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

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Generalization to a system with n degrees of freedom:

if all the generalized coordinates are

independent

for k holonomic constraints:REVIEW

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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

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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!

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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

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If there are only time-independent potentials and time-independent constraints, then the hamiltonian represents the total energy.

Proof:

{

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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:

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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:

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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)

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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)

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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

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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)

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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

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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:

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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:

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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 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!

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