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Hamilton’s principle
Hamiltonian Dynamics
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based on FW-18
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the action is stationary under small
virtual displacements about the
actual motion of the system
Equivalent to Newton’s laws!
fixed initial and final
configurations
Euler-Lagrange equations
New set of coordinates (transformations are assumed nonsingular and invertible):
Variational statement of mechanics:
(for conservative forces)
action
based on FW-32
Hamilton’s principle:
a different function of new coordinates and velocities
Hamilton’s principle for the new set of coordinates:
the particle takes the path
that minimizes the integrated
difference of the kinetic and
potential energies
Lagrange’s equations remain invariant under the point transformations!
we can choose any set of generalized coordinates and
Lagrange’s equations will correctly describe the dynamics
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Generalization to a system with n degrees of freedom:
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if all the generalized
coordinates are
independent
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Generalized momenta and the Hamiltonian
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based on FW-20
Let’s define generalized momentum (canonical momentum):
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for independent generalized coordinates
Lagrange’s equations can be written as:
for k holonomic constraints:
if the lagrangian does not depend on some
coordinate,
cyclic coordinate
the corresponding momentum is a constant of the
motion, a conserved quantity.
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related to the symmetry of
the problem - the system is
invariant under some
continuous transformation.
For each such symmetry
operation there is a
conserved quantity!
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Hamiltonian Dynamics (coordinates and momenta equivalent variables):
If the lagrangian does not depend explicitly on the time,
then the hamiltonian is a constant of the motion:
generalized momentum:
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relations are assumed
nonsingular and invertible
Hamiltonian:
time shift invariance implies that
the hamiltonian is conserved
Proof:
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Legendre transformation
from
to
Hamilton’s equations:
also:
(equivalent to Lagrange’s
equations)
2n coupled first-order differential equations
for coordinates and momenta
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If there are only time-independent potentials and time-independent
constraints, then the hamiltonian represents the total energy.
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Taking time derivative:
{
Proof:
If the lagrangian does not depend explicitly on the time,
then the hamiltonian is a constant of the motion
in addition we saw before, that for a conservative
system with time-independent constraints:
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Modified Hamilton’s principle:
How can we guarantee
independent variables
subject to independent
variations
?
with fixed endpoints:
We can automatically guarantee
this form if we set coefficients of
velocities to 0:
0
and the new Hamiltonian is:
variations of all ps and qs are independent
Hamilton’s equations from Hamilton’s principle:
whenever the transformations can be written
in terms of some F, then the Hamilton’s
equations hold for new coordinates and
momenta with the new Hamiltonian!
F is the generator of the canonical transformation
(in practice, not easy to determine if such a function exists)
Any F generates some canonical transformation!
(the modified Hamilton’s principle may be taken to be the basic
statement of mechanics, equivalent to Newtons laws)
we will use this freedom to construct a transformation so that all
Q and P are cyclic, i.e. constants of the motion!
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Canonical Transformations
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Hamilton-Jacobi theory
based on FW-34
First let’s introduce another function S:
Under what conditions do the transformations to new set of
coordinates and momenta,
based on FW-35
Legendre transformation
relations are assumed
nonsingular and invertible
from
to
preserve the form of Hamilton’s equations?
Such transformations should satisfy:
ns
atio
equ
on’s tonian
t
l
i
Ham Hamil
s to
lead ith new
w
(canonical transformations)
the total derivative of
any function can be
added because it will not
contribute to the
modified Hamilton’s
principle
S generates canonical transformation, the
Hamilton’s equations hold for new coordinates
and momenta with the new Hamiltonian!
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We want to use the freedom to choose S so that
!
Hamilton’s principal function S is the action:
Then Hamilton’s eqns. imply that all the P and Q are cyclic, i.e. constants of the motion!
!s are constants
the action evaluated along
the dynamic trajectory
Such S must satisfy:
If the Hamiltonian does not explicitly depend on time, H is constant,
and we can separate off the time dependence:
Hamilton-Jacobi equation
first order partial differential equation in n+1 variables
(can imagine integrating it one variable at a time,
keeping remaining variables fixed,
introducing an integration constant each time)
General form of S:
Hamilton-Jacobi equation for
Hamilton’s characteristic function W:
overall integration constant
(irrelevant)
Sometimes the solution W can be separated in a sum of independent additive
functions:
any n independent non-additive integration constants
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General form of S:
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Example (a particle in one-dimensional potential):
overall integration constant
(irrelevant)
any n independent non-additive integration constants
Hamilton-Jacobi equation:
Let’s look at a particular solution:
1
1
Hamilton’s principal function
By assumption:
Hamiltonian is independent of time so
we can look for a solution of the form:
It generates following transformation:
inv
Solution to the mechanical problem:
er
tin
g
Hamilton-Jacobi equation for Hamilton’s characteristic function:
Any set of !s in S represents n constants of motion;
derivatives with respect to !s determine "s,
another set of n constants of motion
Solution:
2n constants, !s and "s, are determined from 2n initial conditions
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Example (a particle in one-dimensional potential) continued:
at this point the trajectory is not determined
The 2nd constant of motion:
provides relation between q and t
(constants ! and " determined from initial conditions)
For harmonic oscillator:
as expected
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Connection with quantum mechanics:
wave function
Schrödinger equation
We seek a wave-like solution:
real function
=0
Hamilton-Jacobi equation
The phase of the semiclassical wave function is the
classical action evaluated along the path of motion!
Separating off time dependance corresponds to looking for stationary states, and problem
often allow a separation of variables:
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