Lagrangian equations of motion. Conservation laws

This chapter addresses the invariance of the Lagrangian equations of motion under the coordinate to transformation, the transformation of the energy and generalised momenta under the coordinate transformation. The integrals of motion for a particle moving in the field with a given symmetry to the Noether’s theorem, the Lagrangian functions, and the Lagrangian equations of motion for the electromechanical system. The authors also discuss the influence of constraints and friction on the motion of a system, the virial theorem and its generalization in the presents of a magnetic field, and an additional integral of motion for a system of three interacting particles.

Integration of one-dimensional equations of motions

If the potential energy is independent of time, the energy of the system remains constant during the motion of a closed system. A system with one degree of freedom allows for the determination of the law of motion in quadrature. In this chapter, the authors consider motion of the particles in the one-dimensional fields. They discuss also how the law and the period of a particle moving in the potential field change due to adding to the given field a small correction.

Small oscillations of systems with several degrees of freedom

This chapter addresses the free and forced oscillations of simple systems (with two or three degrees of freedom), the free oscillations of systems with the degenerate frequencies, and the eigen-oscillations of the electromechanical systems. This chapter also studies the oscillations of more complex systems using orthogonality of eigenoscillations and the symmetry properties of the system, the free oscillations of an anisotropic charged oscillator moving in a uniform constant magnetic field, and the perturbation theory adapted for the small oscillations. Finally, the chapter addresses oscillations of systems in which gyroscopic forces act and the eigen-oscillations of the simple molecules.

Adiabatic invariants

This chapter addresses the adiabatic invariant for a mathematical pendulum, a model of a “gas” consisting of a single molecule in a piston, adiabatic approximation, and a simplified model of an ion H2+. The chapter also discusses the connection between the volume and the pressure of a gas consisting of particles inside an elastic cube, the adiabatic invariants for a charged anisotropic harmonic oscillator in a uniform magnetic field, a magnetic trap, and the action and angle variables for the simple systems.

Canonical transformations

This chapter addresses the canonical transformation defined by the given generating function, the rotation in the phase space as a canonical transformation, and themovement of the system as a canonical transformation. The chapter also discusses using the canonical transformations for solving the problems of the anharmonic oscillations and using the canonical transformation to diagonalize the Hamiltonian function of an anisotropic charged harmonic oscillator in a magnetic field. Finally, the chapter addresses the canonical variables which reduce the Hamiltonian function of the harmonic oscillator to zero and using them for consideration of the system of the harmonic oscillators with the weak nonlinear coupling.

The Hamiltonian equations of motion. Poisson brackets

This chapter addresses invariance of the Hamiltonian function under a given transformation and the conservation law, the Hamiltonian function for the beam of light, the motion of a charged particle in a nonuniform magnetic field, and the motion of electrons in a metal or semiconductor. The chapter also discusses the Poisson brackets and the model of the electron and nuclear paramagnetic resonances, the Poisson brackets for the components of the particle velocity, and the “hidden symmetry” of the hydrogen atom.

Oscillations of linear chains

This chapter addresses chain of particles connected by springs as the simplest models used in theory of solids, the travelling and standing waves on a chain, and the free and forced oscillations of N particles which are connected by springs and which can move either along a straight line or along a ring. The chapter also addresses the free and forced oscillations of 2N particles, alternating either with masses or with elastic constants; the free and forced oscillations of the artificial line with inductances and capacitors; and the elastic rod as the limiting case of the system of N particles connected by spring in the limit N tends to infinity.

Small oscillations of systems with one degree of freedom

This chapter addresses the small free oscillations for particle moving near the minimum of the potential energy, an oscillator with friction under action of a given force, and the stable oscillations of an oscillator which is acted upon by a periodic force. The authors also discuss the differential cross section for the oscillator which excited to an given energy by a fast particle and a harmonic oscillator in the field of the travelling wave.

Scattering in a given field. Collision between particles

This chapter addresses the differential and total cross section for the scattering of particles by central field, the scattering of particles by the fixed to ellipsoid, and the small angles scattering of particles by central field as well as a dipol. The authors also discuss the cross section for the process where a particle falls towards the centre of the field, decay of particles and the distribution of the secondary particle, and the change in intensity of a beam of particles travelling through a volume filled with absorbing centres.

The Hamilton–Jacobi equation

This chapter discusses the motion of particles which are scattered by and fall towards the center of the dipol, the motion of a particle in the Coulomb and the constant electric fields, and a particle inside a smooth elastic ellipsoid. The chapter also addresses the trajectory of a particle moving in the field of two Coulomb centres and a beam of electrons inside a short magnetic lens.