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Classical mechanics

Classical mechanics is a physical theory describing the motion of objects such as projectiles, parts of machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics involved substantial change in the methods and philosophy of physics.[1] The qualifier classical distinguishes this type of mechanics from physics developed after the revolutions in physics of the early 20th century, all of which revealed limitations in classical mechanics.[2]

For the textbooks, see Classical Mechanics (Goldstein) and Classical Mechanics (Kibble and Berkshire).

The earliest formulation of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on the 17th century foundational works of Sir Isaac Newton, and the mathematical methods invented by Gottfried Wilhelm Leibniz, Leonhard Euler and others to describe the motion of bodies under the influence of forces. Later, methods based on energy were developed by Euler, Joseph-Louis Lagrange, William Rowan Hamilton and others, leading to the development of analytical mechanics (which includes Lagrangian mechanics and Hamiltonian mechanics). These advances, made predominantly in the 18th and 19th centuries, extended beyond earlier works; they are, with some modification, used in all areas of modern physics.


If the present state of an object that obeys the laws of classical mechanics is known, it is possible to determine how it will move in the future, and how it has moved in the past. Chaos theory shows that the long term predictions of classical mechanics are not reliable. Classical mechanics provides accurate results when studying objects that are not extremely massive and have speeds not approaching the speed of light. With objects about the size of an atom's diameter, it becomes necessary to use quantum mechanics. To describe velocities approaching the speed of light, special relativity is needed. In cases where objects become extremely massive, general relativity becomes applicable. Some modern sources include relativistic mechanics in classical physics, as representing the field in its most developed and accurate form.

Branches[edit]

Traditional division[edit]

Classical mechanics was traditionally divided into three main branches. Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in equilibrium with its environment.[3] Kinematics describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move.[4][5][3] Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics.[6][7][8] Dynamics goes beyond merely describing objects' behavior and also considers the forces which explain it. Some authors (for example, Taylor (2005)[9] and Greenwood (1997)[10]) include special relativity within classical dynamics.

Forces vs. energy[edit]

Another division is based on the choice of mathematical formalism. Classical mechanics can be mathematically presented in multiple different ways. The physical content of these different formulations is the same, but they provide different insights and facilitate different types of calculations. While the term "Newtonian mechanics" is sometimes used as a synonym for non-relativistic classical physics, it can also refer to a particular formalism based on Newton's laws of motion. Newtonian mechanics in this sense emphasizes force as a vector quantity.[11]


In contrast, analytical mechanics uses scalar properties of motion representing the system as a whole—usually its kinetic energy and potential energy. The equations of motion are derived from the scalar quantity by some underlying principle about the scalar's variation. Two dominant branches of analytical mechanics are Lagrangian mechanics, which uses generalized coordinates and corresponding generalized velocities in configuration space, and Hamiltonian mechanics, which uses coordinates and corresponding momenta in phase space. Both formulations are equivalent by a Legendre transformation on the generalized coordinates, velocities and momenta; therefore, both contain the same information for describing the dynamics of a system. There are other formulations such as Hamilton–Jacobi theory, Routhian mechanics, and Appell's equation of motion. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called the principle of least action. One result is Noether's theorem, a statement which connects conservation laws to their associated symmetries.

By region of application[edit]

Alternatively, a division can be made by region of application:

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 is the same in any inertial reference frame)

F′ = F (the force on a particle is the same in any inertial reference frame)

the is not a constant in classical mechanics, nor does the special position given to the speed of light in relativistic mechanics have a counterpart in classical mechanics.

speed of light

Alonso, M.; Finn, J. (1992). Fundamental University Physics. Addison-Wesley.

(1999). The Feynman Lectures on Physics. Perseus Publishing. ISBN 978-0-7382-0092-7.

Feynman, Richard

Feynman, Richard; Phillips, Richard (1998). Six Easy Pieces. Perseus Publishing.  978-0-201-32841-7.

ISBN

; Charles P. Poole; John L. Safko (2002). Classical Mechanics (3rd ed.). Addison Wesley. ISBN 978-0-201-65702-9.

Goldstein, Herbert

; Berkshire, Frank H. (2004). Classical Mechanics (5th ed.). Imperial College Press. ISBN 978-1-86094-424-6.

Kibble, Tom W.B.

Kleppner, D.; Kolenkow, R.J. (1973). . McGraw-Hill. ISBN 978-0-07-035048-9.

An Introduction to Mechanics

Landau, L.D.; Lifshitz, E.M. (1972). . Franklin Book Company. ISBN 978-0-08-016739-8.

Course of Theoretical Physics, Vol. 1 – Mechanics

Morin, David (2008). (1st ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-87622-3.

Introduction to Classical Mechanics: With Problems and Solutions

; Jack Wisdom (2001). Structure and Interpretation of Classical Mechanics. MIT Press. ISBN 978-0-262-19455-6.

Gerald Jay Sussman

O'Donnell, Peter J. (2015). Essential Dynamics and Relativity. CRC Press.  978-1-4665-8839-4.

ISBN

Thornton, Stephen T.; Marion, Jerry B. (2003). Classical Dynamics of Particles and Systems (5th ed.). Brooks Cole.  978-0-534-40896-1.

ISBN

Crowell, Benjamin. (an introductory text, uses algebra with optional sections involving calculus)

Light and Matter

Fitzpatrick, Richard. (uses calculus)

Classical Mechanics

Hoiland, Paul (2004).

Preferred Frames of Reference & Relativity

Horbatsch, Marko, "".

Classical Mechanics Course Notes

Rosu, Haret C., "". Physics Education. 1999. [arxiv.org : physics/9909035]

Classical Mechanics

Shapiro, Joel A. (2003).

Classical Mechanics

Sussman, Gerald Jay & Wisdom, Jack & Mayer, Meinhard E. (2001).

Structure and Interpretation of Classical Mechanics

Tong, David. (Cambridge lecture notes on Lagrangian and Hamiltonian formalism)

Classical Dynamics


Movies and photos of hundreds of working mechanical-systems models at Cornell University. Also includes an e-book library of classic texts on mechanical design and engineering.

Kinematic Models for Design Digital Library (KMODDL)

Free videos of actual course lectures with links to lecture notes, assignments and exams.

MIT OpenCourseWare 8.01: Classical Mechanics

Alejandro A. Torassa,

On Classical Mechanics