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Small Coupled Oscillations

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Types of motion Each multi-particle body has different types of degrees of freedom: translational, rotational and oscillatory

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Formulation of the problem Let us consider a many-particle Lagrangian The system is stable, if each particle has a stable equilibrium position We assume small deviations from equilibrium η i 6.1

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Formulation of the problem Kinetic energy of the system Potential energy of the system 6.1

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Formulation of the problem We assume that this term does not vanish The Lagrangian of the system Equations of motion 6.1

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Normal coordinates We have a system of linear ordinary differential equations of the 2 nd order A natural choice of a trial solution Equations of motion result in 6.2 6.3

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Normal coordinates Let us consider diagonal terms l = k 6.2 6.3

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Normal coordinates If real α and β are assumed to be some velocities, then this expression has a form of a kinetic energy, which is always positively defined Thus if 6.2 6.3

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Normal coordinates Equations of motion do not have exponentially growing solutions This can be true only for two diagonal matrices We have a freedom of normalization for matrix a ; let us impose the following normalization: Recall Then 6.2 6.3

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Normal coordinates Equations of motion : We completely diagonalized our problem We have a generalized eigen-value problem Eigen-values of the problem are solutions of the secular equation: Eigen-vectors: 6.2 6.3

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Normal coordinates Secular equation As the number of generalized coordinates increases, the power of the secular equation grows For very large systems, there are two ways to calculate eigen-values: analytical application of the group theory and computer calculations Modern applications: molecular vibrational spectroscopy, solid-state vibrational spectroscopy, etc. 6.2 6.3

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Example: longitudinal oscillations of a CO 2 molecule CO 2 is a linear molecule; we will model it as follows: The Lagrangian 6.4

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Example: longitudinal oscillations of a CO 2 molecule Secular equation: 6.4

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Example: longitudinal oscillations of a CO 2 molecule Eigen-vectors: 6.4

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Example: longitudinal oscillations of a CO 2 molecule Eigen-vectors: 6.4

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Example: longitudinal oscillations of a CO 2 molecule Normal coordinates: 6.4

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Forced oscillations For open systems, we introduce generalized forces For each generalized coordinate, there is a component of a force We can introduce modified generalized forces for each normal coordinate Total work done Equations of motion: 6.5

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Forced oscillations Let us consider a periodic external force We look for a solution in the following form: After substitution into the equation of motion For generalized coordinates Resonance 6.5

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Questions?

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Normal coordinates 6.2 6.3

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The independent coordinates of a rigid body Let us consider a many-particle Lagrangian The system is stable, if each particle has a stable equilibrium position We assume small deviations from equilibrium 6.1

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The independent coordinates of a rigid body Let us consider a many-particle Lagrangian The system is stable, if each particle has a stable equilibrium position 6.1

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