# Classical mechanics

Classical mechanics is the part of physics that deals with motion and forces. In its most well-known formulation it is known as Newtonian mechanics, named after Isaac Newton.

As any scientific concept, the classical mechanics has limited range of validity. This range is pretty large and covers the most of situations the civilian people meet in the common life. For example, the model works well for everyday situations such as a car changing lanes on a motorway or a football flying through the air. For very small objects however, quantum mechanics must be applied for accurate results. Similarly, the behaviour of objects which travel at speeds approaching the speed of light or in a strong gravitational field can not be described by classical mechanics alone. For such situations, relativity must be applied.

## Contents

### Basic concepts

The classical mechanics deals with bodies, or physical bodies; such a body can be called also a material point. The bodies are described in terms of their coordinates in some specific reference frame. One can imagine the reference frame as a ruler with marks that give values of the coordinates. In the 3-dimensional space, the coordinates of each body can be treated as 3-vectors.

The coordinate of some body in some reference frame can be expressed as sum of two vectors: vector of coordinates in the old reference frame and the vector of coordinates of the old reference frame in a new reference frame.

Also, in the Newtonian mechanics, it is assumed that there exist time, universal for all the frame references, and the coordinates of all the bodies are smooth functions of time. The description of movements of bodies in terms of time-dependent coordinates is called kinematics.

The bodies are allowed to interact. The interaction is characterized with forces. The forces are treated as 3-dimensional vectors. If several forces act on the body, they affect in a way, equivalent to the force which is a vector sum of all forces applied.

Each body is attributed some positive real number called mass. The mass of each body determines, how the body reacts to the force by other bodies.

### Motion

Any body that moves from one point to another has an average velocity (vav) which is a measure of the rate of change of displacement (Δx) with time. In equation form:

v = Δx / Δt

The instantaneous velocity is then the limit of the average as the time interval (Δt) approaches zero:

v = dx / dt

In a one dimensional system the term speed could be used instead of velocity however in more dimensions the difference between a vector quantity (like velocity which has a magnitude and a direction) and a scalar quantity (such as speed which only has a magnitude) is very important.

If the velocity of a body changes with time the body has acceleration (a). Acceleration is related to velocity in the same way as velocity is to displacement:

$a = \Delta v/\Delta t\,\!$ , and
a = dv / dt

One of Newton's inventions, calculus, which was simultaneously and independently invented by Gottfried Wilhelm Leibniz, is useful in mechanics. Acceleration is the derivative of velocity (with respect to time), which is the derivative of displacement (with respect to time).

### Newton's laws of motion

Newton's laws of motion help to analyze the principles of dynamics, the relationship of motion to the forces that cause it. These three laws were first published in 1687 in Philosophiae Naturalis Principia Mathematica. The following is an English translation of the laws:

• First Law: There exist such a reference frame, in which any body that does not interact with other bodies moves with acceleration zero.

Such a reference frame is called inertial reference frame. Any reference frame, that moves with constant velocity with respect to some reference frame is also inertial frame reference. This property of bodies in the classical mechanics is called inertia, a tendency to keep moving in the same direction until another force causes it to stop or change direction. By default, the frame references are assumed to be inertial.

• Second Law: If a net force acts on a body, the body accelerates. The force equals the mass of the body multiplied by the acceleration.

This relation of force and motion is a fundamental law of nature. The acceleration of an object is directly proportional to the net force on the object and inversely proportional to the mass of the object. This can be written in equation form as:

$\textbf{F} = m \textbf{a}$

where F is the net force needed to cause an acceleration a in a body of mass m. Note that F and a are vectors, thus a change in the direction of motion is also a form of acceleration.

• Third Law: If body A exerts a force on body B, then body B exerts a force on body A. This force will have an equal magnitude and opposite direction.

This is less formally stated as; every action has an equal and opposite reaction. It's important to remember these two forces act on different bodies. For example a ball thrown in the air is being pulled towards the centre of the earth by a force due to gravity and is exerting a force of equal magnitude pulling the earth towards the ball. The acceleration on the earth is negligible because it has a much larger mass as stated in the second law. A useful example is attempting a tug of war on ice skates. No matter who is stronger, the person with the largest mass will inevitably win.

These laws are only valid in an inertial frame of reference or, as Newton called it, in an absolute space. While Newton's laws can be stated very easily, it can be hard to apply them to real-world situations where there are many different forces acting on an object. When two objects interact in contact with each other there are contact forces in action. Usually a normal (perpendicular) force and a friction force. The friction force always acts in a direction opposite to the direction of the force (it opposes the change).

### Laws of conservation

Many basic laws of conservation follow from the Laws of Newton:

The center of mass of any isolated system of bodies is linear function of time;

The momentum of any isolated system of bodies remains constant;

The angular momentum of any isolated system remains constant;

The Energy of any isolated system remains constant.

### Formalization of the classical mechanics and the extensions

Apart from Newton's formulation, classical mechanics can also be expressed in the Lagrangian and Hamiltonian formalisms. Hamiltonian mechanics is the starting point for canonical quantum mechanics, while Hamilton's path integral version of quantum field theory begins with Lagrangian mechanics.

Such a formalism becomes important making the extension of the newtonian mechanics. The consideration of system of elementary points with fixed distances between them leads to the mechanic of rigid body. The consideration of deformable body leads to the theory of elastic body and the consideration of liquids and gases is called mechanics of fluids.

In addition, the Lagrangian or Hamiltonian formalism applies to the systems with holomonic constrains, that have application in the mechanical engineering.

### Range of validity

The concepts of classical mechanics such as coordinate, universal time have limited range of validity.

#### Quantum effects

If some system changes its state during time t during an evolution, then, this change can be characterized with action

A = Et

where E is the range of energies the system has. The time can be related with a period or quasi-period, if the system shows a periodic or quasi-periodic motion. Usually, for the applicability of the Laws of Newton, it is sufficient to have

$A \gg \hbar$

where $\hbar \approx 10^{-27} \rm erg\cdot s$ is the Planck's constant.

At the consideration of movements with action of order of Planck's constant or smaller, the concept of coordinate becomes a rough approximation or just invalid; and the quantum mechanics should be used for the description of the system. Usually, this refers to the atomic and molecular systems. However, some pretty macroscopic objects (for example, the electromagnetic field in a superconducting cavity or in a system with counting of individual photons) may require the quantum description.

#### Relativistic effects

The classical mechanics works for slow objects; in some reference frame, the speeds or all the objects should be small compared to the speed of light. This limit follows from the postulate that no one material object can move with speed of light or faster. If some spacecraft during long time moves with the same acceleration, say, g, then, within the Newtonian mechanics, owe would expect its speed to be linear function of time. In principle the acceleration may reman g in the so-called joint frame reference (i.e., the frame reference, specific for each moment t of time, characterized in that the speed of the body in this reference frame is zero just at the moment t). However, the acceleration defined as second derivative of the coordinate with respect to time may be something different, in such a way that the velocity of that spacecraft approaches c, but never reaches it.

The branch of physics that deals with such velocity is called relativistic mechanics. The basic laws of conservation in the relativistic mechanics are the same as in the classical mechanics, but such concepts as momentum and energy have to be redefined; the momentum of a particle cannot be expressed as product of its mass to its velocity; the same refers also to the energy.

#### Quantum field theory

In the case when the speed is high, and the action is small, the creation of new material objects becomes possible. In this case one cannot use the concepts of mechanics in the common sense: not only the coordinates of the particles cannot be correctly determined, but even the number of particle cannot be characterized with an integer number. This case revers to the Quantum field theory.

#### Homogeneity of space

The isotropy and homogeneity of the space–time is basic concept of mechanics, both classical and quantum and both relativistic and non-relativistic. However, this homogeneity is only approximation: the gravitational interaction disturbs the metrics of the space. This disturbance may be significant due to huge masses in vicinity of the physical system. The space has also the quantum fluctuations of metrics and even topology. In such a way, the smooth and flat space and time, used in the classical mechanics, it is just very good approximation. It is expected to be valid, while the distances are much larger than the Planck distance, and the time intervals considered are larger than the Planck time. These scales of space and time are really small; at the beginning of the 21st century, no one experiment may approach this scale measuring time and distance.

#### Technology is far from the limits

The classical mechanics has fundamental limits. The devices for measurement of distance cannot be very small and very light; otherwise, they, by themselves, become quantum objects. The devices cannot be too heavy; otherwise, they disturb the metrics of space due to the gravitational interaction. In this sense there is no physical limit, in which the classical mechanics is "mathematically exact". However, the range of masses and velocities and the precision of the physical devices, is many orders of magnitude far from these fundamental limits. In particular, all the mechanical machines, including the spacecrafts and satellites, are described with the classical mechanics, and extremely precise measurements are necessary to see the deviation from the laws of Newton due to the relativistic effects. At least for the beginning of the 21st century, such a deviation appears only in the second order of the perturbation theory with respect to v/c, id est, the deviation from the Newtonian mechanics is of order of (v/c)2. Even at the cosmic speeds, say, v/c = 10-5, the relative deviation is at the level of 10-10 and is barely detectable. The similar estimates can be done for the quantum effects; yet, they do not limit the precision of the mechanical devices. In this sense, the classical mechanics is very precise science.