Thinking about Hamiltonians at Hamilton services

Every time we go through Hamilton Services on the M74 I end up thinking Physics.

I know that the town is not connected with the great Irish mathematician William Rowan Hamilton, but it has given me the kick I need to try out a Hamiltonian. The Hamiltonian is a function which describes the total energy of a system and this can evolve over time so that results in mechanics can be calculated. I decided to see what happens in this type of analysis for a falling object. The object is to be of mass m and dropped from height h. First I used the suvat equations of uniformly accelerated motion to calculate an expression for the distance of fall, which I have called x, and for the speed v at that height.

 Then I used these expressions to calculate the kinetic energy T and potential energy V.

 The Hamiltonian turns out to be the constant value mgh which is good because that's what it should be if air resistance is neglected. Maybe the Hamiltonian will be a good analytical tool if we decide to include a dissipative force.

L is the Lagrangian which is the difference between the energies. I have a problem. My further analysis using the Lagrangian is out by a factor of 2. There may be a flaw in my analysis. I'll think about it before writing more about this.