Dynamics

Dynamics is the branch of applied mathematics (specifically classical mechanics) concerned with the study of forces and torques and their effect on motion, as opposed to kinematics, which studies the motion of objects without reference to these forces. Isaac Newton defined the fundamental physical laws which govern dynamics in physics, especially his second law of motion.

History
One of the first reflections on the causes of movement is due to the Greek philosopher Aristotle; which defined the movement, the dynamic, as:.:

The realization act, of a capacity or possibility of being power, while it is being updated.

On the other hand, unlike the current approach, Aristotle reverses the study of kinematics and dynamics, studying first the causes of movement and then the movement of bodies. This approach hindered the advance in the knowledge of the movement phenomenon until, in the first instance, St. Albert the Great, who was the one who pointed out this difficulty, and ultimately to Galileo Galilei and Isaac Newton. In fact, Thomas Bradwardine, in 1328, presented in his De proportionibus velocitatum in motibusa mathematical law that linked speed with the ratio of motives to resistance forces; his work influenced the medieval dynamic for two centuries, but, for what has been called a mathematical accident in the definition of “increase”, his work was discarded and was not given historical recognition in his day.

Galileo’s experiments on uniformly accelerated bodies led Newton to formulate his fundamental laws of motion, which he presented in his main work Philosophiae Naturalis Principia Mathematica.

Current scientists believe that Newton’s laws give the right answers to most problems related to moving bodies, but there are exceptions. In particular, equations for describing movement are not suitable when a body travels at high speeds with respect to the speed of light or when objects are extremely small in size comparable to sizes.

Principles
Generally speaking, researchers involved in dynamics study how a physical system might develop or alter over time and study the causes of those changes. In addition, Newton established the fundamental physical laws which govern dynamics in physics. By studying his system of mechanics, dynamics can be understood. In particular, dynamics is mostly related to Newton’s second law of motion. However, all three laws of motion are taken into account because these are interrelated in any given observation or experiment.

Linear and rotational dynamics
The study of dynamics falls under two categories: linear and rotational. Linear dynamics pertains to objects moving in a line and involves such quantities as force, mass/inertia, displacement (in units of distance), velocity (distance per unit time), acceleration (distance per unit of time squared) and momentum (mass times unit of velocity). Rotational dynamics pertains to objects that are rotating or moving in a curved path and involves such quantities as torque, moment of inertia/rotational inertia, angular displacement (in radians or less often, degrees), angular velocity (radians per unit time), angular acceleration (radians per unit of time squared) and angular momentum (moment of inertia times unit of angular velocity). Very often, objects exhibit linear and rotational motion.

For classical electromagnetism, Maxwell’s equations describe the kinematics. The dynamics of classical systems involving both mechanics and electromagnetism are described by the combination of Newton’s laws, Maxwell’s equations, and the Lorentz force.

Force
From Newton, force can be defined as an exertion or pressure which can cause an object to accelerate. The concept of force is used to describe an influence which causes a free body (object) to accelerate. It can be a push or a pull, which causes an object to change direction, have new velocity, or to deform temporarily or permanently. Generally speaking, force causes an object’s state of motion to change.

Newton’s laws
Newton described force as the ability to cause a mass to accelerate. His three laws can be summarized as follows:

First law: If there is no net force on an object, then its velocity is constant. Either the object is at rest (if its velocity is equal to zero), or it moves with constant speed in a single direction.
Second law: The rate of change of linear momentum P of an object is equal to the net force Fnet, i.e., dP/dt = Fnet.
Third law: When a first body exerts a force F1 on a second body, the second body simultaneously exerts a force F2 = −F1 on the first body. This means that F1 and F2 are equal in magnitude and opposite in direction.

Newton’s Laws of Motion are valid only in an inertial frame of reference.

Calculation in dynamics
In classic mechanics and relativistic mechanics, by means of the concepts of displacement, velocity and acceleration it is possible to describe the movements of a body or object without considering how they have been produced, a discipline known as kinematics. On the contrary, mechanics deals with the study of the movement of bodies subjected to the action of forces. In quantum systems the dynamics requires a different approach due to the implications of the uncertainty principle.

The dynamic calculation is based on the equation approach of the movement and its integration. For extremely simple problems the equations of Newtonian mechanics directly aided by conservation laws are used. In classical and relativistic mechanics, the essential equation of dynamics is Newton’s second law (or Newton-Euler law) in the form:

where F is the sum of the forces and p the amount of movement. The above equation is valid for a rigid particle or solid, for a continuous medium you can write an equation based on it that must be fulfilled locally. In theory of general relativity it is not trivial to define the concept of force resulting from the curvature of space time. In non-relativistic quantum mechanics, if the system is conservative the fundamental equation is the Schrödinger equation:

Conservation laws
Conservation laws can be formulated in terms of theorems that establish under what concrete conditions a given quantity is “conserved” (that is, it remains constant in value over time as the system moves or changes over time). In addition to the law of conservation of energy, the other important conservation laws take the form of vectorial theorems. These theorems are:

The theorem of the momentum, which for a system of point particles requires that the forces of the particles depend only on the distance between them and are directed according to the line that joins them. In mechanics of continuous media and mechanics of the rigid solid, vectorial theorems of conservation of momentum can be formulated.

The kinetic moment theorem establishes that under conditions similar to the previous vectorial theorem, the sum of moments of force with respect to an axis is equal to the temporal variation of the angular momentum. In particular, the Lagrangian of the system.

These theorems establish under what conditions the energy, the amount of movement or the kinetic moment are conserved magnitudes. These conservation laws sometimes allow to find in a simpler way the evolution of the physical state of a system, often without the need to directly integrate the differential equations of the movement.

Equations of movement
There are several ways of proposing equations of movement that allow predicting the evolution over time of a mechanical system based on the initial conditions and the acting forces. In classical mechanics there are several possible formulations to propose equations:

The Newtonian mechanics that resorts to write directly ordinary differential equations of second order in terms of forces and in Cartesian coordinates. This system leads to equations difficult to integrate by elementary means and is only used in extremely simple problems, usually using inertial reference systems.

The Lagrangian mechanics, this method also uses ordinary second – order differential equations, but allows the use of totally general coordinates, called generalized coordinates, which are better suited to the geometry of the problem. Furthermore, the equations are valid in any reference system, whether this is inertial or not. In addition to obtaining more easily integrable systems, Noether’s theorem and coordinate transformations, we can find integrals of movement, also called conservation laws, more simply than the Newtonian approach.
The Hamiltonian mechanics is similar to the previous one but in it the equations of motion are ordinary differential equations of the first order. In addition, the range of allowable coordinate transformations is much wider than in Lagrangian mechanics, which makes it even easier to find movement integrals and conserved quantities.

The Hamilton-Jacobi method is a method based on the resolution of a differential equation in partial derivatives by the method of separation of variables, which is the simplest means when an appropriate set of movement integrals are known.

In relativistic mechanics the last three approaches are possible, in addition to a direct approach to simple problems that is analogous to many methods of Newtonian mechanics. Likewise, the mechanics of continuous media admit lagrangian and Hamiltonian approaches, although the underlying formalism is a classical or relativistic system, it is notably more complicated than in the case of rigid particle and solid systems (the latter have a finite number of degrees). freedom, unlike a continuous medium). Finally, quantum mechanics, both non-relativistic and relativistic, also requires a remarkably more complex mathematical formalism that usually involves the use of Hilbert spaces even for systems with a finite number of degrees of freedom.

Dynamics of mechanical systems
In physics there are two important types of physical systems: finite particle systems and fields. The evolution in time of the first can be described by a finite set of ordinary differential equations, which is why it is said to have a finite number of degrees of freedom. On the other hand, the evolution in time of the fields requires a set of complex equations. In partial derivatives, and in a certain informal sense they behave like a system of particles with an infinite number of degrees of freedom.

Most mechanical systems are of the first type, although there are also mechanical systems that are described more simply as fields, as with fluids or deformable solids. It also happens that some mechanical systems ideally formed by an infinite number of material points, such as rigid solids, can be described by a finite number of degrees of freedom.

Dynamics of the particle
The dynamics of the material point is a part of Newtonian mechanics in which systems are analyzed as point particle systems and instantaneous forces are exerted at a distance.

In the theory of relativity it is not possible to treat a set of charged particles in mutual interaction, simply using the positions of the particles at each instant, since in said frame it is considered that remote actions violate physical causality. In these conditions the force on a particle, due to the others, depends on the past positions of the same.

Dynamics of the rigid solid
The mechanics of a rigid solid is one that studies the movement and balance of material solids ignoring their deformations. It is, therefore, a mathematical model useful to study a part of the mechanics of solids, since all real solids are deformable. Rigid solid is understood as a set of points of space that move in such a way that the distances between them are not altered, whatever the acting force (mathematically, the movement of a rigid solid is given by a uniparamtric group of isometries).

Continuous media dynamics and field theory
In physics there are other entities such as continuous media (deformable and fluid solids) or fields (gravitational, electromagnetic, etc.) that can not be described by a finite number of coordinates that characterize the state of the system. In general, defined functions are required over a four-domain domain or region. The treatment of classical mechanics and the relativistic mechanics of continuous media requires the use of differential equations in partial derivatives, which causes analytical difficulties much more noticeable than those found in systems with a finite number of coordinates or degrees of freedom (which often they can be treated as systems ofordinary differential equations).

Concepts related to dynamics

Inertia
Inertia is the property of bodies not to modify their state of rest or uniform movement, if they are not influenced by other bodies or if the action of other bodies is compensated.

In physics it is said that a system has more inertia when it is more difficult to achieve a change in the physical state of it. The two most frequent uses in physics are mechanical inertia and thermal inertia. The first of them appears in mechanics and is a measure of difficulty to change the state of movement or rest of a body. The mechanical inertia dependent on the amount of mass and inertia tensor of the body. Thermal inertia measures the difficulty with which a body changes its temperature by being in contact with other bodies or being heated. Thermal inertia depends on theamount of mass and heat capacity.

The so-called inertial forces are fictitious or apparent forces for an observer in a non-inertial reference system.

The inertial mass is a measure of the resistance of a mass to the change in velocity in relation to an inertial reference system. In classical physics the inertial mass of point particles is defined by means of the following equation, where particle one is taken as the unit ():


where m i is the inertial mass of particle i, and i1 is the initial acceleration of particle i, in the direction of particle i towards particle 1, in a volume occupied only by particles i and 1, where both particles are initially at rest and at a distance unit. There are no external forces but the particles exert forces on each other.

Work and energy
The work and energy displayed by the mechanical energy theorems. The principal, and from which the other theorems are derived, is the kinetic energy theorem. This theorem can be stated in a differential version or in an integral version. From now on, reference will be made to the Theorem of kinetic energy as TEC.

Thanks to TEC, a relationship can be established between mechanics and other sciences, such as chemistry and electrical engineering, from which it derives its vital importance.

Strength and potential

The mechanics of particles or continuous media have slightly different formulations in classical mechanics, relativistic mechanics and quantum mechanics. In all of them, the causes of change are represented by forces or derived concepts such as the potential energy associated with the system of forces. In the first two the concept of force is used fundamentally, whereas in quantum mechanics it is more frequent to pose the problems in terms of potential energy. The resulting force about a classical mechanical system is related to the variation of the amount of movement by the simple relationship:

When the mechanical system is also conservative the potential energy is related to kinetic energy Associated with movement through the relationship:

In relativistic mechanics the above relations are not valid if t refers to the temporal component measured by any observer, but if t is interpreted as the observer’s own time then they are valid. In classical mechanics, given the absolute character of time, there is no real difference between the observer’s own time and its temporal coordinate.

Dynamic systems
The theory of dynamic systems is a branch of mathematics, closely related to the theory of differential equations and the theory of chaos that studies the qualitative properties of the equations of dynamic evolution.

Source from Wikipedia

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