What Do Hamiltonians and Hamilton's Equations Tell Us About Dimensions?

[Nabeshin]

Hamiltonians, and in general Hamilton's equations, make no reference to dimensionality at all. You can apply them to an arbitrary system.

What we're trying to say is as simple as this: When calculating integrals in a phase space, for example of one particle with momentum, you integrate over three spatial coordinates and three momenta to perform the integral over all of phase space. This is a six-dimensional integral, everyone calls it that, and these are the 'Dimensions' that we're familiar with. They might not share the same status as the 3+1 that we know about, but they certainly serve a similar mathematical function, and that's why we use the same word for both.

Your final comment about space expanding and gravitation makes little sense, everyone (who understands GR) knows that spacetime is warped by stress/energy/momentum, not simply mass. And the phrase 'each particle is a dimension' was never uttered anywhere except for in your post, and I can not make anything of it.

Finally, indeed challenging a mentor is not grounds for immediate banning, as you seem to think. Being obnoxious and adversarial, however, is not going to contribute any brownie points. I suggest you spend less time complaining about your imminent ban and more time trying to understand.

 

[Max™]

^Trufax.

 

[BadBrain]

Hamiltonians, and in general Hamilton's equations, make no reference to dimensionality at all. You can apply them to an arbitrary system.

What we're trying to say is as simple as this: When calculating integrals in a phase space, for example of one particle with momentum, you integrate over three spatial coordinates and three momenta to perform the integral over all of phase space. This is a six-dimensional integral, everyone calls it that, and these are the 'Dimensions' that we're familiar with. They might not share the same status as the 3+1 that we know about, but they certainly serve a similar mathematical function, and that's why we use the same word for both.

Your final comment about space expanding and gravitation makes little sense, everyone (who understands GR) knows that spacetime is warped by stress/energy/momentum, not simply mass. And the phrase 'each particle is a dimension' was never uttered anywhere except for in your post, and I can not make anything of it.

Finally, indeed challenging a mentor is not grounds for immediate banning, as you seem to think. Being obnoxious and adversarial, however, is not going to contribute any brownie points. I suggest you spend less time complaining about your imminent ban and more time trying to understand.

What I'm saying is that Hamilton was definitely ahead of his time, and that he suspected that something else was out there beyond the physics of his own understanding.

The three spatial dimensions are obvious. The three moments I see as functions of the three spatial dimensions with respect to time. In other words, his calculations respected time as a dimension which interacts with space, even though he never phrased it in that manner. Moments necessarily imply motion, and, in the absence of time, there is no motion, only a dead stasis which makes nonsense of the Hamiltonians, as kinetic energy is impossible and potential energy is irrelevant. He was, in fact, describing space-time eighty years before Einstein, having access to more advanced knowledge than that available to Hamilton, formalized the concept.