Reality → Base → Physics → Classic
Based on meticulous mathematical evaluation of astronomical observations, Kepler discovered in the early 17th century laws that for the first time correctly described the motion of planets around the Sun. Kepler's laws are: 1. the planetary orbits are ellipses; 2. the line joining Sun and planet sweeps equal areas in equal time; and 3. the square of the orbital period is proportional to the cube of the orbit’s semi-major axis. It took about 100 years until the genius of Newton presented a comprehensive physical explanation with his laws of motion and gravitation [1] . It then became evident that Kepler’s second law is a consequence of the conservation of angular momentum and the third law reflects the equilibrium of gravitational and centrifugal forces. Kepler’s and Newton’s discoveries allow accurate calculation and prediction of celestial orbits and events, as well as of trajectories of artificial satellites and space vehicles. However, despite the formulas’ astounding accuracy and immense usefulness, puzzling questions of why and how matter (or mass) causes gravity remain unanswered.
Newton formulated the relation between a body’s mass and its movement in his famous three laws of motion. Simplified, they describe: 1. inertia - without a force, a body is either at rest or moves with constant velocity; 2. force - equals mass times acceleration (or change of momentum); and 3. reaction - every force occurs together with an equal force in opposite direction. These simple and most powerful laws revolutionized physics [2] . Many physical quantities, including work, power, kinetic and potential energy, are directly related to Newton's laws [3] . Kinematics and dynamics explain many natural phenomena, including microscopic motions and oscillations and waves, and provide analytical tools in many areas of physics, including acoustics, thermodynamics, and even atomic, nuclear and particle physics .
In an oscillation, physical quantities change periodically. A sine function describes the displacement motion of a simple harmonic oscillator. Amplitude, velocity (first time derivative) and acceleration (second time derivative) equally follow harmonic changes (described by sine and cosine functions) [4] . Any number of sine functions with different frequencies, amplitudes, and phases can be combined by superposition to result in any periodic functions. Reversely, any periodic function can be decomposed into basic sine/cosine functions by Fourier transformation [5] . Waves are oscillations that move forward in space. The momentary amplitude is not only a function of time but also of distance [6] . The wave’s speed of propagation depends on the stiffness (resistance to deformation) and the density of the medium through which it travels. The momentary displacement caused by a wave can be transversal (perpendicular to the direction of the wave’s propagation) or longitudinal (in direction of the wave’s propagation) [7] . Reflexion, refraction, diffraction, and interference are phenomena associated with waves [8] . Electromagnetic waves propagate also in vacuum, i.e. in 'free space' without the presence of any carrier medium (see also Speed of light). An even weirder type of waves are de Broglie’s matter waves.
The ideal gas law defines the relation between pressure, volume, and temperature of a gas, i.e., the product of pressure and volume (a mechanical work quantity) is proportional to the absolute temperature, where the gas constant is the proportionality factor. The law, originally derived from empirical evidence, is central to thermodynamics and physical chemistry [9] . Statistics and probability theory explain thermodynamic properties with the erratic motion of atoms and molecules [10] . A multitude of engineering formulas and tables based on thermodynamics are used in the design of combustion engines and turbines, thermal power plants, and heating, air conditioning and refrigeration systems.
Newton's law of universal gravitation postulates that the attractive force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them, where the constant of proportionality is the universal gravitational constant G ( 6.7 × 10-11 N m2 kg-2). G must not be confused with g ( 9.81 m s-2), the average acceleration applied to all objects exposed to Earth’s gravitational field at or near Earth’s surface. G applies to the mutual attraction of any masses anywhere (see a sample Calculation for Earth-Moon gravity).
Before Newton, it was widely believed that force was needed to maintain velocity, a view apparently closer to daily experience, but without the insight that the counteracting friction is a force that decelerates (i.e. causes negative acceleration). Newton’s insights opened the way for unprecedented progress in science and technology.
The math of harmonic oscillations also applies to the electric LC circuit and its resonant frequency, a discovery that led to creation and reception of artificial radio waves, with immensely useful applications in RF engineering and, later, also in laser technologies.
See also Electromagnetic radiation, Note 1 (and Sheet for some simple equations).
Fourier analysis can be applied to design rectangular or other electric pulses used in signal processing. For this and other problems, the math can be eased by using complex numbers to convert trigonometric functions into exponential functions (see also Mathematical constants, Notes 3 and 4).
The wave equation expresses that the second partial differential with respect to time equals the second partial differential with respect to distance times the square of the wave’s speed of propagation. The solution of this equation is a sine function dependent on time and distance (see Sheet).
In seismology, transversal waves are called shear waves. They can only occur in solids, as liquids and gases have no shear resistance. The more common mechanical waves are longitudinal, i.e., compression, shock, and acoustic waves which travel through all three phases (solid, liquid, gas) of matter.
A very helpful tool to better understand wave phenomena is the Huygens-Fresnel principle. It postulates that each point of a wave front becomes the origin of a new (secondary) wave and that the new wave front is the resultant of of all secondary waves from points already passed. The refraction of a ray of light is described by the principle of least time and Snell's law.
The ideal gas law is a combination of the experimental gas laws of Boyle, Charles, and Gay-Lussac. Most gases behave like the 'ideal' gas if temperature is high or pressure low, i.e., if the distance between molecules compared to their diameter is large, so that the influence of intermolecular forces is negligible. The molecular hypothesis is also reflected in the gas constant's close relationship with the Avogadro and Boltzmann constants (see Sheet).
The kinetic theory of gases relates macroscopic gas properties to the assumed free and random movement of gas molecules (e.g., temperature is described by the mean velocity, pressure by the impact, and heat energy by the kinetic energy of molecules). Inclusion of intermolecular forces (e.g., by the van der Waals equation) improves accuracy and even explains phase transition and properties of fluids. Boltzmann's entropy formula describes the relationship between entropy and the statistical distribution of locations and impulses of all atoms or molecules of the system. The term 'statistical thermodynamics' is often used as a synonym for ' statistical mechanics', a branch of theoretical physics. The amazing, difficult to understand connection between random events at the invisible micro-level and concrete, highly predictable phenomena at the macro-level is repeated in quantum mechanics.