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Hamilton’s principlebased on FW-18
Variational statement of mechanics: (for conservative forces)
actionthe particle takes the path
that minimizes the integrated difference of the kinetic and
potential energiesEquivalent to Newton’s laws!
REVIEW
174
Generalization to a system with n degrees of freedom:
if all the generalized coordinates are
independent
for k holonomic constraints:REVIEW
175
Hamiltonian Dynamicsbased on FW-32Hamilton’s principle:
the action is stationary under small virtual displacements about the actual motion of the system
fixed initial and final configurations
Euler-Lagrange equations
New set of coordinates (transformations are assumed nonsingular and invertible):
a different function of new coordinates and velocities
Hamilton’s principle for the new set of coordinates:
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
176
Generalized momenta and the Hamiltonianbased on FW-20
Let’s define generalized momentum (canonical momentum):
for independent generalized coordinates
Lagrange’s equations can be written as:
if the lagrangian does not depend on some coordinate,
cyclic coordinatethe corresponding momentum is a constant of the motion, a conserved quantity.
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!
REVIEW
177
Proof:
If the lagrangian does not depend explicitly on the time, then the hamiltonian is a constant of the motion:
time shift invariance implies that the hamiltonian is conserved
REVIEW
178
If there are only time-independent potentials and time-independent constraints, then the hamiltonian represents the total energy.
Proof:
{
REVIEW
179
generalized momentum:
from to
Hamiltonian Dynamics (coordinates and momenta equivalent variables):
Hamiltonian:
relations are assumed nonsingular and invertible
Legendre transformation
Hamilton’s equations:
2n coupled first-order differential equations for coordinates and momenta
(equivalent to Lagrange’s equations)
also:
180
If the lagrangian does not depend explicitly on the time, then the hamiltonian is a constant of the motion
Taking time derivative:
in addition we saw before, that for a conservative system with time-independent constraints:
181
Modified Hamilton’s principle:
independent variables subject to independent
variationswith fixed endpoints:
0
variations of all ps and qs are independent
Hamilton’s equations from Hamilton’s principle:(the modified Hamilton’s principle may be taken to be the basic statement of mechanics, equivalent to Newtons laws)
182
Canonical Transformationsbased on FW-34
Under what conditions do the transformations to new set of coordinates and momenta,
preserve the form of Hamilton’s equations?
relations are assumed nonsingular and invertible
Such transformations should satisfy:
the total derivative of any function can be
added because it will not contribute to the
modified Hamilton’s principle
leads to Hamilton’s equations
with new Hamiltonian
(canonical transformations)
183
How can we guarantee ?
We can automatically guarantee this form if we set coefficients of velocities to 0:
and the new Hamiltonian is: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!
(in practice, not easy to determine if such a function exists)F is the generator of the canonical transformation
Any F generates some canonical transformation!we will use this freedom to construct a transformation so that all
Q and P are cyclic, i.e. constants of the motion!184
Hamilton-Jacobi theorybased on FW-35First let’s introduce another function S:
S generates canonical transformation, the Hamilton’s equations hold for new coordinates
and momenta with the new Hamiltonian!
from to
Legendre transformation
185
We want to use the freedom to choose S so that !Then Hamilton’s eqns. imply that all the P and Q are cyclic, i.e. constants of the motion!
Such S must satisfy:
Hamilton-Jacobi equationfirst order partial differential equation in n+1 variables
General form of S:
any n independent non-additive integration constants
overall integration constant (irrelevant)
(can imagine integrating it one variable at a time, keeping remaining variables fixed,
introducing an integration constant each time)
186
any n independent non-additive integration constants
General form of S:overall integration constant
(irrelevant)
Let’s look at a particular solution:
11
Hamilton’s principal function
It generates following transformation:By assumption:
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 to the mechanical problem:
2n constants, !s and "s, are determined from 2n initial conditions
inve
rting
187
Sometimes the solution W can be separated in a sum of independent additive functions:
Hamilton’s principal function S is the action:
!s are constants
the action evaluated along the dynamic trajectory
If the Hamiltonian does not explicitly depend on time, H is constant, and we can separate off the time dependence:
Hamilton-Jacobi equation for Hamilton’s characteristic function W:
188
Example (a particle in one-dimensional potential):
Hamilton-Jacobi equation:
Hamiltonian is independent of time so we can look for a solution of the form:
Hamilton-Jacobi equation for Hamilton’s characteristic function:
Solution:
189
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 expected190
Connection with quantum mechanics:
Schrödinger equation
wave function
Separating off time dependance corresponds to looking for stationary states, and problem often allow a separation of variables:
We seek a wave-like solution:
real function
Hamilton-Jacobi equation= 0
The phase of the semiclassical wave function is the classical action evaluated along the path of motion!
191