Appendix G
Hamiltonian Mechanics
G.1 Generalized Coordinates and Momenta
Generalized coordinates: any set of coordinates by means of which the position of the
particles in a system may be specified.
While it is possible to set up Newton’s equations of motion in terms of Cartesian
components for each specific system with which we deal and then change over into
generalized coordinates, it is both desirable and, ultimately, convenient to have a
method for setting up equations of motion directly in terms of some convenient set
of generalized coordinates, as was first accomplished by Lagrange in the eighteenth
century.
Obviously, the number of coordinates in the new system must be the same as the
number of Cartesian coordinates for all of the particles involved in the system of N
particles under consideration, i.e., 3N. Traditionally, each coordinate is designated
by q i , so that a set of 3N generalized coordinates (three coordinates for each
particle) can be written as q 1 , q 2 , · · · , q 3N .
Example A particle confined to move in a plane may be described by two generalized coordinates (q 1 , q 2 ), which could be specifically Cartesian coordinates (x, y) or
polar coordinates (r, θ ) or, similarly, for a particle moving in the full 3-space, with
generalized coordinates (q 1 , q 2 , q 3 ), or specifically, Cartesian coordinates (x, y, z),
spherical polar coordinates (r, θ, φ), or cylindrical coordinates (r, φ, z).
For every configuration of an N-particle system, the generalized coordinates
must have a definite set of values, each coordinate q i will be a function of the set of
Cartesian coordinates, plus time (for moving coordinate systems), so that we may
write q i as
q i = q i (x 1 , y 1 , z 1 , · · · , z N ; t) ,
i = 1, · · · , 3N .
(G.1.1)
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9
657
Hamiltonian Mechanics
G.1 Generalized Coordinates and Momenta
Generalized coordinates: any set of coordinates by means of which the position of the
particles in a system may be specified.
While it is possible to set up Newton’s equations of motion in terms of Cartesian
components for each specific system with which we deal and then change over into
generalized coordinates, it is both desirable and, ultimately, convenient to have a
method for setting up equations of motion directly in terms of some convenient set
of generalized coordinates, as was first accomplished by Lagrange in the eighteenth
century.
Obviously, the number of coordinates in the new system must be the same as the
number of Cartesian coordinates for all of the particles involved in the system of N
particles under consideration, i.e., 3N. Traditionally, each coordinate is designated
by q i , so that a set of 3N generalized coordinates (three coordinates for each
particle) can be written as q 1 , q 2 , · · · , q 3N .
Example A particle confined to move in a plane may be described by two generalized coordinates (q 1 , q 2 ), which could be specifically Cartesian coordinates (x, y) or
polar coordinates (r, θ ) or, similarly, for a particle moving in the full 3-space, with
generalized coordinates (q 1 , q 2 , q 3 ), or specifically, Cartesian coordinates (x, y, z),
spherical polar coordinates (r, θ, φ), or cylindrical coordinates (r, φ, z).
For every configuration of an N-particle system, the generalized coordinates
must have a definite set of values, each coordinate q i will be a function of the set of
Cartesian coordinates, plus time (for moving coordinate systems), so that we may
write q i as
q i = q i (x 1 , y 1 , z 1 , · · · , z N ; t) ,
i = 1, · · · , 3N .
(G.1.1)
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9
657