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At the beginning of this semester, when my prof in classical mechanics derived Lagrange's Equations, he introduced the concept of "generalised coordinates" to develop the ideas. These coordinates he called
[tex] {q}_j \ \ \text{and} \ \ \dot{q}_j [/tex]
j = 1, 2, ... , 3N
N is the number of particles in the system. He wrote down the Lagrangian as
[tex] L(q, \dot{q}, t) [/tex]
But I was always confused by this. Those are not independent variables, I thought. Surely the genearlised position and velocity coordinates are both functions of time! He sort of glossed over this in class. The only thing I got from his explanation was that we consider them to be arbitrary, independent parameters until will solve the equations of motion and arrive at a solution for each, seeing that they are related. But to me this was more a description of what we were doing than a justifcation for it. Can anybody explain what's going on, especially in light of these observations (which just represent my understanding of what's going on...they may have errors)?
1. [itex] q [/itex] is always related to [itex] \dot{q} [/itex] by the diff. eq. [itex] \dot{q} = dq/dt [/itex]. So they are never independent. In fact, for all t for which [itex] q [/itex] and [itex] \dot{q} [/itex] are both defined, there is one value of the latter for every value of the former. So would it be correct to say that [itex] \dot{q} [/itex] is a function of q? Is this true in general for a function and its time derivative?
2. I understand how it might be "safest" or most "rigorous" and "general" NOT to make any assumptions about the nature of the generalised coordinates (just to pretend they are arbitrary, independent parameters that describe the system), but is all this care really necessary in light of 1? We know that q and q' are functions of time, and no solution of the equations of motion obtained by Lagrangian methods will result in q' independent of q...that would be totally unphysical, wouldn't it?
Totally confused...
[tex] {q}_j \ \ \text{and} \ \ \dot{q}_j [/tex]
j = 1, 2, ... , 3N
N is the number of particles in the system. He wrote down the Lagrangian as
[tex] L(q, \dot{q}, t) [/tex]
But I was always confused by this. Those are not independent variables, I thought. Surely the genearlised position and velocity coordinates are both functions of time! He sort of glossed over this in class. The only thing I got from his explanation was that we consider them to be arbitrary, independent parameters until will solve the equations of motion and arrive at a solution for each, seeing that they are related. But to me this was more a description of what we were doing than a justifcation for it. Can anybody explain what's going on, especially in light of these observations (which just represent my understanding of what's going on...they may have errors)?
1. [itex] q [/itex] is always related to [itex] \dot{q} [/itex] by the diff. eq. [itex] \dot{q} = dq/dt [/itex]. So they are never independent. In fact, for all t for which [itex] q [/itex] and [itex] \dot{q} [/itex] are both defined, there is one value of the latter for every value of the former. So would it be correct to say that [itex] \dot{q} [/itex] is a function of q? Is this true in general for a function and its time derivative?
2. I understand how it might be "safest" or most "rigorous" and "general" NOT to make any assumptions about the nature of the generalised coordinates (just to pretend they are arbitrary, independent parameters that describe the system), but is all this care really necessary in light of 1? We know that q and q' are functions of time, and no solution of the equations of motion obtained by Lagrangian methods will result in q' independent of q...that would be totally unphysical, wouldn't it?
Totally confused...