- #1

BadBrain

- 196

- 1

http://en.wikipedia.org/wiki/Hamiltonian_mechanics

I refer specifically to this passage:

"Basic physical interpretation

The simplest interpretation of the Hamilton equations is as follows, applying them to a one-dimensional system consisting of one particle of mass m under time-independent boundary conditions: The Hamiltonian represents the energy of the system (provided that there are NO external forces, or additional energy added to the system), which is the sum of kinetic and potential energy, traditionally denoted T and V, respectively. Here q is the x coordinate and p is the momentum, mv. Then

H = T + V, T = p^2/2m, V = V(q) = V(x)

Note that T is a function of p alone, while V is a function of x (or q) alone.

Now the time-derivative of the momentum p equals the Newtonian force, and so here the first Hamilton equation means that the force on the particle equals the rate at which it loses potential energy with respect to changes in x, its location. (Force equals the negative gradient of potential energy.)

The time-derivative of q here means the velocity: the second Hamilton equation here means that the particle’s velocity equals the derivative of its kinetic energy with respect to its momentum. (Because the derivative with respect to p of p2/2m equals p/m = mv/m = v.)"

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My problem here is the statement to the effect that "the force on the particle equals the rate at which it loses potential energy with respect to changes in x, its location." I feel that a more correct statement would be that the force on the particle equals the rate at which its potential energy is being converted into kinetic energy (assuming it's falling within a vacuum, eliminating the need to account for frictional atmospheric resistance and consequent draining of gravitational potential energy into heat-generastion through the process of overcoming aerodynamic drag). Furthermore, seeing as we're only speaking of mono-dimemsional motion here, why use the phrase "changes in x, its location", as opposed to the standard term "translational displacement"?