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. The model holds 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.

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.

Motion

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

$v=\Delta x/\Delta t$

The instantaneous velocity is then the limit of the average as the time interval ($\Delta$ t) 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: A body acted on by no net force moves with constant velocity and zero acceleration.

This means that a body in motion has a property called inertia, a tendency to keep moving in the same direction until another force causes it to stop or change direction.

• 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.

An illustration of forces acting on an object. g is the acceleration due to the Earth's gravitational field. The object is at rest.

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).

Inertial propulsion

The apparatus for an inertial propulsion engine violates the First Law of Newton. Such an engine is reported to have been installed in the Russian satellite "Jubilee" (Юбилейный) which was launched on May 23, 2008, as reported by the Russian newspaper Pravda and quoted below:

Specialists of the Scientific Research Institute for Space Systems (SRISS) conducted successful tests of a perpetual motion machine in space. Valery Menshikov, the director of the institute, said that the machine was installed in the Jubilee satellite which was launched into orbit almost a year ago. The satellite can now move from one orbit to another with the help of the engine, which discharges no reaction mass. The first tests were conducted in June and July of 2008. The tests revealed some problems that need further developments of the machine, but the orbital experiment was conducted successfully in general.