- #1
Rasalhague
- 1,387
- 2
(1) How can a generalized velocity function, [itex]\dot{q}[/itex], be "independent" of the corresponding generalized position function, [itex]q[/itex]. One is the derivative of the other.
(2) How can any Lagrangian function be "time-independent", given that its component functions are defined as functions that depend on time, [itex](q^1,...,q^n)[/itex], and their time derivatives, [itex](\dot{q}^1,...,\dot{q}^n[/itex]? Is dependence not a transitive relation?
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Is it that on a specific path through [itex]\mathcal{T}(\mathcal{C})[/itex], the tangent bundle of a configuration space, velocity can perhaps be parameterized by position, and therefore made dependent on position, somewhat as as a path in [itex]\mathbb{R}^2[/itex] can be parameterized by either one of its two component functions (so long as it's not parallel to the other axis for any way), or by some other parameter, call it [itex]t[/itex]--but to name an arbitrary point in the set of all possible states of a system (before any constraints are specified, and thus before any curve is defined) requires independent coordinate values for position and velocity, since either could take any value for all we know yet (given the degrees of freedom), just as, in [itex]\mathbb{R}^2[/itex], it takes two real numbers to unambiguously identify a point, given all the possible points that exist in that set.
And in answer to 2, I'm guessing dependence is indeed not transitive, since the domain of a function, [itex]f[/itex], is not necessarily the same as the domain of the composition, [itex]f \circ g[/itex] of that function with another, [itex]g[/itex]. I suppose an analogy would be containment, which is transitive in everyday English, but not in set theory: [itex]x \in S[/itex] and [itex]S \subseteq T[/itex] does not imply [itex]x \in T[/itex].
(2) How can any Lagrangian function be "time-independent", given that its component functions are defined as functions that depend on time, [itex](q^1,...,q^n)[/itex], and their time derivatives, [itex](\dot{q}^1,...,\dot{q}^n[/itex]? Is dependence not a transitive relation?
*
Is it that on a specific path through [itex]\mathcal{T}(\mathcal{C})[/itex], the tangent bundle of a configuration space, velocity can perhaps be parameterized by position, and therefore made dependent on position, somewhat as as a path in [itex]\mathbb{R}^2[/itex] can be parameterized by either one of its two component functions (so long as it's not parallel to the other axis for any way), or by some other parameter, call it [itex]t[/itex]--but to name an arbitrary point in the set of all possible states of a system (before any constraints are specified, and thus before any curve is defined) requires independent coordinate values for position and velocity, since either could take any value for all we know yet (given the degrees of freedom), just as, in [itex]\mathbb{R}^2[/itex], it takes two real numbers to unambiguously identify a point, given all the possible points that exist in that set.
And in answer to 2, I'm guessing dependence is indeed not transitive, since the domain of a function, [itex]f[/itex], is not necessarily the same as the domain of the composition, [itex]f \circ g[/itex] of that function with another, [itex]g[/itex]. I suppose an analogy would be containment, which is transitive in everyday English, but not in set theory: [itex]x \in S[/itex] and [itex]S \subseteq T[/itex] does not imply [itex]x \in T[/itex].