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. A particle of mass m in 3D is subject to a potential V(x, y, z) = ²x Write the Lagrangian and the Lagrange Eqns. Very briefly, and without any further calculation find the small oscillation frequencies, sketch the corresponding motion for each, indicate which ones of them are zero, and discuss physically why they are zero. (This problem should take no more than 5-6 lines, plus drawings.)

. A particle of mass m in 3D is subject to a potential V(x, y, z) = ²x Write the Lagrangian and the Lagrange Eqns. Very briefly, and without any further calculation find the small oscillation frequencies, sketch the corresponding motion for each, indicate which ones of them are zero, and discuss physically why they are zero. (This problem should take no more than 5-6 lines, plus drawings.)

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mes in the other columns, plus time spent in office hours). 1. A particle of mass m in 3D is subject to a potential V(x, y, z)=x² Write the Lagrangian and the Lagrange Eqns. Very briefly, and without any further calculation find the small oscillation frequencies, sketch the corresponding motion for each, indicate which ones of them are zero, and discuss physically why they are zero. (This problem should take no more than 5-6 lines, plus drawings.) 1 57°F Cloudy 2. A thin hoop of radius R and mass M oscillates in its own vertical plane with one point of the hoop fixed. Attached to the hoop is a point mass M constrained to move without friction along the hoop. 7 3 (a) Write the transformation equations for the center of the hoop and the position of the bead, and their time derivatives. Find the kinetic energy, the potential energy, and the full lagrangian for the system. (b) Expand the lagrangian to 2nd order in small angular deviations from equilibrium. Find the matrices T and V. (c) Find the normal mode frequencies (d) Find the normal mode eigenvectors. Sketch the motion corresponding to each one of them. You don't need to find the normalization constants for the eigenvectors. A Thumbtack on an inclined alone @ U UUGOV F 3 2 O J F4 % Q Search 5 FS € ^ 6 F6 P R PRE 7 A rigid body in the shape of a thumbtack formed from a thin X 8 FB ( 9 PS 1 S F10 F11 8:10 PM 11/19/2023 BACKSPACE
Transcribed Image Text:mes in the other columns, plus time spent in office hours). 1. A particle of mass m in 3D is subject to a potential V(x, y, z)=x² Write the Lagrangian and the Lagrange Eqns. Very briefly, and without any further calculation find the small oscillation frequencies, sketch the corresponding motion for each, indicate which ones of them are zero, and discuss physically why they are zero. (This problem should take no more than 5-6 lines, plus drawings.) 1 57°F Cloudy 2. A thin hoop of radius R and mass M oscillates in its own vertical plane with one point of the hoop fixed. Attached to the hoop is a point mass M constrained to move without friction along the hoop. 7 3 (a) Write the transformation equations for the center of the hoop and the position of the bead, and their time derivatives. Find the kinetic energy, the potential energy, and the full lagrangian for the system. (b) Expand the lagrangian to 2nd order in small angular deviations from equilibrium. Find the matrices T and V. (c) Find the normal mode frequencies (d) Find the normal mode eigenvectors. Sketch the motion corresponding to each one of them. You don't need to find the normalization constants for the eigenvectors. A Thumbtack on an inclined alone @ U UUGOV F 3 2 O J F4 % Q Search 5 FS € ^ 6 F6 P R PRE 7 A rigid body in the shape of a thumbtack formed from a thin X 8 FB ( 9 PS 1 S F10 F11 8:10 PM 11/19/2023 BACKSPACE
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