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. A massless spring of equilibrium length b and spring constant k connects two particles of masses m₁ and m2. The system rests on a large frictionless horizontal surface and may translate, oscillate and rotate. (a) Write the Lagrangian, simplifying the problem by separating the motion of the center of mass and the motion with respect to the center of mass, i.e. T = Tem +T'. Find the generalized momenta. Determine Lagrange's equations of motion. Which generalized momenta are associated with cyclic coordinates? (b) Determine the Hamiltonian and Hamilton's equations of motion. (c) Find all possible conditions under which the system is in dynamical equilibrium (i.e. the second time derivative of all generalized coordinates is zero).

. A massless spring of equilibrium length b and spring constant k connects two particles of masses m₁ and m2. The system rests on a large frictionless horizontal surface and may translate, oscillate and rotate. (a) Write the Lagrangian, simplifying the problem by separating the motion of the center of mass and the motion with respect to the center of mass, i.e. T = Tem +T'. Find the generalized momenta. Determine Lagrange's equations of motion. Which generalized momenta are associated with cyclic coordinates? (b) Determine the Hamiltonian and Hamilton's equations of motion. (c) Find all possible conditions under which the system is in dynamical equilibrium (i.e. the second time derivative of all generalized coordinates is zero).

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V Draw 37°F Clear Read aloud 2. A massless spring of equilibrium length b and spring constant k connects two particles of masses m₁ and m2. The system rests on a large frictionless horizontal surface and may translate, oscillate and rotate. FE (a) Write the Lagrangian, simplifying the problem by separating the motion of the center of mass and the motion with respect to the center of mass, i.e. T = Tcm +T'. Find the generalized momenta. Determine Lagrange's equations of motion. Which generalized momenta are associated with cyclic coordinates? 2 (b) Determine the Hamiltonian and Hamilton's equations of motion. (c) Find all possible conditions under which the system is in dynamical equilibrium (i.e. the second time derivative of all generalized coordinates is zero). H (d) Find the frequencies and normal modes of oscillation around dynamical equilibrium. (The standard small ascillations formalism will not work here because you are expanding around dynamical, not static, equilibrium. You will need to expand the lagrangian up to quadratic order in small deviations n, from dynamical equilibrium, get the equations of motion in that approximation, and propose a solution of the form n; = aje(wt+5).) If there is any set of conditions under which a normal mode has w²<0, explain what this means about stability and also explain physically why it happens under this set of conditions. (e) Sketch the normal modes. If a mode has zero frequency, explain briefly and in physical terms why that happens. Q Search of 3 FS P QD PRE F7 * G FB FB 10 M F11 + - FR Q10 S 8 563 8:51 PM 11/27/2023 BACKSPACE o D 13 O W +
Transcribed Image Text:V Draw 37°F Clear Read aloud 2. A massless spring of equilibrium length b and spring constant k connects two particles of masses m₁ and m2. The system rests on a large frictionless horizontal surface and may translate, oscillate and rotate. FE (a) Write the Lagrangian, simplifying the problem by separating the motion of the center of mass and the motion with respect to the center of mass, i.e. T = Tcm +T'. Find the generalized momenta. Determine Lagrange's equations of motion. Which generalized momenta are associated with cyclic coordinates? 2 (b) Determine the Hamiltonian and Hamilton's equations of motion. (c) Find all possible conditions under which the system is in dynamical equilibrium (i.e. the second time derivative of all generalized coordinates is zero). H (d) Find the frequencies and normal modes of oscillation around dynamical equilibrium. (The standard small ascillations formalism will not work here because you are expanding around dynamical, not static, equilibrium. You will need to expand the lagrangian up to quadratic order in small deviations n, from dynamical equilibrium, get the equations of motion in that approximation, and propose a solution of the form n; = aje(wt+5).) If there is any set of conditions under which a normal mode has w²<0, explain what this means about stability and also explain physically why it happens under this set of conditions. (e) Sketch the normal modes. If a mode has zero frequency, explain briefly and in physical terms why that happens. Q Search of 3 FS P QD PRE F7 * G FB FB 10 M F11 + - FR Q10 S 8 563 8:51 PM 11/27/2023 BACKSPACE o D 13 O W +
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