quiz02_PHYS3318

# quiz02_PHYS3318 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Physics Course: 8.09 Classical Mechanics Term: Fall 2006 Quiz 2 November 15, 2006 Instructions Do not start until you are told to do so. Solve all problems. Put your name on the covers of all notebooks you are using. Show all work neatly in the white book, label the problem you are working on. Mark the ﬁnal answers. Books and notes are not to be used. You may use your calculator.

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2 Useful Formulae Newton and Basic Kinematics: F ± = ˙ = m±a for v ± c t = t ±v = ±v 0 + dt ± ±a t =0 t = t t = t ± r = ± r 0 + ±v 0 t + dt ± dt ±± F ± ( t ±± ) /m t =0 t =0 Gravitational Law: F ± = Gm 1 m 2 r ˆ 12 2 r 12 Lagrangian and Hamiltonian: ∂q L ( q, q ˙ )= T U ; H ( p, q )= T + U = p L ∂t Hamilton Equation of Motion: ∂H ∂H = p ˙; q ∂q ∂p Generating function: ∂F ( Q, q ) ∂F ( Q, q ) = p ; = P ∂q ∂Q Poisson Brackets: ∂g ∂f ∂g ∂f [ g, f ]= ∂q ∂p ∂p ∂q Euler-Lagrange (without and with constraints): ∂L d ∂L ∂L d ∂L ∂g =0 ; + λ =0 ∂x dt ∂x ˙ ∂x dt ∂x ˙ ∂x Polar Coordinates: x = r sin θ cos φ ; y = r sin θ sin φ ; z = r cos θ Orbit Equation: µ 1 u ±± + u = F ( u )w i t h u = ²
± = ± ± ± ± ± ± 3 Eﬀective Potential: ² 2 V ( r )= U ( r )+ 2 µr 2 Keplerian Orbits: k α U ( r )= ; = ε cos θ +1 r r α = ² 2 ; ε = 1+ 2 2 ; τ 2 = 4 π 2 µ a 3 µk µk 2 k α α r min = a (1 ε )= ; r max = a (1 + ε )= 1+ ε 1 ε Spherical Coordinates: x = r sin θ cos φ ; y

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## This note was uploaded on 03/29/2010 for the course PHYS 3318 taught by Professor Flanagan during the Spring '08 term at Cornell.

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quiz02_PHYS3318 - Massachusetts Institute of Technology...

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