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5. A particle of mass m is projected upward with a velocity vo at an angle a to the horizontal in the uniform gravitational field of the earth as shown in figure. Ignore air resistance and take the potential energy U(y = 0) = 0. Using the Cartesian coordinate system, answer the following questions. (a) Find the Lagrangian in terms of x and y and identify cyclic coordinates . (b) Find the conjugate momenta, identify them and discuss which are conserved and why. (c) Using the Lagrange's equations, find the x- and y- components of the velocity as functions of time. (d) Find the Hamiltonian. (e) Ignoring air resistance, use Hamiltonian dynamics with the coordinates shown, to find the x- and y- components of the velocity as functions of time. parabola

5. A particle of mass m is projected upward with a velocity vo at an angle a to the horizontal in the uniform gravitational field of the earth as shown in figure. Ignore air resistance and take the potential energy U(y = 0) = 0. Using the Cartesian coordinate system, answer the following questions. (a) Find the Lagrangian in terms of x and y and identify cyclic coordinates . (b) Find the conjugate momenta, identify them and discuss which are conserved and why. (c) Using the Lagrange's equations, find the x- and y- components of the velocity as functions of time. (d) Find the Hamiltonian. (e) Ignoring air resistance, use Hamiltonian dynamics with the coordinates shown, to find the x- and y- components of the velocity as functions of time. parabola

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5. A particle of mass m is projected upward with a velocity vo at an angle a to the horizontal in the uniform gravitational field of the earth as shown in figure. Ignore air resistance and take the potential energy U(y = 0) = 0. Using the Cartesian coordinate system, answer the following questions. %3D (a) Find the Lagrangian in terms of x and y and identify cyclic coordinates. (b) Find the conjugate momenta, identify them and discuss which are conserved and why. (c) Using the Lagrange's equations, find the x- and y- components of the velocity as functions of time. (d) Find the Hamiltonian. (e) Ignoring air resistance, use Hamiltonian dynamics with the coordinates shown, to find the r- and y- components of the velocity as functions of time. parabola Vo
Transcribed Image Text:5. A particle of mass m is projected upward with a velocity vo at an angle a to the horizontal in the uniform gravitational field of the earth as shown in figure. Ignore air resistance and take the potential energy U(y = 0) = 0. Using the Cartesian coordinate system, answer the following questions. %3D (a) Find the Lagrangian in terms of x and y and identify cyclic coordinates. (b) Find the conjugate momenta, identify them and discuss which are conserved and why. (c) Using the Lagrange's equations, find the x- and y- components of the velocity as functions of time. (d) Find the Hamiltonian. (e) Ignoring air resistance, use Hamiltonian dynamics with the coordinates shown, to find the r- and y- components of the velocity as functions of time. parabola Vo
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