Energy conservation

William Hamilton (1805-1865) introduced the Hamiltonian systems. He introduced also the quaternion algebra. His motivation was to find a multiplication for 4 vectors. In some sense it introduces the dot and cross product because (0,a,b,c) * (0,u,v,w) = (-A.U, A x U) where X=(a,b,c) and Y = (u,v,w). This was in 1843 and might have been one of the first occasions, where the dot and cross product implicitly appeared in mathematics. The Hamiltonian system description is not only beautiful, it is a useful step to find the quantum version of a system.

Problem A: Energy conservation

Mechanical systems are determined by the energy function H(x,y), a function of two variables. Here, x is the position and y is the momentum. The equations of motion for the curve r(t) = (x(t),y(t)) are

     x'(t)  =  H_y(x,y) 
     y'(t)  = -H_x(x,y)

which are called Hamilton equations. These equations tell what the velocity vector r'(t) = (x'(t),y'(t)) is at a given point.
a) (4 points) Using the chain rule, verify that in full generality, the energy of a Hamiltonian system is preserved: for every path r(t) = (x(t),y(t)) solving the system, we have

  H(x(t),y(t)) = const

b) (2 points) What is the relation between the level curves of the function H(x,y) and the solution curves r(t) = (x(t),y(t)) of the system?
c) (2 points) Determine whether the Hamiltonian system with energy H(x,y) = x⁴ + y⁴ can have paths which go to infinity.
d) (2 points) Determine whether the Hamiltonian system with energy H(x,y) = x⁴ - y⁴ has solution paths for which the position goes to infinity.

Here are two examples of mechanical systems.

The harmonic oscillator	The pendulum
H(x,y) = x²/2 + y²/2 The Hamilton equations are dx/dt = y dy/dt = - x It has the solution x(t) = sin(t),y(t) = cos(t).	H(x,y) = -cos(x) + y²/2 The Hamilton equations are dx/dt = y dy/dt = -sin(x) Its solution can not be described by elementary functions.

The harmonic oscillator

The pendulum

H(x,y) = x²/2 + y²/2

The Hamilton equations are

    dx/dt  = y 
    dy/dt  = - x

It has the solution x(t) = sin(t),y(t) = cos(t).

 
H(x,y) = -cos(x) + y²/2

The Hamilton equations are

     dx/dt = y
     dy/dt = -sin(x)

Its solution can not be described by elementary functions.