Recent content by RoyLB

The book Advanced Calculus by David Widder has a power series derivation of the Laplace Transform very much like Prof Mattuck's. I like the idea of the LT as an (infinite dimensional) dot product with exp(-st). The Fourier transform is similar - just use exp(j omega t). This approach can be...

Hurkyl, Thanks for the link. I really should look more closely at wikipedia. According to http://en.wikipedia.org/wiki/Fractional_Fourier_transform#Interpretation_of_the_Fractional_Fourier_Transform and http://en.wikipedia.org/wiki/Linear_canonical_transformation, the Fourier...

Hurkyl, Thanks for responding. Perhaps my question was too literal. If this relation is true, is there some was to understand the transform as a rotation, besides the Parseval theorem being analogous to preserving a "length" in some infinite dimensional space? What if we restricted...

From my undergraduate textbook: Circuits, Signals, and Systems by Siebert, p 453 ==================================================== Consider the two principal waveform representations schemes ... x(t) = \int x(\tau)\delta(t - \tau)d\tau x(t) = \int X(f)e^{j2\pi f t}df...

Studiot, Nevermind the math (unless its truly novel). Does he opine at all as to the nature of energy? - Roy

Studiot For control engineering, I'd agree. For my own knowledge, its all relevant :smile: So what does he say about what energy is? - Roy

GRDixon, I do indeed enjoy the Lectures and Feynman's other books. I was very disappointed that the great Feynman gave up when it came to explaining energy :smile: He explained quantum electrodynamics to the layman, but he couldn't explain energy to Caltech freshmen! IIRC, he promised in the...

Cleonis, We may agree. I was lamenting the fact that energy appeared to be just a bookkeeping device in Newtonian physics, when I knew it was more than that. I think to really appreciate what energy is, you have to go beyond the approximation that is Newtonian physics and turn to...

All, What do you guys think of this? I just found it in, from all places, Wikipedia: (http://en.wikipedia.org/wiki/Energy#Regarding_applications_of_the_concept_of_energy) In classical physics energy is considered a scalar quantity, the canonical conjugate to time. In special relativity...

Studiot, Thanks for the advice and the recommendation. I will look for it. - Roy

Cleonis I think what you're saying is that to really appreciate energy, we have to go beyond the simplification of our Newtonian worldview and embrace relativity. Forget about this ineffable "potential energy stored in a spring". The compressed spring has more mass (imperceptible as it is to...

Cleonis Perhaps it only works for quadratic functions of the velocity coordinates, and not the position ones I like this. As you indicate, potential energy is a function of the relative configuration in space of the components of the system. Kinetic energy is a function of the relative...

Cleonis, My initial objection was that this seemed like pure mathematics to me. Acceleration, distance, and velocity are kinematically related (or if you prefer, are derivatives of one other), so of course acceleration times distance will yield velocity squared terms. Now that I've embraced...

Gerenuk, I don't dispute that the dot product makes sense from a mathematical standpoint. I would have thought it was obvious from what I had posted several times now, but my intended audience is not mathematicians, physicists nor engineers. - Roy

In the book "Mathematical Aspects of Classical and Celestial Mechanics", 3rd edition, by Vladimir I. Arnold, Valery V. Kozlov, and Anatoly I. Neishtadt, this paragraph appears on page 19(!) ------------ Example 1.1. A natural mechanical system is a triplet (M, T, V ), where M is a smooth...