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
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± = ± ± ± ± ± ± 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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quiz02_PHYS3318 - Massachusetts Institute of Technology...

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