Analytical mechanics

In mathematical physics, Analytical mechanics is a term used for a refined, mathematical form of classical mechanics, constructed from the 18th century onwards as a formulation of the subject as founded by Isaac Newton and Galileo Galilei. Often the term vectorial mechanics is applied to the form based on Newton's work, to contrast it with analytical mechanics which uses two scalar properties of motion, the kinetic and potential energies, instead of vector forces, to analyze the motion.^[1]

The subject has two principal parts: Lagrangian mechanics and Hamiltonian mechanics.

Formalism

d'Alembert's principle

The foundation which the subject is built on is d'Alembert's principle.

This principle states that infinitesimal virtual work done by a force is zero, which is the work done by a force consistent with the constraints of the system. The idea of a constraint is key - since this limits what the system can do. By analogy with Fermat's principle, which is the variational principle in geometric optics, Maupertuis' principle was discovered in classical mechanics, though from a "divine" conception. Lagrange, Euler, and Hamilton later deduced the theory logically using mathematics.

Generalized coordinates and quantities

Generalized coordinates incorporate constraints on the system. From these, generalized velocities, momenta and forces can be calculated.

Lagrangian and Hamiltonian mechanics

Using generalized coordinates and generalized forces, Lagrange's equations can be obtained. Using the Legendre transformation, the exact calculation of generalized momentum can be determined and so can Hamilton's equations.

The Lagrangian formulation identifies the actual path followed by the motion as a selection of the path over which the time integral of kinetic energy is least, assuming the total energy to be fixed, and imposing no conditions on the time of transit. The Hamiltonian formulation is more general, allowing time-varying energy, identifying the path followed to be the one with stationary action (the integral over the path of the difference between kinetic and potential energies), holding the departure and arrival times fixed.^[1][2] These approaches underlie the path integral formulation of quantum mechanics.

Hamilton's canonical equations provides integral, while Lagrange's equation provides differential equations. Finally we may derive the Hamilton–Jacobi equation.

Hamiltonian-Jacobi mechanics

The study of the solutions of the Hamilton-Jacobi equations leads naturally to the study of symplectic manifolds and symplectic topology.^[3][4] In this formulation, the solutions of the Hamilton–Jacobi equations are the integral curves of Hamiltonian vector fields.

Beyond classical mechanics

Although analytical mechanics was primarily developed to extend the scope of classical mechanics, the concepts lead theoretical physicists to the development of quantum mechanics (and its refinement quantum field theory), and was used in General relativity.

References and notes

1. ^ ^{a b} Cornelius Lanczos (1970). The variational principles of mechanics (4rth Edition ed.). New York: Dover Publications Inc.. Introduction, pp. xxi–xxix. ISBN 0-486-65067-7. http://books.google.com/books?id=ZWoYYr8wk2IC&pg=PR4&dq=isbn=0486650677#PPR21,M1. 
2. ^ The term action has various meanings. This definition is only one, and corresponds specifically to an integral of the system Lagrangian.
3. ^ VI Arnolʹd (1989). Mathematical methods of classical mechanics (2nd Edition ed.). Springer. Chapter 8. ISBN 978-0-387-96890-2. http://books.google.com/books?id=Pd8-s6rOt_cC&printsec=frontcover&dq=isbn=9780387968902#PPT18,M1. 
4. ^ C Doran & A Lasenby (2003). Geometric algebra for physicists. Cambridge University Press. p. §12.3, pp. 432–439. ISBN 978-0-521-71595-9. http://www.worldcat.org/search?q=9780521715959&qt=owc_search. 

See also

• Action (physics)
• Applied mechanics
• Classical mechanics
• Dynamics
• Hamilton–Jacobi equation
• Hamilton's principle
• Kinematics
• Kinetics (physics)

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