Classical mechanics: Encyclopedia II - Classical mechanics - Description of the theory

Classical mechanics - Description of the theory

The following introduces the basic concepts of classical mechanics. For simplicity, it uses point particles, objects with negligible size. The motion of a point particle is characterized by a small number of parameters: its position, mass, and the forces applied to it. Each of these parameters is discussed in turn.

In reality, the kind of objects which classical mechanics can describe always have a non-zero size. True point particles, such as the electron, are normally better described by quantum mechanics. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom—for example, a baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made up of a large number of interacting point particles. The center of mass of a composite object behaves like a point particle.

Classical mechanics - Position and its derivatives

The position of a point particle is defined with respect to an arbitrary fixed point in space, which is sometimes called the origin, O. It is defined as the vector r from O to the particle. In general, the point particle need not be stationary, so r is a function of t, the time elapsed since an arbitrary initial time. In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute in all reference frames. In addition to relying on absolute time, classical mechanics uses Euclidean geometry ^[1].

The velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time or

In classical mechanics, velocities are directly additive and subtractive. For example, if one car traveling East at 60 km/h passes another car traveling East at 50 km/h, then from the perspective of the slower car, the faster car is traveling East at 60−50 = 10 km/h. Whereas, from the perspective of the faster car, the slower car is moving 10 km/h to the West. What if the car is traveling north? Velocities are directly additive as vector quantities; they must be dealt with using vector analysis.

Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector u = ud and the velocity of the second object by the vector v = ve where u is the speed of the first object, v is the speed of the second object, and d and e are unit vectors in the directions of motion of each particle respectively, then the velocity of the first object as seen by the second object is:

Similarly:

When both objects are moving in the same direction, this equation can be simplified to:

Or, by ignoring direction, the difference can be given in terms of speed only:

The acceleration, or rate of change of velocity, is the derivative of the velocity with respect to time or

The acceleration vector can be changed by changing its magnitude, changing its direction, or both. If the magnitude of v decreases, this is sometimes referred to as deceleration or retardation; but generally any change in the velocity, including deceleration, is simply referred to as acceleration.

The following consequences can be derived about the perspective of an event in two reference frames, S and S, where S is traveling at a relative velocity of u to S.

• v'' = v - u (the velocity v' of a particle from the perspective of S is slower by u than its velocity v from the perspective of S)
• a' = a (the acceleration of a particle remains the same regardless of reference frame)
• F' = F (since F = ma) (the force on a particle remains the same regardless of reference frame; see Newton's law)
• the speed of light is a constant
• the form of Maxwell's equations is not preserved across reference frames

Classical mechanics - Forces; Newton's second law

Newton's second law relates the mass and velocity of a particle to a vector quantity known as the force. If m is the mass of a particle and F is the vector sum of all applied forces (i.e. the net applied force), Newton's second law states that

The quantity mv is called the momentum. The net force on a particle is ,thus, equal to rate change of momentum of the particle with time. Typically, the mass m is constant in time, and Newton's law can be written in the simplified form

where is the acceleration. It is not always the case that m is independent of t. For example, the mass of a rocket decreases as its propellant is ejected. Under such circumstances, the above equation is incorrect and the full form of Newton's second law must be used.

Newton's second law is insufficient to describe the motion of a particle. In addition, it requires a value for F, obtained by considering the particular physical entities with which the particle is interacting. For example, a typical resistive force may be modelled as a function of the velocity of the particle, for example:

with λ a positive constant. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation, which is called the equation of motion. Continuing the example, assume that friction is the only force acting on the particle. Then the equation of motion is

This can be integrated to obtain

where v₀ is the initial velocity. This means that the velocity of this particle decays exponentially to zero as time progresses. This expression can be further integrated to obtain the position r of the particle as a function of time.

Important forces include the gravitational force and the Lorentz force for electromagnetism. In addition, Newton's third law can sometimes be used to deduce the forces acting on a particle: if it is known that particle A exerts a force F on another particle B, it follows that B must exert an equal and opposite reaction force, -F, on A. The strong form of Newton's third law requires that F and -F act along the line connecting A and B, while the weak form does not. Illustrations of the weak form of Newton's third law are often found for magnetic forces.

Classical mechanics - Energy

If a force F is applied to a particle that achieves a displacement Δs, the work done by the force is the scalar quantity

If the mass of the particle is constant, and ΔW_total is the total work done on the particle, obtained by summing the work done by each applied force, from Newton's second law:

where E_k is called the kinetic energy. For a point particle, it is defined as

For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.

A particular class of forces, known as conservative forces, can be expressed as the gradient of a scalar function, known as the potential energy and denoted E_p:

If all the forces acting on a particle are conservative, and E_p is the total potential energy, obtained by summing the potential energies corresponding to each force

This result is known as conservation of energy and states that the total energy,

is constant in time. It is often useful, because many commonly encountered forces are conservative.

Classical mechanics - Beyond Newton's Laws

Classical mechanics also includes descriptions of the complex motions of extended non-pointlike objects. The concepts of angular momentum rely on the same calculus used to describe one-dimensional motion.

There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. They are equivalent to Newtonian mechanics, but are often more useful for solving problems. These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, for describing mechanical systems.

Classical mechanics - Classical transformations

Consider two reference frames S and S' . For observers in each of the reference frames an event has space-time coordinates of (x,y,z,t) in frame S and (x' ,y' ,z' ,t' ) in frame S' . Assuming time is measured the same in all reference frames, and if we require x = x' when t = 0, then the relation between the space-time coordinates of the same event observed from the reference frames S' and S, which are moving at a relative velocity of u in the x direction is:

This set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform). This type of transformation is a limiting case of Special Relativity when the velocity u is very small compared to c, the speed of light.

Adapted from the Wikipedia article "Description of the theory", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki