1. The Hamiltonian method
Amanda Burke

2. outline Historical background Deriving Euler’s Equation
Euler Lagrange Equation
Lagrange Generalized Momenta
Deriving Hamiltonian
Using the Hamiltonian
Ex. Falling body
Ex. Particle bound to surface of cylinder
Conclusion
Sources

3. Historical background
In the 1750s both Euler and Lagrange were working on the tautochrone problem, trying to determine the shape of a curve such a that a weighted particle falls to a certain point in a certain amount of time
In 1756 Lagrange found a solution to the problem and sent it to Euler
Together they worked to further Lagrange’s method in application to mechanics, leading to Lagrangian mechanics
Their correspondence led to the development of the calculus of variations, whose name was coined in 1766 by Euler
In 1833, William Rowan Hamilton had been playing with Lagrangian mechanics and formulated Hamiltonian mechanics which used canonical conjugates to describe the motion of systems

4. Deriving euler’s equation:
Setting up some postulates before we proceed:
x is some generalized coordinate
α is some variation from the path

5. Move the operator inside the integral because they are equivalent statements
Use product rule and chain rule
Solve for a and b in a form which we can use

6. Substitute a and b back in
Differentiate by parts
Term is 0 because η(x1) = η(x2) = 0
Substituting back in

7. Euler’s Equation Pull out the common factor Now set α = 0
Because η(x) is some continuous function, Inside term must be making the integrand 0
*** If the function which we consider is the Lagrangian, we must take the partial derivative of the Lagrangian with respect to α in each spatial dimension

8. Euler-lagrange equation
Set x to be time qj is some generalized coordinate Time derivative of qj is the generalized velocity of each coordinate The Lagrangian is a function of generalized coordinate and generalized velocity

9. Lagrange’s equations of motion: generalized momenta
The Lagrangian is L = T – U We are considering a system in which potential energy does not depend on velocity (only conservative forces) Hence, generalized momenta can be derived from the partial of the Lagrangian with respect to generalized velocity Time derivate of each side gives force on the system in each generalized direction ****This second equation is rarely considered

10. Developing the Hamiltonian
***The Lagrangian is not constant in time, we’d like to get to something which is conserved to more easily analyze dynamic systems
Take the sum of the partials of the Lagrangian with respect to position and velocity for each dimension
Use some mathematical
trickery treating an operator as
a fraction
Finally get down to something
which is constant in time

11. Hamilton’s equations of motion
Substitute a Lagrange Equation of Motion into the Hamiltonian Doing a very similar derivation to that of the Lagrange Equations, one can derive the Hamiltonian Equations of Motion You can see that the Hamiltonian Equations of Motion are in terms of the time derivative of the position in each dimension and the momentum in each of those dimensions

12. Free body in a uniform gravitational field – 1D
Find the Lagrangian Equations of Motion in the y dimension.
Find the Hamiltonian Equations of Motion in the y dimension.
In agreeance with Newton’s Laws, the Hamiltonian Method gives back that the momentum of the body is mass times velocity and the force on the body is the gravitational force on that object.
It can also be noted that two of the equations of motion give the same value.

13. a particle bound to the surface of a cylinder – 2Dz
The restoring force is given, R is constant
From force, derive the potential energy and transform into the correct coordinate system
Solve the kinetic energy in the correct coordinate system
Set up the Lagrangian
Use Lagrange generalized momenta
Find the momentum in each of the two dimensions (z direction, Θ direction) R r y Θ x

14. Set up the Hamiltonian
Have Hamiltonian in terms of velocity and position, but want it is terms of momentum and position to make substitutions
Square the momentums and make substitutions
Find an equation of motion for the particle bound to the surface of the cylinder
Harmonic motion in the z dimension
Constant motion in the Θ dimension
Use the Hamiltonian Equations of Motion
Find the force and velocity in each dimension

15. conclusion
It can be shown that the Lagrangian and Hamiltonian equations agree with Newton’s Laws of Motion
The Hamiltonian is a useful tool for finding complicated equations of motion
Compared with the Lagrangian instead of getting one second order differential equation, with the Hamiltonian you get two first order differential equation

16. Sources http://nm.mathforcollege.com/anecdotes/lagrange.pdf
Hand, L. N.; Finch, J. D. (2008). Analytical Mechanics. Cambridge University Press. ISBN 
Marion, Thornton. Classical Dynamics of Particles and Systems, 5th edition, Harcourt Brace & Co.