Generalized coordinates

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I said earlier on in these lectures that, later on, we would define generalized coordinates. I never gave you the formal definition for the prior examples since it wasn’t really necessary to know the technical definition in order to solve the problems. In this section, I’ll give a brief explanation of exactly what is meant by generalized coordinates. This formal definition will be necessary for solving the double-pendulum problem which we’ll discuss next. We’ll define a set of generalized coordinates $$q_j(x)$$ where $$j=1,2,\text{ … },n$$ as any coordinates which satisfy all of the following conditions: they must be complete, independent, and the physical system must be holonomic. Coordinates are complete if when you restrict a system’s range of motion along all degrees of freedom except for one )this is accomplished by setting all coordinate values equal to a constant except for on which is treated as a variable), the system can still move freely along that one degree of freedom and the variable coordinate can take on any value. Coordinates are independent if they can specify the location of every particle composing the system. Lastly, a system is holonomic if its number of degrees of freedom equal the number of coordinates. Let’s hit the point home by considering an example of a coordinate system which satisfies all three conditions. Let’s use the two angular coordinates $$(\theta_,\theta_2)$$ (where both angular coordinates are measured counter-clockwise from the vertical and $$\theta_1$$ and $$\theta_2$$ represent the angular position in $$m_1$$ and $$m_2$$, respectively) to specify the locations of all the particles composing a rigid double-pendulum as in Figure 1. If we set either one of the coordinates equal to a constant (imagine holding one of the pendulums in place), then the motion of the other pendulum is still entirely unrestricted along a degree of freedom and the other coordinate can thus take on any value. Therefore, the two angular coordinates $$(\theta_1,\theta_2)$$ are complete. Since we are idealizing the double-pendulum to be absolutely rigid, it follows that the coordinate $$\theta_1$$ specifies the angular position of every particle (the ones composing the string and the mass $$m_1$$ attached at the end of the string) along the length $$l_1$$ and, analogously, the coordinate $$\theta_2$$ specifies every particle along the length $$l_2$$. Thus, the coordinates $$(\theta_1,\theta_2)$$ are independent. Lastly, we can see that each pendulum has one degree of freedom; therefore the system as a whole has two degrees of freedom. Since the number of coordinates necessary to specify the location o fall the particles is equal to the number of degrees of freedom, it follows that the system is holonomic. Since our choice of coordinates $$(\theta_1,\theta_2)$$ satisfy all three of these conditions, they are generalized coordinates. In the case of a rigid double-pendulum, there is only one pair of generalized coordinates. But in general, there are many possible sets of generalized coordinates which can be used to describe the configuration of the system.