Mathematic Methods HW Solutions 46

# Mathematic Methods HW Solutions 46 - m r ¨ θ 2˙ r ˙ θ...

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Chapter 9 2.1 ( y - b ) 2 = 4 a 2 ( x - a 2 ) 2.2 x 2 + ( y - b ) 2 = a 2 2.3 ax = sinh( ay + b ) 2.4 ax = cosh( ay + b ) 2.5 y = ae x + be - x or y = A cosh( x + B ), etc. 2.6 x + a = 4 3 ( y 1 / 2 - 2 b )( b + y 1 / 2 ) 1 / 2 2.7 e x cos( y + b ) = C 2.8 K 2 x 2 - ( y - b ) 2 = K 4 2.9 x = ay 2 + b 2.10 y = Ax 3 / 2 - ln x + B 3.1 dx/dy = C ( y 3 - C 2 ) - 1 / 2 3.2 dx/dy = Cy 2 (1 - C 2 y 4 ) - 1 / 2 3.3 x 4 y p 2 = C 2 (1 + x 2 y p 2 ) 3 3.4 dx dy = C y ( y 4 - C 2 ) 1 / 2 3.5 y 2 = ax + b 3.6 x = ay 3 / 2 - 1 2 y 2 + b 3.7 y = K sinh( x + C ) = ae x + be - x , etc., as in Problem 2.5 3.8 r cos( θ + α ) = C 3.9 cot θ = A cos( φ - α ) 3.10 s = be at 3.11 a ( x + 1) = cosh( ay + b ) 3.12 ( x - a ) 2 + y 2 = C 2 3.13 ( x - a ) 2 = 4 K 2 ( y - K 2 ) 3.14 r = be 3.15 r cos( θ + α ) = C , in polar coordinates; or, in rectangular coordinates, the straight line x cos α - y sin α = C . 3.17 Intersection of the cone with r cos p θ + C 2 P = K 3.18 Geodesics on the sphere: cot θ = A cos( φ - α ) . (See Problem 3.9) Intersection of z = ax + by with the sphere: cot θ = a cos φ + b sin φ . 4.4 42 . 2 min; 5 . 96 min 4.5 x = a (1 - cos θ ), y = a ( θ - sin θ ) + C 4.6 x = a ( θ - sin θ ) + C , y = 1 + a (1 - cos θ ) 4.7 x = a (1 - cos θ ) - 5 2 , y = a ( θ - sin θ ) + C 5.2 L = 1 2 m r 2 + r 2 ˙ θ 2 + ˙ z 2 ) - V ( r,θ,z ) m r - r ˙ θ 2 ) = - ∂V/∂r
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Unformatted text preview: m ( r ¨ θ + 2˙ r ˙ θ ) =-(1 /r )( ∂V/∂θ ) m ¨ z =-∂V/∂z Note: The equations in 5.2 and 5.3 are in the form m a = F =-∇ V . 5.3 L = 1 2 m (˙ r 2 + r 2 ˙ θ 2 + r 2 sin 2 θ ˙ φ 2 )-V ( r,θ,φ ) m (¨ r-r ˙ θ 2-r sin 2 θ ˙ φ 2 ) =-∂V/∂r m ( r ¨ θ + 2˙ r ˙ θ-r sin θ cos θ ˙ φ 2 ) =-(1 /r )( ∂V/∂θ ) m ( r sin θ ¨ φ + 2 r cos θ ˙ θ ˙ φ + 2 sin θ ˙ r ˙ φ ) =-(1 /r sin θ )( ∂V/∂φ ) 5.4 L = 1 2 ml 2 ˙ θ 2-mgl (1-cos θ ) l ¨ θ + g sin θ = 0 46...
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