semi-axes are decreased.
Show that, if \(T\) is the period of \(\mathbf{f}\), then \(g^{T+s}=g^{s} \cdot g^{T}\) and, in particular, \(g^{n T}=\left(g^{T}\right)^{n}\), so that the mappings \(g^{n T}\) ( \(n\) an integer) form a group.
Show that linearization is a well-defined operation: the operator \(A\) does not depend on the coordinate system. The advantage of the linearized system is that it is linear and therefore easily solved: \(\mathbf{y}(t)=e^{A t} \mathbf{y}(0), \quad\) where \(e^{A t}=E+A t+\frac{A^{2} t^{2}}{2 !}+\cdots .\)
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