Editing Analytical mechanics (section)

==Motivation==
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The goal of mechanical theory is to solve mechanical problems, such as arise in physics and engineering. Starting from a physical system—such as a mechanism or a star system—a [[mathematical model]] is developed in the form of a differential equation. The model can be solved numerically or analytically to determine the motion of the system.

Newton's vectorial approach to mechanics describes motion with the help of [[vector (mathematics and physics)|vector]] quantities such as [[force]], [[velocity]], [[acceleration]]. These quantities characterise the [[motion]] of a body idealised as a [[mass point geometry|"mass point"]] or a "[[particle]]" understood as a single point to which a mass is attached. Newton's method has been successfully applied to a wide range of physical problems, including the motion of a particle in [[Earth]]'s [[gravitational field]] and the motion of planets around the Sun. In this approach, Newton's laws describe the motion by a differential equation and then the problem is reduced to the solving of that equation.

When a mechanical system contains many particles, however (such as a complex mechanism or a [[fluid]]), Newton's approach is difficult to apply. Using a Newtonian approach is possible, under proper precautions, namely isolating each single particle from the others, and determining all the forces acting on it. Such analysis is cumbersome even in relatively simple systems. Newton thought that [[Newton's third law|his third law]] "action equals reaction" would take care of all complications.{{citation needed|date=February 2023}} This is false even for such simple system as [[rotation]]s of a [[solid body]].{{clarify|date=February 2023}} In more complicated systems, the vectorial approach cannot give an adequate description.

The analytical approach simplifies problems by treating [[mechanical system]]s as ensembles of particles that interact with each other, rather considering each particle as an isolated unit. In the vectorial approach, forces must be determined individually for each particle, whereas in the analytical approach it is enough to know one single function which contains implicitly all the forces acting on and in the system. 
Such simplification is often done using certain kinematic conditions which are stated ''a priori''. However, the analytical treatment does not require the knowledge of these forces and takes these kinematic conditions for granted.{{citation needed|date=February 2023}}

Still, deriving the equations of motion of a complicated mechanical system requires a unifying basis from which they follow.{{clarify|date=February 2023}} This is provided by various [[variational principle]]s: behind each set of equations there is a principle that expresses the meaning of the entire set. Given a fundamental and universal quantity called [[action (physics)|''action'']], the principle that this action be stationary under small variation of some other mechanical quantity generates the required set of differential equations. The statement of the principle does not require any special [[coordinate system]], and all results are expressed in [[generalized coordinates]]. This means that the analytical equations of motion do not change upon a [[coordinate transformation]], an [[invariant (physics)|invariance]] property that is lacking in the vectorial equations of motion.<ref name=Lanczos1>{{cite book |title=The variational principles of mechanics |last=Lanczos |first=Cornelius |pages=3–6 |edition=4th |publisher=Dover Publications Inc. |location= New York |isbn=978-0-486-65067-8 |year=1970 |url=https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PR4}}</ref>

It is not altogether clear what is meant by 'solving' a set of differential equations. A problem is regarded as solved when the particles coordinates at time ''t'' are expressed as simple functions of ''t'' and of parameters defining the initial positions and velocities. However, 'simple function' is not a [[well-defined]] concept: nowadays, a [[function (mathematics)|function]] ''f''(''t'') is not regarded as a formal expression in ''t'' ([[elementary function]]) as in the time of Newton but most generally as a quantity determined by ''t'', and it is not possible to draw a sharp line between 'simple' and 'not simple' functions. If one speaks merely of 'functions', then every mechanical problem is solved as soon as it has been well stated in differential equations, because given the initial conditions and ''t'' determine the coordinates at ''t''. This is a fact especially at present with the modern methods of [[computer simulation|computer modelling]] which provide arithmetical solutions to mechanical problems to any desired degree of accuracy, the [[differential equation]]s being replaced by [[difference equation]]s.

Still, though lacking precise definitions, it is obvious that the [[two-body problem]] has a simple solution, whereas the [[three-body problem]] has not. The two-body problem is solved by formulas involving parameters; their values can be changed to study the class of all solutions, that is, the [[mathematical structure]] of the problem. Moreover, an accurate mental or drawn picture can be made for the motion of two bodies, and it can be as real and accurate as the real bodies moving and interacting. In the three-body problem, parameters can also be assigned specific values; however, the solution at these assigned values or a collection of such solutions does not reveal the mathematical structure of the problem. As in many other problems, the mathematical structure can be elucidated only by examining the differential equations themselves.

Analytical mechanics aims at even more: not at understanding the mathematical structure of a single mechanical problem, but that of a class of problems so wide that they encompass most of mechanics. It concentrates on systems to which Lagrangian or Hamiltonian equations of motion are applicable and that include a very wide range of problems indeed.<ref>{{Cite book |last=Synge |first=J. L. |url=http://link.springer.com/10.1007/978-3-642-45943-6 |title=Principles of Classical Mechanics and Field Theory / Prinzipien der Klassischen Mechanik und Feldtheorie |date=1960 |publisher=Springer Berlin Heidelberg |isbn=978-3-540-02547-4 |editor-last=Flügge |editor-first=S. |series=Encyclopedia of Physics / Handbuch der Physik |volume=2 / 3 / 1 |location=Berlin, Heidelberg |chapter=Classical dynamics |doi=10.1007/978-3-642-45943-6 |oclc=165699220}}</ref>

Development of analytical mechanics has two objectives: (i) increase the range of solvable problems by developing standard techniques with a wide range of applicability, and (ii) understand the mathematical structure of mechanics. In the long run, however, (ii) can help (i) more than a concentration on specific problems for which methods have already been designed.