Appendix A
Classical Mechanics Concepts
In this appendix, we review some basic concepts from classical mechanics that are
used in this book.
A.1 Newton’s Equations
Newton’s equations of motion describe the behavior of collections of N point
particles in a three-dimensional space (3N degrees of freedom) in terms of 3N
coupled second-order differential equations (the number of equations can be
reduced if constraints are present),
d 2 (m α r α )
dt 2
= F α ,
(A.1)
where α = 1, . . . , N, m α is the mass, r α is the displacement of the αth particle, and
F α is the net force on the αth particle due to the other particles and any external fields
that might be present. Equations (A.1) only have simple structure in inertial frames
of reference and for Cartesian coordinates. For noninertial frames and general
orthogonal curvilinear coordinate systems, they rapidly become complicated and
nonintuitive.
Because Newton’s equations are second order, the state of a collection of
N particles at time t is determined once the velocities v α =
dr α
dt = ˙
r α and
displacements r α are specified at time t. Newton’s equations allow one to determine
the state of the system at time t uniquely in terms of the state at time t = 0. Thus
a system composed of N point particles evolves in a phase space composed of 3N
velocity and 3N position coordinates.
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3
397
Classical Mechanics Concepts
In this appendix, we review some basic concepts from classical mechanics that are
used in this book.
A.1 Newton’s Equations
Newton’s equations of motion describe the behavior of collections of N point
particles in a three-dimensional space (3N degrees of freedom) in terms of 3N
coupled second-order differential equations (the number of equations can be
reduced if constraints are present),
d 2 (m α r α )
dt 2
= F α ,
(A.1)
where α = 1, . . . , N, m α is the mass, r α is the displacement of the αth particle, and
F α is the net force on the αth particle due to the other particles and any external fields
that might be present. Equations (A.1) only have simple structure in inertial frames
of reference and for Cartesian coordinates. For noninertial frames and general
orthogonal curvilinear coordinate systems, they rapidly become complicated and
nonintuitive.
Because Newton’s equations are second order, the state of a collection of
N particles at time t is determined once the velocities v α =
dr α
dt = ˙
r α and
displacements r α are specified at time t. Newton’s equations allow one to determine
the state of the system at time t uniquely in terms of the state at time t = 0. Thus
a system composed of N point particles evolves in a phase space composed of 3N
velocity and 3N position coordinates.
© Springer Nature Switzerland AG 2021
L. Reichl, The Transition to Chaos, Fundamental Theories of Physics 200,
https://doi.org/10.1007/978-3-030-63534-3
397