- #1
jjustinn
- 164
- 3
What would the Hamiltonian for a system of two classical point particles, with no interaction except for an elastic collision between them at a point look like?
My gut says it's the usual T + V, with T = p12/2m1 + p22/2m2 and
V = Kδ(r1-r2)
With K approaching infinity -- each particle acts to the other like an infinite potential barrier.
I am bothered by the infinity multiplying the (infinite) delta function...it doesn't seem absolutely necessary that it's infinite; if I'm not mistaken, it should only need to be later than the kinetic energy , but I wanted to avoid momentum-dependent forces if possible: I'm reminded of the "normal force" of intro mechanics, which is exerted by an immovable barrier on anything acting on it.
I'm also bothered that I cannot find this Hamiltonian anywhere, since it 'a such a fundamental system...but maybe my search terms are just no good ( Hamiltonian "elastic collision" "interaction term")
My gut says it's the usual T + V, with T = p12/2m1 + p22/2m2 and
V = Kδ(r1-r2)
With K approaching infinity -- each particle acts to the other like an infinite potential barrier.
I am bothered by the infinity multiplying the (infinite) delta function...it doesn't seem absolutely necessary that it's infinite; if I'm not mistaken, it should only need to be later than the kinetic energy , but I wanted to avoid momentum-dependent forces if possible: I'm reminded of the "normal force" of intro mechanics, which is exerted by an immovable barrier on anything acting on it.
I'm also bothered that I cannot find this Hamiltonian anywhere, since it 'a such a fundamental system...but maybe my search terms are just no good ( Hamiltonian "elastic collision" "interaction term")
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