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I think the definition of generalised coordinates is something along the following lines:

A set of parameters that discribe the configuration of a system with respect to some refrence configuration.

I am however, confused about one thing. Does the number of generalised coordinates in a system have to equal the number of degrees of freedom, or can we have more?

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  • $\begingroup$ You can have as many coordinates as you want: the question is, why would you need more? $\endgroup$ – Demosthene Mar 2 '15 at 12:40
  • $\begingroup$ @Demosthene You may want more if reducing the number to the number of degrees of freedom lead to a very 'nasty' expression, but I see your point. $\endgroup$ – Quantum spaghettification Mar 2 '15 at 12:47
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The minimum number of independent generalized coordinates is given by the number of degrees of freedom. Nothing prevents you from using more coordinates (dependent) if you so desire.

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You can definitely have more generalised coordinates than the degrees of freedom. Consider a particle in space which is constrained to move on a straight horizontal rod. You can describe such a system by choosing the $x,y,z$ coordinate of the particle in space as generalized coordinates. But the constraint is reducing the degrees of freedom from 3 to 1, so that you will get an equivalent description with just one generalized coordinate, if chosen wisely.

In the above example, for a particle to move on a horizontal rod, say the $x$ axis, the constraints are $y=z=0$, which can be introduced, say, in the Hamiltonian through Lagrange multipliers, viz. $$H = \frac{\Vert\mathbf p\Vert^2}{2m} + \lambda y + \mu z.$$ Since the constraint are time independent you will have to enforce $$\{y,H\} = \{z,H\} = 0,$$ from which you get the secondary constraints $p_y = p_z = 0$, which express the fact that the particle cannot move along the $y$ and the $z$ direction because of the primary constraints. You can then forget about the $y$ and $z$ coordinates and just describe your system with 1 generalised coordinate, namely $x$, and simply write $$H = \frac{p_x^2}{2m}$$ which is defined on the reduced phase space $\mathbb R\times\mathbb R$, rather than the larger (and, in some sense, redundant) $\mathbb R^3\times\mathbb R^3$.

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The minimum number of independent coordinate or variable which is required to describe the motion of a dynamical system is known as generalized coordinate.

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