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- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/14%3A_Coupled_Linear_Oscillators/14.12%3A_Collective_Synchronization_of_Coupled_OscillatorsCollective synchronization of coupled oscillators is a multifaceted phenomenon where large ensembles of coupled oscillators, with comparable natural frequencies, self synchronize leading to coherent c...Collective synchronization of coupled oscillators is a multifaceted phenomenon where large ensembles of coupled oscillators, with comparable natural frequencies, self synchronize leading to coherent collective modes of motion. Biological examples include congregations of synchronously flashing fireflies, crickets that chirp in unison, an audience clapping at the end of a performance, networks of pacemaker cells in the heart, as well as neural networks in the brain and spinal cord.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/14%3A_Coupled_Linear_Oscillators/14.11%3A_Damped_Coupled_Linear_OscillatorsIn general, dissipative forces are non linear which greatly complicates solving the equations of motion for damped coupled oscillator systems. However, for some systems the dissipative forces depend l...In general, dissipative forces are non linear which greatly complicates solving the equations of motion for damped coupled oscillator systems. However, for some systems the dissipative forces depend linearly on velocity which allows use of the Rayleigh dissipation function.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/14%3A_Coupled_Linear_Oscillators/14.S%3A_Coupled_linear_oscillators_(Summary)It is observed that the eigenvalue corresponding to the most coherent motion of the coupled oscillators corresponds to the most collective motion and its eigenvalue is displaced the most in energy fro...It is observed that the eigenvalue corresponding to the most coherent motion of the coupled oscillators corresponds to the most collective motion and its eigenvalue is displaced the most in energy from the remaining eigenvalues.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/14%3A_Coupled_Linear_Oscillators/14.10%3A_Discrete_Lattice_ChainA crystalline lattice comprises thousands of coupled oscillators in a three dimensional matrix. A classical treatment of lattice dynamics of is of interest since classical mechanics underlies many fea...A crystalline lattice comprises thousands of coupled oscillators in a three dimensional matrix. A classical treatment of lattice dynamics of is of interest since classical mechanics underlies many features of the motion of atoms in a crystalline lattice. The linear discrete lattice chain is the simplest example of many-body coupled oscillator systems that can illuminate the physics underlying a range of interesting phenomena in solid-state physics.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/14%3A_Coupled_Linear_Oscillators/14.E%3A_Coupled_linear_oscillators_(Exercises)Two particles, each with mass \(m\), move in one dimension in a region near a local minimum of the potential energy where the potential energy is approximately given by \[U = \frac{1}{2} k (7x^2_1 + 4...Two particles, each with mass \(m\), move in one dimension in a region near a local minimum of the potential energy where the potential energy is approximately given by \[U = \frac{1}{2} k (7x^2_1 + 4x^2_2 + 4x_1x_2)\nonumber\] where \(k\) is a constant. Four identical masses \(m\) are connected by four identical springs, spring constant \(\kappa\), and constrained to move on a frictionless circle of radius \(b\) as shown on the left in the figure.
