Handout_3 - Classical Dynamics and Fluids P 45 R UTHERFORD...

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Unformatted text preview: Classical Dynamics and Fluids P 45 R UTHERFORD SCATTERING Classical Dynamics and Fluids P 46 R UTHERFORD SCATTERING II Classical Dynamics and Fluids P 47 H YPERBOLIC O RBITS IN R EPULSIVE P OTENTIAL A NOTHER W AY O C P impact parameter semi-latus rectum semi-major axis f f Path for repulsive force b a r r 8 We can simply use the results for the attractive potential case and let exceed . The radius r is then negative and the particle traces out the repulsive branch, getting closest to O at = , so that r ( ) = a (1 + e ) . This works because the potential energy A r is negative when r < and so represents a repulsive potential. This approach has the considerable advantage that no sign changes are needed, but it has something of a through the looking glass character to it. Classical Dynamics and Fluids P 48 T HE T HREE-B ODY P ROBLEM Use a computer. . . Some hierarchical systems can be stable indefinitely e.g. Sun, Earth and the Moon. A general 3-body encounter can be very complicated, but a general feature emerges. If 3 bodies are allowed to attract each other from a distance (a), they will speed up and interact strongly (b). Eventually the interaction is likely to form a close binary (negative gravitational binding energy) releasing kinetic energy, which may be enough for the bodies to escape to infinity (c). This mechanism is responsible for evaporation of stars from star clusters. (maybe also invalidates the virial theorem?) The planet Pluto has a close companion Charon, and has an eccentric orbit which takes it inside Neptunes orbit. A 3-body collision amongst Neptunes moons is the most likely cause. Classical Dynamics and Fluids P 49 R IGID B ODY D YNAMICS A rigid body is a special case of the many-particle system we have already studied, in which all the interparticle distances are fixed. The location of all the particles in the body can therefore be described by 6 coordinates: the position R of the centre of mass, and 3 angles ( ,, ) (the Euler angles), which describe the attitude of the body with respect to the spatial ( x,y,z )-axes. We will define these angles later. More importantly, the velocity of any particle in the body is determined by the velocity v of the CoM and a single angular velocity . The dynamics of the rigid body is then determined by its total mass and the inertia tensor that relates the angular momentum J to the angular velocity . This inertia tensor is the generalisation of the moment of inertia of a body rotating about a fixed axis. For a body spinning about a fixed axis (say e z ), the moment of inertia I = X m ( x 2 + y 2 ) relates the angular momentum J to the angular velocity via J = I ....
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Handout_3 - Classical Dynamics and Fluids P 45 R UTHERFORD...

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