phys105-problems-1

phys105-problems-1 - × ( r × v )] = r ( v 2 + r · a )-v...

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Problem Set 1 Physics 105: Analytical Mechanics, Spring 2008 Reading assignment: read Taylor chapter 1. Problems: 1. Consider the following matrices: A = 1 2 - 1 0 3 1 2 0 1 , B = 2 1 0 0 - 1 2 1 1 3 , C = 2 1 4 3 1 0 , Find the following: a) det( AB ), b) AC , c) AB - B t A t , d) A ( BC ) - ( AB ) C 2. Prove that a × ( b × c ) = b ( a · c ) - c ( a · b ) where a , b and c are vectors. 3. Find the values of α for which λ is an orthogonal transformation: λ = 1 0 0 0 α - α 0 α α 4. Show that a) i,j ± ijk δ ij = 0 b) j,k ± ijk ± ljk = 2 δ il c) i,j,k ± ijk ± ijk = 6 5. Assume that r = r (t), v (t) = ˙ r and a (t) = ¨ r are vectors and then show that d dt [ r
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Unformatted text preview: × ( r × v )] = r ( v 2 + r · a )-v ( r · v )-r 2 a 6. Show that R (2 a r · ˙r + 2 b ˙r · ¨ r ) dt = a r 2 + b ˙r 2 + const. where r = ( x 1 , x 2 , x 3 ) and depends on t , and a and b are constants. 7. Derive the velocity and acceleration in spherical coordinates ( r, θ, φ ) where r = p x 2 + y 2 + z 2 , θ = arccos z r and φ = arctan y x . 8. Show that ∇ ( φψ ) = φ ∇ ψ + ψ ∇ φ where ψ and φ are scalars. 1...
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This homework help was uploaded on 04/05/2008 for the course PHYSICS 105 taught by Professor Edgarknobloch during the Spring '07 term at Berkeley.

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