Hamilton's Equations

Hamilton's equations are a set of differential equations that describe the motion of a system in terms of its generalized coordinates and momenta. In order to obtain Hamilton's equations for the projectile motion of a particle in the gravitational field, we first need to define the generalized coordinates and momenta.

Generalized Coordinates
In the case of projectile motion, the particle's position can be described by its horizontal coordinate x and vertical coordinate y. Therefore, the generalized coordinates are given by q1 = x and q2 = y.

Generalized Momenta
The generalized momenta are defined as the derivatives of the Lagrangian with respect to the velocities. In this case, the Lagrangian for the particle in the gravitational field is given by:

L = T - U = (m/2)(ẋ² + ẏ²) - mgy

where T is the kinetic energy, U is the potential energy, m is the mass of the particle, ẋ and ẏ are the velocities in the x and y directions respectively, and g is the acceleration due to gravity.

The generalized momenta are therefore given by p1 = ∂L/∂ẋ = mẋ and p2 = ∂L/∂ẏ = mẏ.

Hamilton's Equations
Hamilton's equations are given by:

dq1/dt = (∂H/∂p1)
dq2/dt = (∂H/∂p2)
dp1/dt = - (∂H/∂q1)
dp2/dt = - (∂H/∂q2)

where H is the Hamiltonian, defined as the sum of the generalized momenta multiplied by the velocities minus the Lagrangian:

H = Σ(piqi) - L

Projectile Motion in the Gravitational Field
In the case of projectile motion, the Hamiltonian is given by:

H = p1ẋ + p2ẏ - L = (mẋ² + mẏ²)/2 + mgy

Using the expressions for the generalized coordinates and momenta, we can substitute them into the Hamiltonian:

H = (p1² + p2²)/(2m) + mgy

Now, taking the derivatives with respect to the generalized coordinates and momenta, we obtain:

dq1/dt = (∂H/∂p1) = p1/m
dq2/dt = (∂H/∂p2) = p2/m
dp1/dt = - (∂H/∂q1) = 0
dp2/dt = - (∂H/∂q2) = -mg

Cyclic Coordinate and Time of Flight
A cyclic coordinate is a generalized coordinate whose corresponding momentum is conserved. In the case of projectile motion, the horizontal coordinate x is cyclic, since its corresponding momentum p1 = mẋ is conserved.

If the point of projection is the origin (x = 0), then the horizontal momentum p1 = mẋ is zero at all times. This implies that the horizontal coordinate x remains constant throughout the motion, making it a cyclic coordinate.

The time of flight of the projectile can be defined as the time it takes for the particle