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(f) Hamiltonian solution The generalized momentum is Po = 01² m The Hamiltonian is H(0, pe) = = 咯 21² m - glm cos(0) The Hamilton equations of motion are pe = glm sin(0) ė Ре Pm Differentiating the second equation Po=1² mö and replacing this into the first, we get g sin(0) + 1 = 0 which is the Lagrangian equation of motion for 0. Pendulum moving in a horizontal plain Consider a simple pendulum, a mass m suspended on a massless string of fixed length 1. For simplicity assume that the string does not bend and that the motion takes place in a vertical plane. Obtain and solve the equation(s) of motion. The problem can be solved in many different ways and most of these ways offer some lessons. m g (g) You learned that the Hamiltonian is the generator of time translations, thus f = [f, H] for any phase- space functions. Use this knowledge to derive the equation of motion for 0. (g) Poisson method f= Applying the Poisson relation, f = [f, H], for f = 0, we get ė = Ре Pm Applying the Poisson relation for f = pe, we get P₁ = −g l m sin(0) Combining the time derivative of the & equation with the p. equation we get 0" (t): g sin(0(t))

(f) Hamiltonian solution The generalized momentum is Po = 01² m The Hamiltonian is H(0, pe) = = 咯 21² m - glm cos(0) The Hamilton equations of motion are pe = glm sin(0) ė Ре Pm Differentiating the second equation Po=1² mö and replacing this into the first, we get g sin(0) + 1 = 0 which is the Lagrangian equation of motion for 0. Pendulum moving in a horizontal plain Consider a simple pendulum, a mass m suspended on a massless string of fixed length 1. For simplicity assume that the string does not bend and that the motion takes place in a vertical plane. Obtain and solve the equation(s) of motion. The problem can be solved in many different ways and most of these ways offer some lessons. m g (g) You learned that the Hamiltonian is the generator of time translations, thus f = [f, H] for any phase- space functions. Use this knowledge to derive the equation of motion for 0. (g) Poisson method f= Applying the Poisson relation, f = [f, H], for f = 0, we get ė = Ре Pm Applying the Poisson relation for f = pe, we get P₁ = −g l m sin(0) Combining the time derivative of the & equation with the p. equation we get 0" (t): g sin(0(t))

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(f) Hamiltonian solution The generalized momentum is Po = 01² m The Hamiltonian is H(0, pe) = = 咯 21² m - glm cos(0) The Hamilton equations of motion are pe = glm sin(0) ė Ре Pm Differentiating the second equation Po=1² mö and replacing this into the first, we get g sin(0) + 1 = 0 which is the Lagrangian equation of motion for 0.
Transcribed Image Text:(f) Hamiltonian solution The generalized momentum is Po = 01² m The Hamiltonian is H(0, pe) = = 咯 21² m - glm cos(0) The Hamilton equations of motion are pe = glm sin(0) ė Ре Pm Differentiating the second equation Po=1² mö and replacing this into the first, we get g sin(0) + 1 = 0 which is the Lagrangian equation of motion for 0.
Pendulum moving in a horizontal plain Consider a simple pendulum, a mass m suspended on a massless string of fixed length 1. For simplicity assume that the string does not bend and that the motion takes place in a vertical plane. Obtain and solve the equation(s) of motion. The problem can be solved in many different ways and most of these ways offer some lessons. m g (g) You learned that the Hamiltonian is the generator of time translations, thus f = [f, H] for any phase- space functions. Use this knowledge to derive the equation of motion for 0. (g) Poisson method f= Applying the Poisson relation, f = [f, H], for f = 0, we get ė = Ре Pm Applying the Poisson relation for f = pe, we get P₁ = −g l m sin(0) Combining the time derivative of the & equation with the p. equation we get 0" (t): g sin(0(t))
Transcribed Image Text:Pendulum moving in a horizontal plain Consider a simple pendulum, a mass m suspended on a massless string of fixed length 1. For simplicity assume that the string does not bend and that the motion takes place in a vertical plane. Obtain and solve the equation(s) of motion. The problem can be solved in many different ways and most of these ways offer some lessons. m g (g) You learned that the Hamiltonian is the generator of time translations, thus f = [f, H] for any phase- space functions. Use this knowledge to derive the equation of motion for 0. (g) Poisson method f= Applying the Poisson relation, f = [f, H], for f = 0, we get ė = Ре Pm Applying the Poisson relation for f = pe, we get P₁ = −g l m sin(0) Combining the time derivative of the & equation with the p. equation we get 0" (t): g sin(0(t))
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