A point particle with mass m moves in the potential V =...

Name: A point particle with mass m moves in the potential V = a/r^2 - bz/r^3 - c(x^2 + y^2)/r^4, where r^2 = x^2 + y^2 + z^2, a, b, c are constants. Find its Hamilton's principal function and characteristic function.
Uploaded: 2023-05-29T16:54:00-08:00
Duration: 7 min 4 s
Channel: Sam Stansfield
Description: A point particle with mass m moves in the potential V = a/r^2 - bz/r^3 - c(x^2 + y^2)/r^4, where r^2 = x^2 + y^2 + z^2, a, b, c are constants. Find its Hamilton's principal function and characteristic function.

A point particle with mass m moves in the potential V = a/r^2 - bz/r^3 - c(x^2 + y^2)/r^4, where r^2 = x^2 + y^2 + z^2, a, b, c are constants. Find its Hamilton's principal function and characteristic function.

Transcript

00:01 So we're given this hamiltonian, just t plus v.

00:07 It's in three dimensions, so our p represents the three -dimensional moment.

00:12 I'm gonna write it in spherical coordinates because we're already halfway there anyway.

00:21 And my h looks like this.

00:32 But the interesting thing about the potential is that it goes like one over r squared.

00:38 All the terms go like one over r squared.

00:44 So that means that really it's kind of some function of theta divided by r squared.

00:54 That makes it possible to separate the hamilton -jacobi equation.

00:59 So for the principal function, s, we have an equation that looks like this, okay? and if i make the substitution, we make s a sum of terms, each of which is a function of only one variable.

01:38 And substitute that in, we can see how that separation is gonna work, okay? and i'm gonna make an observation right away.

02:14 This is equal to zero.

02:19 So if i make, right away i can see that t and phi are fairly simple.

02:34 I'm gonna call it et.

02:35 In a minute, i'm gonna replace that with minus e times t because written this way, the energy is negative.

02:42 M is the z component of angular momentum, capital m.

02:48 Little m is the mass.

02:53 Here's my theta equation.

03:11 Capital l is the total angular momentum.

03:22 Then my radial equation looks like this.

03:27 So then i gotta put in the negative e.

03:30 And i can see why i wanna do that.