Write down the Hamiltonian function and Hamilton's canonical equations for a simple pendulum.
To derive the Hamiltonian function and Hamilton's canonical equations for a simple pendulum, we first need to establish the system's Lagrangian.
The Lagrangian (L) of the simple pendulum can be derived using the kinetic and potential energy of the system. The kinetic energy (T) is given by the formula T = (1/2) m l^2 θ̇^2, where m is the mass of the pendulum bob, l is the length of the pendulum arm, and θ̇ is the angular velocity of the pendulum. The potential energy (V) is given by V = m g l (1 - cos θ), where g is the acceleration due to gravity.
Using the Lagrangian (L = T - V), we have L = (1/2) m l^2 θ̇^2 - m g l (1 - cos θ).
To find the Hamiltonian function (H), we need to perform a Legendre transformation, which involves finding the generalized momenta (p) associated with the generalized coordinates (q). In this case, the generalized coordinate is θ, and its conjugate momentum is defined as p = ∂L/∂θ̇.
Differentiating the Lagrangian with respect to θ̇, we have p = m l^2 θ̇.
Next, we can express θ̇ in terms of p to eliminate θ̇ from the Lagrangian.
θ̇ = p/(m l^2).
Now, we can write the Hamiltonian function (H) as H = p θ̇ - L.
Substituting the values of θ̇ and L, we get H = (p^2)/(2 m l^2) + m g l (1 - cos θ).
Now, let's determine Hamilton's canonical equations using the Hamiltonian function.
The first canonical equation is dp/dt = -∂H/∂θ, which represents the rate of change of momentum.
Differentiating the Hamiltonian (H) with respect to θ, we get ∂H/∂θ = -m g l sin θ.
Therefore, dp/dt = m g l sin θ.
The second canonical equation is dq/dt = ∂H/∂p, which represents the rate of change of the generalized coordinate.
Differentiating the Hamiltonian (H) with respect to p, we get ∂H/∂p = p/(m l^2).
Therefore, dq/dt = p/(m l^2).
In summary, for a simple pendulum, the Hamiltonian function is H = (p^2)/(2 m l^2) + m g l (1 - cos θ), and the Hamilton's canonical equations are dp/dt = m g l sin θ and dq/dt = p/(m l^2).