Vibration in Several Degrees of Freedom

The equations of motion, by Hamilton s formulation, follow from applying Equations 7.5 and 7.6 with this Hamiltonian. 

Double harmonic oscillator. This consists of two particles, 1 and 2, that move only in the direction of the x-axis. The masses of 1 and 2 are m, and njj, respectively, and their positions are x, and Xj. Spring a is coimected to an unmovable wall and to particle 1. Spring b connects the two particles. 

These are coupled differential equations that reflect the fact that the two springs are physically coupled. They are connected at particle 1. Solving this type of problem or this set of coupled differential equations is considered briefly later in this chapter. For now, the important feature is that systems of several degrees of freedom, in general, have coupled equations of motion. 

There are circumstances where the equations of motion in several degrees of freedom are not coupled. In these cases, the Hamiltonian has a special form referred to as a separable form. Separability arises when a particular Hamiltonian can be written as additive, independent functions  

In Equation 7.19, H is a function only of the first position and momentum coordinates, while H is a separate function involving only the other coordinates. When we apply Equations 7.5 and 7.6 to H in order to find the equations of motion for this system, the equations for q and p are found to be distinct, or mathematically unrelated to any of the equations that involve any of the other coordinates. In fact, the equations for q and p could just as well have been obtained by taking H to be the Hamiltonian for the motion in the Jl-direction, while the other equations of motion could have been obtained directly from the H Hamiltonian. Thus, the additive, independent terms in the Hamiltonian can be separated in order to deduce the equations of motion. Furthermore, since the equations of motion are separable, the actual physical motions of the H system and the H system are unrelated and independent. It is possible that a Hamiltonian can be separable in all variables that is, it may be that 

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