Classical mechanics by Rosu H.C.

By Rosu H.C.

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2 2 57 Comparing with (10), we identify k11 = (m1 + m2)l12 k12 = k21 = 0 k22 = m2 gl2 . 2 . 1 1 T = (m1 + m2 )l12 θ 1 + m2 l22 θ2 +m2l1 l2 θ1 θ 2 . 2 2 Identifying terms from the comparison with (11) we find m11 = (m1 + m2 )l12 m12 = m21 = m2 l1l2 m22 = m2 l22 . Substituting the energies in (12) one obtains the Lagrangian for the double pendulum oscillator and as the final result the equations of motion for this case: m11 m12 m21 m22 .. 1 θ2 + k11 0 0 k22 θ1 θ2 =0. 2 FORCED HARMONIC OSCILLATOR If an external weak force acts on an oscillator system the oscillations of the system are known as forced oscillations.

In this case the axis X3 is the axis of symmetry. The Euler equations projected onto the principal axes of inertia read I1 ω˙ 1 + ω2 ω3 (I2 − I3 ) = N1 (68) I2 ω˙ 2 + ω3 ω1 (I3 − I1 ) = N2 (69) I3 ω˙ 3 + ω1 ω2 (I1 − I2 ) = N3. (70) Since the system we consider here is free of torques N1 = N2 = N3 = 0 , (71) we use I1 = I2 in (71) to get I1 ω˙ 1 + ω2 ω3 (I2 − I3 ) = 0 (72) I2 ω˙ 2 + ω3 ω1 (I3 − I1 ) = 0 (73) I3ω˙ 3 = 0. (74) The equation (74) implies that ω3 = const. The equations (72) and (73) are rewritten as follows: ω˙ 1 = −Ωω2 where Ω = ω3 ω˙ 2 = −Ωω1 .

9 Euler angles As we already know, a rotation can be described by a rotation matrix λ by means of the equation x = λx´. t. the system whose axes are represented by x´. λn. There are many possibilities to choose these λ´s. One of them is the set of angles φ, θ and ϕ called Euler angles. The partial rotations are in this case the following: • A rotation around the X´3 axis of angle ϕ (in the positive trigonometric sense). The corresponding matrix is:   cos ϕ sin ϕ 0   λϕ =  − sin ϕ cos ϕ 0  .

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