
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)) = constb) (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) = x^{4} + y^{4} can have paths which go to infinity. d) (2 points) Determine whether the Hamiltonian system with energy H(x,y) = x^{4}  y^{4} 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) = x^{2}/2 + y^{2}/2The Hamilton equations are dx/dt = y dy/dt =  xIt has the solution x(t) = sin(t),y(t) = cos(t). 
H(x,y) = cos(x) + y^{2}/2The Hamilton equations are dx/dt = y dy/dt = sin(x)Its solution can not be described by elementary functions. 