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15), we can write for the force F(r ) ∇ × F(r ) ≡ curlF(r ) = 0. 17) Thus, the necessary and sufficient condition to ensure the force F(r ) to be a conservative force is its curl to vanish. In this case such a function U (x, y , z ) of the co-ordinates can always be found, so that F (r ) = −∇U . 2 CONSERVATION OF MOMENTUM The law of conservation of momentum in a closed system originates from the homogeneity of space. Consider a translation δr of all particles of the system. e. the Lagrangian function should retain its form.

65b) In a cylindrical co-ordinate system ρ , ϕ , z (Fig. 1-4), the displacement Figure -4. Displacement element in cylindrical co-ordinates. element ds1 on the plane, which is a result of increasing ρ and ϕ by d ρ and dϕ , is the same as in polar co-ordinates. Due to this displacement, the particle moves from P0 to P1 . Taking into account the increase dz in zdirection, the particle displaces into P2 , and the displacement is a diagonal of the rectangle, built on dz and ds1 . Hence, ( ds ) 2 = ( dz ) + ( ds1 ) = ( dz ) + ρ 2 ( dϕ ) + ( d ρ ) .

9) to t 2 ⎛ ∂L ⎞ ∂L δS = ∫ ⎜ δq+ δ q ⎟ dt. 6) it follows that δ q = d δ q , and dt integrating the second term by parts, we obtain t2 t t2 t 2 2 ∂L ∂L ∂L d ∂L  δ qdt = d δ q = δ q − δq dt . 11) is equal to zero. 10) and taking into account Hamilton's principle (the minimum action), we get t 2 ⎛ ∂L d ∂L ⎞ δS =∫ ⎜ − ⎟ δ qdt = 0. 12) As δ q is an arbitrary function, this ( δ S = 0 ) is possible only if d ∂L ∂L − = 0. 13) If the system has s degrees of freedom with generalized co-ordinates q j and generalized velocities q j , Eq.

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