Classical Mechanics and Electrodynamics
2nd Edition by Jon Magne Leinaas,
Chapter 1-15
SOLUTION MANUAL
, Contents
Part 1 Analytical Mechanics 1
Chapter 1: Generalized coordinates 3
Chapter 2: Lagrange’s equations 9
Chapter 3: Hamiltonian dynamics 31
Part 2 Relativity 47
Chapter 4: The four-dimensional space-time 49
Chapter 5: Consequences of the Lorentz transformations 55
Chapter 6: Four-vector formalism and covariant equations 63
Chapter 7: Relativistic kinematics 67
Chapter 8: Relativistic dynamics 77
Part 3 Electrodynamics 89
Chapter 9: Maxwell’s equations 91
Chapter 10: Electromagnetic field dynamics 99
Chapter 11: Maxwell’s equations with stationary sources 105
Chapter 12: Electromagnetic radiation 113
Part 4 Classical Field Theory 129
Chapter 13: Lagrangian and Hamiltonian formulations 131
Chapter 14: Symmetry transformations 139
Chapter 15: Relativistic fields 145
v
, PART 1
Analytical Mechanics
Chapter 1
Generalized coordinates
Problem 1.1
Four mechanical systems are studied. In all cases the number of degrees of
freedom are specified, and an appropriate set of generalized coordinates is
chosen.
a) The first system consists of a pendulum attached to a block which
in turn is attached to a spring. We assume all motion takes place in a two-
dimensional, vertical plane. The block is constrained to move in the hori-
zontal direction, and the pendulum is constrained by the constant length of
the rod. Starting from two degrees of freedom for each of the two objects,
the two constraints reduce the number of degrees of freedom to two, one for
each object. A natural choice of generalized coordinates is the horizon- tal
displacement x of the block and the angle θ of the rod relative to the vertical
direction.
b) The second system consists of a pendulum attached to a vertical disk,
which rotates with a fixed angular frequency. Also here we consider the
motion restricted to a two-dimensional, vertical plane. There is no degree of
freedom related to the rotating disk, since it has an externally determined
angular frequency. The pendulum is again only constrained by the fixed
length of the rod, and the number of degrees of freedom of the system is
therefore one. A natural choice of generalized coordinate is the angle θ
between the pendulum rod and the vertical direction.
c) In the third case a rigid rod can tilt without sliding on the top of
the cylinder, while the cylinder can roll on a horizontal plane. Assuming
again that the motion is restricted to a two-dimensional, vertical plane, the
starting point is three degrees of freedom for each object. For the cylinder
this corresponds to two coordinates for its center of mass and one for its
angle of rotation. For the rod there are two coordinates needed to determine
the position of its center of mass, and one coordinate to determine the angle of
the rod relative to the horizontal (or vertical) direction.
3
,
2nd Edition by Jon Magne Leinaas,
Chapter 1-15
SOLUTION MANUAL
, Contents
Part 1 Analytical Mechanics 1
Chapter 1: Generalized coordinates 3
Chapter 2: Lagrange’s equations 9
Chapter 3: Hamiltonian dynamics 31
Part 2 Relativity 47
Chapter 4: The four-dimensional space-time 49
Chapter 5: Consequences of the Lorentz transformations 55
Chapter 6: Four-vector formalism and covariant equations 63
Chapter 7: Relativistic kinematics 67
Chapter 8: Relativistic dynamics 77
Part 3 Electrodynamics 89
Chapter 9: Maxwell’s equations 91
Chapter 10: Electromagnetic field dynamics 99
Chapter 11: Maxwell’s equations with stationary sources 105
Chapter 12: Electromagnetic radiation 113
Part 4 Classical Field Theory 129
Chapter 13: Lagrangian and Hamiltonian formulations 131
Chapter 14: Symmetry transformations 139
Chapter 15: Relativistic fields 145
v
, PART 1
Analytical Mechanics
Chapter 1
Generalized coordinates
Problem 1.1
Four mechanical systems are studied. In all cases the number of degrees of
freedom are specified, and an appropriate set of generalized coordinates is
chosen.
a) The first system consists of a pendulum attached to a block which
in turn is attached to a spring. We assume all motion takes place in a two-
dimensional, vertical plane. The block is constrained to move in the hori-
zontal direction, and the pendulum is constrained by the constant length of
the rod. Starting from two degrees of freedom for each of the two objects,
the two constraints reduce the number of degrees of freedom to two, one for
each object. A natural choice of generalized coordinates is the horizon- tal
displacement x of the block and the angle θ of the rod relative to the vertical
direction.
b) The second system consists of a pendulum attached to a vertical disk,
which rotates with a fixed angular frequency. Also here we consider the
motion restricted to a two-dimensional, vertical plane. There is no degree of
freedom related to the rotating disk, since it has an externally determined
angular frequency. The pendulum is again only constrained by the fixed
length of the rod, and the number of degrees of freedom of the system is
therefore one. A natural choice of generalized coordinate is the angle θ
between the pendulum rod and the vertical direction.
c) In the third case a rigid rod can tilt without sliding on the top of
the cylinder, while the cylinder can roll on a horizontal plane. Assuming
again that the motion is restricted to a two-dimensional, vertical plane, the
starting point is three degrees of freedom for each object. For the cylinder
this corresponds to two coordinates for its center of mass and one for its
angle of rotation. For the rod there are two coordinates needed to determine
the position of its center of mass, and one coordinate to determine the angle of
the rod relative to the horizontal (or vertical) direction.
3
,
