Chapter 1: Dynamics (mechanics)

There is a physical theory known as classical mechanics that describes the motion of various things, including but not limited to projectiles, bits of equipment, spaceships, planets, stars, and galaxies. When classical mechanics was developed, there was a significant shift in the approaches that were taken and the philosophy that underpinned physics. This type of mechanics is distinguished from physics that was established after the revolutions in physics that occurred in the early 20th century. These revolutions showed limitations in classical mechanics, and the qualifier "classical" distinguishes both types of mechanics.

Newtonian mechanics is a term that is frequently used to refer to the oldest version of classical mechanics. It is comprised of the mathematical methods that were developed by Gottfried Wilhelm Leibniz, Leonhard Euler, and others to describe the motion of bodies under the action of forces. These mathematical methods are composed of the physical notions that are founded on the foundational writings of Sir Isaac Newton, which were published in the 17th century. Later on, Euler, Joseph-Louis Lagrange, William Rowan Hamilton, and other others developed methods that were based on energy. These approaches ultimately led to the development of analytical mechanics, which encompasses Lagrangian mechanics and Hamiltonian mechanics. These developments, which were mostly created in the 18th and 19th centuries, went beyond prior studies; they are currently utilized, albeit with some modifications, in every field of modern physics.

If the current condition of an object that is subject to the principles of classical mechanics is known, then it is possible to ascertain how the thing will move in the future as well as how it has moved in the past. This demonstrates that the predictions of classical mechanics over the long run are not dependable, as demonstrated by chaos theory. When it comes to the study of objects that are not extraordinarily huge and have speeds that are not close to the speed of light, classical mechanics yields correct conclusions. It is vital to make use of quantum mechanics when dealing with objects that are roughly the same size as the diameter of an atom. Special relativity is required in order to provide a description of velocities that are close to the speed of light. The scientific theory of general relativity is relevant in situations in which things reach enormously vast sizes. Relativistic mechanics are included in classical physics by some contemporary sources since they are considered to be the most developed and accurate form of the respective topic.

The field of classical mechanics was traditionally subdivided into three primary branches.

Statics is a subfield of classical mechanics that focuses on the study of force and torque acting on a physical system that does not experience acceleration but rather is in equilibrium with its surroundings. Statics is concerned with the analysis of force and torque acting on a physical system. Kinematics is a branch of physics that examines the motion of points, bodies (objects), and systems of bodies (groups of objects) without taking into account the forces that cause these things to move. Kinematics is a subject of study that is frequently referred to as the "geometry of motion" and is sometimes considered to be a subfield of mathematics. The study of dynamics goes beyond only describing the behavior of objects; it also takes into account the factors that explain that behavior.

There are authors who include special relativity into classical dynamics. Some examples of these authors include Taylor (2005) and Greenwood (1997).

The choice of mathematical formalism provides the basis for yet another method of classification. There is a wide variety of mathematical representations that can be used to present classical mechanics. There is no difference in the actual composition of these various formulations; rather, they offer distinct insights and make it easier to perform various kinds of calculations. Although the phrase "Newtonian mechanics" is frequently used as a synonym for non-relativistic classical physics, it can also be used to refer to a specific formalism that is founded on Newton's equations of motion. According to this interpretation, Newtonian mechanics places considerable emphasis on force as a vector quantity.

Analytical mechanics, on the other hand, makes use of scalar properties of motion; these properties reflect the system as a whole and typically include the system's kinetic energy and potential energy. Through the application of an underlying theory concerning the change of the scalar, the equations of motion are obtained from the scalar quantity used. Lagrangian mechanics, which employs generalized coordinates and corresponding generalized velocities in configuration space, and Hamiltonian mechanics, which employs coordinates and corresponding momenta in phase space, are two of the fundamental branches of analytical mechanics. Both of these branches are considered to be the most prominent. Both formulations are equal because a Legendre transformation is performed on the generalized coordinates, velocities, and momenta. As a result, both formulations include the same information for characterizing the dynamics of a system. The Hamilton-Jacobi theory, Routhian mechanics, and Appell's equation of motion are some of the additional formulations that exist. Using the widely applicable finding known as the principle of least action, it is possible to derive all equations of motion for particles and fields, regardless of the formalism that is being presented. As a consequence, Noether's theorem is a statement that establishes a connection between conservation rules and the symmetries that are connected with them.

A divide can also be created according to the region of application, as an alternative:

Classical physics frequently models real-world things as point particles, which are also known as objects with insignificant size. This is done for the purpose of simplicity. There are only a few characteristics that define the motion of a point particle. These parameters include the particle's position, its mass, and the forces that are applied to it. A further application of classical mechanics is the description of the more complicated motions of extended non-pointlike objects. When it comes to this particular domain, Euler's laws offer expansions to Newton's laws. Both the ideas of angular momentum and the calculus that is used to describe motion in one dimension are dependent on the same framework. The rocket equation is an extension of the concept of the rate of change of an object's momentum, which takes into account the phenomena that occur when an object "loses mass." (These generalizations and extensions are derived from Newton's principles, for example, by breaking down a solid body into a collection of points.)

In the real world, the kinds of things that classical mechanics can describe always have a size that is equal to or greater than zero. (The behavior of very small particles, such as the electron, is more accurately explained by quantum mechanics.) Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom, e.g., a baseball can spin while it is moving. The findings for point particles, on the other hand, can be utilized to investigate such objects by treating them as composite objects, which are composed of a large number of point particles that behave collectively among themselves. In a composite object, the center of mass behaves in a manner similar to that of a point particle.

Traditional mechanics is based on the assumption that matter and energy possess defined and observable characteristics, such as their position in space and their velocity. The assumption that forces act immediately is also made by non-relativistic mechanics (for more information, see also Action at a distance).

According to a coordinate system that is centered on an arbitrary fixed reference point in space known as the origin O, the position of a point particle is defined in relation to that coordinate system. One possible way to express the location of a particle P using a straightforward coordinate system is to use a vector that is denoted by an arrow with the letter r and that extends from the origin O to the point P. According to the general rule, it is not necessary for the point particle to be stationary in relation to O. The value of r is defined as a function of t, which is the time, in situations where P is moving in relation to O. According to the theory of pre-Einstein relativity, also known as Galilean relativity, time is regarded as an absolute. This means that the amount of time that is viewed to pass between any two occurrences is the same for all observers. In addition to relying on absolute time, classical mechanics believes that the structure of space is based on Euclidean geometry.

The derivative of the position with respect to time is the definition of the velocity, which can also be thought of as the rate of change of displacement with time:

A direct additive and subtractive relationship exists between velocities in classical mechanics. For instance, if one vehicle is heading east at a speed of sixty kilometers per hour and another vehicle is moving in the same direction at fifty kilometers per hour, the slower vehicle will perceive the faster vehicle as traveling east at sixty kilometers per hour less fifty kilometers per hour, which is equal to ten kilometers per hour. The slower car, on the other hand, is driving 10 kilometers per hour to the west, which is commonly represented as -10 kilometers per hour, where the sign indicates that the car is moving in the other...