THE HAMILTONIAN METHOD
AMANDA BURKE
OUTLINE
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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
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
DERIVING EULER’S EQUATION:
Setting up some postulates before we proceed:
x is some generalized coordinate
α is some variation from the path
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
Substitute a and b back in
Differentiate by parts
Term is 0 because η(x1) = η(x2) = 0
Substituting back in
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
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
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
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
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
FREE BODY IN A UNIFORM GRAVITATIONAL FIELD – 1D
g
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.
A PARTICLE BOUND TO THE SURFACE OF A CYLINDER – 2D
z
The restoring force is given, R is constant
From force, derive the potential energy
and transform into the correct coordinate
system
R
r
Θ
y
Solve the kinetic energy in the correct
coordinate system
x
Set up the Lagrangian
Use Lagrange generalized momenta
Find the momentum in each of the two
dimensions (z direction, Θ direction)
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
Use the Hamiltonian
Equations of Motion
Find the force and
velocity in each
dimension
Harmonic motion in the z dimension
Constant motion in the Θ dimension
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
SOURCES
• http://nm.mathforcollege.com/anecdotes/lagrange.pdf
• Hand, L. N.; Finch, J. D. (2008). Analytical Mechanics. Cambridge University Press. ISBN 978-0-52157572-0.
• Marion, Thornton. Classical Dynamics of Particles and Systems, 5th edition, 2003. Harcourt Brace & Co.