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Consider a solid cyliner rolling without slipping on a inclined plane; the inclined plane can slide on the horizotnal plane on which it is resting. Idenfity and write down (a) the appropriate generalised coordinates (b) Lagrangian (c) Euler-Lagrange Equations (d) Hamiltonain (e) Hamilton's Canonical Equations of motion

Consider a solid cyliner rolling without slipping on a inclined plane; the inclined plane can slide on the horizotnal plane on which it is resting. Idenfity and write down (a) the appropriate generalised coordinates (b) Lagrangian (c) Euler-Lagrange Equations (d) Hamiltonain (e) Hamilton's Canonical Equations of motion

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(a) appropriate generalised coordinates (b) Lagrangian (c) Euler-Lagrange Equation (d) Solve it 5. Consider a solid cyliner rolling without slipping on a inclined plane; the inclined plane can slide on the horizotnal plane on which it is resting. Idenfity and write down (a) the appropriate generalised coordinates (b) Lagrangian (c) Euler-Lagrange Equations (d) Hamiltonain (e) Hamilton's Canonical Equations of motion 6. A bead can slide on a straight wire which is always in the vertical plane. C is point on wire and wire is moving so the line joining origin O and Point C is rotated in the vertical plane with constant angular velocity w. (a) Find appropriate generalised coordinates, Lagrangian, and the Euler-Lagrange equations and solve it for the position of the bead. (b) Find Hamiltonain. Comment on the total energy of the system. 7. A bead can move freely on a circular loop; the circular loop is vertical and is rotating about its vertical diameter with constant angular velocity w. The bead will reach some equilibrium position, hind oscillartory frequency about this mean equilibrium position. Note that there are two different kinds of regimes; identify them. Find Hamiltonian, use it to graphically describe the two regimes. 8. For a two-body system, use Lagrangian Mechanics and show how the Kepler's Laws follow from them. 9. Considerr family of ant: show thet if a
Transcribed Image Text:(a) appropriate generalised coordinates (b) Lagrangian (c) Euler-Lagrange Equation (d) Solve it 5. Consider a solid cyliner rolling without slipping on a inclined plane; the inclined plane can slide on the horizotnal plane on which it is resting. Idenfity and write down (a) the appropriate generalised coordinates (b) Lagrangian (c) Euler-Lagrange Equations (d) Hamiltonain (e) Hamilton's Canonical Equations of motion 6. A bead can slide on a straight wire which is always in the vertical plane. C is point on wire and wire is moving so the line joining origin O and Point C is rotated in the vertical plane with constant angular velocity w. (a) Find appropriate generalised coordinates, Lagrangian, and the Euler-Lagrange equations and solve it for the position of the bead. (b) Find Hamiltonain. Comment on the total energy of the system. 7. A bead can move freely on a circular loop; the circular loop is vertical and is rotating about its vertical diameter with constant angular velocity w. The bead will reach some equilibrium position, hind oscillartory frequency about this mean equilibrium position. Note that there are two different kinds of regimes; identify them. Find Hamiltonian, use it to graphically describe the two regimes. 8. For a two-body system, use Lagrangian Mechanics and show how the Kepler's Laws follow from them. 9. Considerr family of ant: show thet if a
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