Classical mechanics

Classical mechanics (often called "Newtonian mechanics" after Isaac Newton, who made major fundamental contributions to the theory) is the physics of forces acting on bodies. It is subdivided into statics, which deals with objects in equilibirium, and dynamics, which deals with objects in disequilibrium.

Classical mechanics produces very accurate results within the domain of everyday experience. It is superseded by relativistic mechanics for systems moving at large velocities, quantum mechanics for systems at small distance scales, and relativistic quantum field theory for systems with both properties. Nevertheless, classical mechanics is still very useful, because (i) it is much simpler and easier to apply than these other theories, and (ii) it has a very large range of approximate validity. Classical mechanics can be used to describe the motion of human-sized objects (such as tops and baseballs), many astronomical objects (such as planets and galaxies), and even certain microscopic objects (such as organic molecules.)

Classical mechanics is broadly compatible with other "classical" theories such as classical electrodynamics and thermodynamics.

Description of the theory

We will now introduce the basic concepts of classical mechanics. For simplicity, we only deal with a point particle, which is an object with negligible size. The motion of a point particle is characterized by a small number of parameters: its position, mass, and the forces applied on it. We will discuss each of these parameters 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 properly described by quantum mechanics. Objects with non-zero size have more complicated behavior than our hypothetical point particles, because their internal configuration can change - for example, a baseball can spin while it is moving. However, we will be able to use our results for point particles to study such objects by treating them as composite objects, made up of a large number of interacting point particles. In fact, it can be shown that most of the results for point particles also hold for composite objects, provided their size is small compared to the distance scales of the problem.

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. The velocity, or the rate of change of position with time, is defined as

v = dr/dt

The acceleration, or rate of change of velocity, is

a = dv/dt

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; but generally any change in the velocity, including deceleration, is simply referred to as acceleration.

Forces

Newton's second law relates the mass and velocity of a body to a vector quantity known as the force. Suppose m is the mass of the body and F the vector sum of all applied forces (or net applied force.) Then

F = d (m v) / dt

m is not necessarily independent of t. For example, a rocket ejects propellant, so its mass decreases with time. The quantity m v is called the momentum.

When m is independent of t (as is often the case), the above equation becomes

F = m a

The exact form of F is obtained from independent considerations, and depends on the particular situation. Newton's third law gives a particular prescription for F: if a body A exerts a force F on another body B, then B exerts an equal and opposite reaction force, -F, on A.

Another example of a force is the force of friction, which is generally a function of the velocity of the particle. For example:

F_friction = - λ v

where λ is some constant positive factor. If we have an independent relation for F such as the one above, it can be substituted into Newton's second law to obtain an ordinary differential equation, called the equation of motion. If friction is the only force acting on an object, the equation of motion is

- λ v = m a = m dv/dt

This can be integrated to obtain

v = v₀ exp (- λ t / m)

where v₀ is the initial velocity. This tells us that the velocity of this body decays exponentially to zero. This expression can be further integrated to obtain r.

Other important forces include the gravitational force and the Lorentz force for electromagnetism.

Energy

If a force F is applied to a body that has achieves a displacement δr, the work done by the force is the scalar quantity

δW = F · δr

Suppose the mass of the body is constant, and δW_total is the total work done on the body, obtained by summing the work done by each applied force. From Newton's second law, we can show that

δW_total = δT

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

T = ¹/₂ m |v|²

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

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 V:

F = - ∇ V

Suppose all the forces acting on a body are conservative, and V is the total potential energy, obtained by summing the potential energies corresponding to each force. Then

F · δr = - ∇ V · δr = - δ V

⇒ - δ V = δ T

⇒ δ (T + V) = 0

This is a frequently useful result, known as the conservation of energy: the total energy E = T + V is constant in time.

Further results

Newton's laws provide many important results for composite bodies. See angular momentum.

There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics. They are equivalent to Newton's laws, but are often more useful for solving problems.

History

To Be Done