Motion of particle in the central potential in classical physics
- 1. 12/17/2019 1 Motion Of Particle In The Central Potential In Classical Physics Krishna Jangid PhD Student Department of Engineering Physics 1
- 2. Motion in a plane Central force motion is always motion in a plane. Spherical symmetry (angle coordinate for rotation is cyclic) Central force 𝐹 = 𝑓 𝑟 𝑟 Central potential: 𝑉(𝑟) 𝑉(𝑟) depends only on 𝑟 𝐿 = 𝑟 × 𝑝 Angular momentum 𝑟 is always perpendicular to 𝐿 Conservative force 𝑓 𝑟 = − 𝜕𝑉(𝑟) 𝜕𝑟 𝐹 = −𝜕𝑉(𝑟) 𝜕𝑟 𝑟 or 2
- 3. Lagrange’s and Hamilton’s equations ⅆ ⅆ𝑡 𝜕𝐿 𝜕 𝑞 𝑘 − 𝜕𝐿 𝜕𝑞 𝑘 = 0 (𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑒′ 𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛) 𝐻 𝑟, 𝑝 𝑟, 𝑝 𝜃 = 1 2𝑚 𝑝 𝑟 2 + 𝑝 𝜃 2 𝑟2 + 𝑉 𝑟 Lagrangian : 𝐿 = 𝑇 − 𝑉 Kinetic energy, 𝑇 = 1 2 𝑚 𝑟2 + 𝑟 𝜃 2 𝐿 𝑟, 𝜃, 𝑟, 𝜃 = 1 2 𝑚 𝑟2 + 𝑟 𝜃 2 − 𝑉 𝑟 𝑝 𝑘 = 𝜕𝐿 𝜕 𝑞 𝑘 Hamiltonian : 𝐻 𝑞, 𝑝, 𝑡 = 𝑘 𝑝 𝑘 𝑞 𝑘 − 𝐿 𝑞, 𝑞, 𝑡 𝐻 𝑟, 𝜃, 𝑝 𝑟, 𝑝 𝜃 = 𝑝 𝑟 𝑟 + 𝑝 𝜃 𝜃 − 𝐿 𝑟, 𝜃 𝐻 = 𝑝 𝑟 𝑟 + 𝑝 𝜃 𝜃 − 1 2 𝑚 𝑟2 + 𝑟 𝜃 2 + 𝑉 𝑟 3
- 4. Lagrange’s and Hamilton’s equations (cont.) 𝐻 𝑟, 𝑝 𝑟, 𝑝 𝜃 = 1 2𝑚 𝑝 𝑟 2 + 𝑝 𝜃 2 𝑟2 + 𝑉 𝑟 𝑚 𝑟 − 𝑚𝑟 𝜃2 + 𝜕𝑉 𝜕𝑟 = 0 ⅆ ⅆ𝑡 𝑚 𝑟 = 𝑝 𝜃 2 𝑚𝑟3 − 𝜕𝑉 𝑟 𝜕𝑟 𝑝 𝑟 = 𝑝 𝜃 2 𝑚𝑟3 − 𝜕𝑉 𝑟 𝜕𝑟 Equation of motion: 𝑞 𝑘 = 𝜕𝐻 𝜕𝑝 𝑘 𝑟 = 𝜕𝐻 𝜕𝑝 𝑟 𝑝 𝑟 = 𝑚 𝑟 𝜃 = 𝜕𝐻 𝜕𝑝 𝜃 𝜃 = 𝑝 𝜃 𝑚𝑟2 1) 𝑝 𝑘 = − 𝜕𝐻 𝜕𝑞 𝑘 𝑝 𝑟 = − 𝜕𝐻 𝜕𝑟 𝑝 𝑟 = 𝑝 𝜃 2 𝑚𝑟3 − 𝜕𝑉 𝑟 𝜕𝑟 𝑝 𝜃 = − 𝜕𝐻 𝜕𝜃 = 0 2) 4
- 5. Is obtained H solvable? 𝐻 𝑟, 𝑝 𝑟, 𝑝 𝜃 = 1 2𝑚 𝑝 𝑟 2 + 𝑝 𝜃 2 𝑟2 + 𝑉 𝑟 Degrees of freedom: 2 Constants of the motion required: 2 Liouville-Arnold criterion for integrability If there exists independent Constants of Motion (C.O.M) such that they Poisson commutes with each other then the Hamilton’s equations are completely integrable. Constants of the motion in our problem: ⅆ𝐻 ⅆ𝑡 = 0 1) For an autonomous system: Hence, H is the C.O.M Hence, 𝒑 𝜽 is the other C.O.M 2) 𝑝 𝜃 = − 𝜕𝐻 𝜕𝜃 = 0 ⅆ𝑝 𝜃 ⅆ𝑡 = 0 𝐻, 𝑝 𝜃 = 0 Hence, C.O.M Poisson commutes each other H is solvable 5
- 6. Stability and Closure of orbit under central force 1 2 𝑚 𝑟2 + 𝑝 𝜃 2 2𝑚𝑟2 + 𝑉 = 𝐸 𝐸 = 𝑉′ + 1 2 𝑚 𝑟2 𝑉′ = 𝑉 + 𝑝 𝜃 2 2𝑚𝑟2 𝑓 𝑟 = − 𝑘 𝑟2 𝑉 = − 𝑘 𝑟 (1) 6Potential for attractive inverse-square law of force Nature of orbits for bounded motion
- 7. Stability and Closure of orbit under central force 𝑉 𝑟 = − 𝑘 𝑟3 𝑓 𝑟 = − 3𝑘 𝑟4 𝑓 𝑟 = −𝑘𝑟 𝑉 𝑟 = 1 2 𝑘𝑟2 𝑓 𝑟 = −𝑘𝑟 𝑛 𝑛 > −3Stability condition (2) (3) 7 Potential for attractive inverse-fourth law of force Potential for linear restoring force
- 8. References • Classical mechanics by Goldstein. • Classical mechanics by Jc upadhyaya. • NPTEL videos