83 lines
3.5 KiB
Markdown
83 lines
3.5 KiB
Markdown
---
|
|
title: 40-hamiltons-equations
|
|
---
|
|
THE formulation of the laws of mechanics in terms of the Lagrangian, and
|
|
of Lagrange's equations derived from it, presupposes that the mechanical
|
|
state of a system is described by specifying its generalised co-ordinates and
|
|
velocities. This is not the only possible mode of description, however. A
|
|
number of advantages, especially in the study of certain general problems of
|
|
mechanics, attach to a description in terms of the generalised co-ordinates
|
|
and momenta of the system. The question therefore arises of the form of
|
|
the equations of motion corresponding to that formulation of mechanics.
|
|
The passage from one set of independent variables to another can be
|
|
effected by means of what is called in mathematics Legendre's transformation.
|
|
In the present case this transformation is as follows. The total differential
|
|
of the Lagrangian as a function of co-ordinates and velocities is
|
|
dL =
|
|
This expression may be written
|
|
(40.1)
|
|
since the derivatives aL/dqi are, by definition, the generalised momenta, and
|
|
aL/dqi = pi by Lagrange's equations. Writing the second term in (40.1) as
|
|
= - Eqi dpi, taking the differential d(piqi) to the left-hand
|
|
side, and reversing the signs, we obtain from (40.1)
|
|
The argument of the differential is the energy of the system (cf. §6);
|
|
expressed in terms of co-ordinates and momenta, it is called the Hamilton's
|
|
function or Hamiltonian of the system:
|
|
(40.2)
|
|
t The reader may find useful the following table showing certain differences between the
|
|
nomenclature used in this book and that which is generally used in the English literature.
|
|
Here
|
|
Elsewhere
|
|
Principle of least action
|
|
Hamilton's principle
|
|
Maupertuis' principle
|
|
Principle of least action
|
|
Maupertuis' principle
|
|
Action
|
|
Hamilton's principal function
|
|
Abbreviated action
|
|
Action
|
|
- -Translators.
|
|
131
|
|
132
|
|
The Canonical Equations
|
|
§40
|
|
From the equation in differentials
|
|
dH =
|
|
(40.3)
|
|
in which the independent variables are the co-ordinates and momenta, we
|
|
have the equations
|
|
=
|
|
(40.4)
|
|
These are the required equations of motion in the variables P and q, and
|
|
are called Hamilton's equations. They form a set of 2s first-order differential
|
|
equations for the 2s unknown functions Pi(t) and qi(t), replacing the S second-
|
|
order equations in the Lagrangian treatment. Because of their simplicity and
|
|
symmetry of form, they are also called canonical equations.
|
|
The total time derivative of the Hamiltonian is
|
|
Substitution of qi and pi from equations (40.4) shows that the last two terms
|
|
cancel, and so
|
|
dH/dt==Hoo.
|
|
(40.5)
|
|
In particular, if the Hamiltonian does not depend explicitly on time, then
|
|
dH/dt = 0, and we have the law of conservation of energy.
|
|
As well as the dynamical variables q, q or q, P, the Lagrangian and the
|
|
Hamiltonian involve various parameters which relate to the properties of the
|
|
mechanical system itself, or to the external forces on it. Let A be one such
|
|
parameter. Regarding it as a variable, we have instead of (40.1)
|
|
dL
|
|
and (40.3) becomes
|
|
dH =
|
|
Hence
|
|
(40.6)
|
|
which relates the derivatives of the Lagrangian and the Hamiltonian with
|
|
respect to the parameter A. The suffixes to the derivatives show the quantities
|
|
which are to be kept constant in the differentiation.
|
|
This result can be put in another way. Let the Lagrangian be of the form
|
|
L = Lo + L', where L' is a small correction to the function Lo. Then the
|
|
corresponding addition H' in the Hamiltonian H = H + H' is related to L'
|
|
by
|
|
(H')p,a - (L')
|
|
(40.7)
|
|
It may be noticed that, in transforming (40.1) into (40.3), we did not
|
|
include a term in dt to take account of a possible explicit time-dependence
|