Consider a point mass m moving in the (r, 2) plane on the parabola z =...

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Consider a point mass m moving in the (r, 2) plane on the parabola z = with a > 0 and subject to the constant gravitational field g = (0, -9). (a) Are there conserved quantities? If so, what are they? x/(2a) (b) Give the kinetic and potential energies for the point mass. What is the Lagrangian, if one chooses r as the generalized coordinate? (c) Derive the canonical momentum p conjugate to r as well as the Lagrange equation for r(t). (d) Determine the Hamiltonian function H. (e) Assuminga, obtain for H a quadratic form in x and p. Write down the canonical equations of motion for this approximate Hamiltonian function, and solve them assuming the initial condition (0) = 0, p(0) Po- Consider a point mass m moving in the (r, 2) plane on the parabola z = with a > 0 and subject to the constant gravitational field g = (0, -9). (a) Are there conserved quantities? If so, what are they? x/(2a) (b) Give the kinetic and potential energies for the point mass. What is the Lagrangian, if one chooses r as the generalized coordinate? (c) Derive the canonical momentum p conjugate to r as well as the Lagrange equation for r(t). (d) Determine the Hamiltonian function H. (e) Assuminga, obtain for H a quadratic form in x and p. Write down the canonical equations of motion for this approximate Hamiltonian function, and solve them assuming the initial condition (0) = 0, p(0) Po-