Editing Analytical mechanics

{{short description|Overview of mechanics based on the least action principle}}
{{Classical mechanics|cTopic=Formulations}}

In [[theoretical physics]] and [[mathematical physics]], '''analytical mechanics''', or '''theoretical mechanics''' is a collection of closely related formulations of [[classical mechanics]]. Analytical mechanics uses ''[[Scalar (physics)|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 [[calculus of variations|variation]]. 

Analytical mechanics was developed by many scientists and mathematicians during the 18th century and onward, after [[Newtonian mechanics]]. Newtonian mechanics considers [[Euclidean vector|vector]] quantities of motion, particularly [[acceleration]]s, [[Momentum|momenta]], [[force]]s, of the constituents of the system; it can also be called ''vectorial mechanics''.<ref name=Lanczos>{{cite book |title=The variational principles of mechanics |last=Lanczos |first=Cornelius |page=Introduction, pp. xxi–xxix |edition=4th |publisher=Dover Publications Inc. |location= New York |isbn=0-486-65067-7 |year=1970 |url=https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PR4 |no-pp=true}}</ref> A scalar is a quantity, whereas a vector is represented by quantity and direction. The results of these two different approaches are equivalent, but the analytical mechanics approach has many advantages for complex problems.

Analytical mechanics takes advantage of a system's ''constraints'' to solve problems. The constraints limit the [[Degrees of freedom (physics and chemistry)|degrees of freedom]] the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as [[generalized coordinates]]. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics allows numerous mechanical problems to be solved with greater efficiency than fully vectorial methods. It does not always work for non-[[conservative force]]s or dissipative forces like [[friction]], in which case one may revert to Newtonian mechanics.

Two dominant branches of analytical mechanics are [[Lagrangian mechanics]] (using generalized coordinates and corresponding generalized velocities in [[Configuration space (physics)|configuration space]]) and [[Hamiltonian mechanics]] (using coordinates and corresponding momenta in [[phase space]]). Both formulations are equivalent by a [[Legendre transformation#Hamilton–Lagrange mechanics|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 law]]s to their associated [[Symmetry (physics)|symmetries]].

Analytical mechanics does not introduce new physics and is not more general than Newtonian mechanics. Rather it is a collection of equivalent formalisms which have broad application. In fact the same principles and formalisms can be used in [[relativistic mechanics]] and [[general relativity]], and with some modifications, [[quantum mechanics]] and [[quantum field theory]].

Analytical mechanics is used widely, from fundamental physics to [[applied mathematics]], particularly [[chaos theory]].

The methods of analytical mechanics apply to discrete particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom. The definitions and equations have a close analogy with those of mechanics.

==Motivation==
{{Multiple issues|section=yes|
{{Overly detailed|section|date=February 2023}}
{{More citations needed|section|date=February 2023}}
}}

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.

==Intrinsic motion==
===Generalized coordinates and constraints===
{{Main | Generalized coordinates}}
In [[Newtonian mechanics]], one customarily uses all three [[Cartesian coordinates]], or other 3D [[coordinate system]], to refer to a body's [[position (vector)|position]] during its motion. In physical systems, however, some structure or other system usually constrains the body's motion from taking certain directions and pathways. So a full set of Cartesian coordinates is often unneeded, as the constraints determine the evolving relations among the coordinates, which relations can be modeled by equations corresponding to the constraints. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motion's geometry, reducing the number of coordinates to the minimum needed to model the motion. These are known as ''generalized coordinates'', denoted ''q<sub>i</sub>'' (''i'' = 1, 2, 3...).<ref>Kibble, Tom, and Berkshire, Frank H. "Classical Mechanics" (5th Edition). Singapore, World Scientific Publishing Company, 2004.</ref>{{rp|231}}

===Difference between [[Curvilinear coordinates|curvillinear]] and [[generalized coordinates]]===
Generalized coordinates incorporate constraints on the system. There is one generalized coordinate ''q<sub>i</sub>'' for each [[Degrees of freedom (physics and chemistry)|degree of freedom]] (for convenience labelled by an index ''i'' = 1, 2...''N''), i.e. each way the system can change its [[Configuration space (physics)|configuration]]; as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates. The number of ''curvilinear'' coordinates equals the [[dimension]] of the position space in question (usually 3 for 3d space), while the number of ''generalized'' coordinates is not necessarily equal to this dimension; constraints can reduce the number of degrees of freedom (hence the number of generalized coordinates required to define the configuration of the system), following the general rule:<ref name="autogenerated1">''Analytical Mechanics'', L.N. Hand, J.D. Finch, Cambridge University Press, 2008, {{ISBN|978-0-521-57572-0}}</ref>{{dubious|date=January 2024}}

{{block indent | em = 1.5 | text = ''['''dimension of position space''' (usually 3)] × [number of '''constituents''' of system ("particles")] − (number of '''constraints''')''}}
{{block indent | em = 1.5 | text = ''= (number of '''degrees of freedom''') = (number of '''generalized coordinates''')''}}

For a system with ''N'' degrees of freedom, the generalized coordinates can be collected into an ''N''-[[tuple]]:
<math display="block">\mathbf{q} = (q_1, q_2, \dots, q_N) </math>
and the [[time derivative]] (here denoted by an overdot) of this tuple give the ''generalized velocities'':
<math display="block">\frac{d\mathbf{q}}{dt} = \left(\frac{dq_1}{dt}, \frac{dq_2}{dt}, \dots, \frac{dq_N}{dt}\right) \equiv \mathbf{\dot{q}} = (\dot{q}_1, \dot{q}_2, \dots, \dot{q}_N) .</math>

===D'Alembert's principle of virtual work===
{{main |  D'Alembert's principle}}
D'Alembert's principle states that infinitesimal ''[[virtual work]]'' done by a force across reversible displacements is zero, which is the work done by a force consistent with ideal constraints of the system. The idea of a constraint is useful – since this limits what the system can do, and can provide steps to solving for the motion of the system. The equation for D'Alembert's principle is:<ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref>{{rp|265}}
<math display="block">\delta W = \boldsymbol{\mathcal{Q}} \cdot \delta\mathbf{q} = 0 \,,</math>
where <math display="block">\boldsymbol\mathcal{Q} = (\mathcal{Q}_1, \mathcal{Q}_2, \dots, \mathcal{Q}_N)</math>
are the [[generalized forces]] (script Q instead of ordinary Q is used here to prevent conflict with canonical transformations below) and {{math|'''q'''}} are the generalized coordinates. This leads to the generalized form of [[Newton's laws]] in the language of analytical mechanics:
<math display="block">\boldsymbol\mathcal{Q} = \frac{d}{dt} \left ( \frac {\partial T}{\partial \mathbf{\dot{q}}} \right ) - \frac {\partial T}{\partial \mathbf{q}}\,,</math>

where ''T'' is the total [[kinetic energy]] of the system, and the notation
<math display="block">\frac {\partial}{\partial \mathbf{q}} = \left(\frac{\partial }{\partial q_1}, \frac{\partial }{\partial q_2}, \dots, \frac{\partial }{\partial q_N}\right)</math>
is a useful shorthand (see [[matrix calculus#Scalar-by-vector|matrix calculus]] for this notation).

===Constraints===
{{Main| Holonomic constraints | Scleronomous | Rheonomous }}
If the curvilinear coordinate system is defined by the standard [[position vector]] {{math|'''r'''}}, and if the position vector can be written in terms of the generalized coordinates {{math|'''q'''}} and time {{mvar|t}} in the form: <math display="block">\mathbf{r} = \mathbf{r}(\mathbf{q}(t),t)</math> and this relation holds for all times {{mvar|t}}, then {{math|'''q'''}} are called ''holonomic constraints''.<ref>McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, {{ISBN|0-07-051400-3}}</ref> Vector {{math|'''r'''}} is explicitly dependent on {{mvar|''t''}} in cases when the constraints vary with time, not just because of {{math|'''q'''(''t'')}}. For time-independent situations, the constraints are also called '''[[Scleronomous|scleronomic]]''', for time-dependent cases they are called '''[[Rheonomous|rheonomic]]'''.<ref name="autogenerated1"/>

==Lagrangian mechanics==
{{Main | Lagrangian mechanics | Euler–Lagrange equations}}

The introduction of generalized coordinates and the fundamental Lagrangian function:

:<math>L(\mathbf{q},\mathbf{\dot{q}},t) = T(\mathbf{q},\mathbf{\dot{q}},t) - V(\mathbf{q},\mathbf{\dot{q}},t)</math>

where ''T'' is the total [[kinetic energy]] and ''V'' is the total [[potential energy]] of the entire system, then either following the [[calculus of variations]] or using the above formula – lead to the [[Euler–Lagrange equations]];

:<math>\frac{d}{dt}\left(\frac{\partial L}{\partial \mathbf{\dot{q}}}\right) = \frac{\partial L}{\partial \mathbf{q}} \,,</math>

which are a set of ''N'' second-order [[ordinary differential equation]]s, one for each ''q<sub>i</sub>''(''t'').

This 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 Lagrangian formulation uses the '''[[Configuration space (physics)|configuration space]]''' of the system, the [[set (mathematics)|set]] of all possible generalized coordinates:

:<math>\mathcal{C} = \{ \mathbf{q} \in \mathbb{R}^N \}\,,</math>

where <math>\mathbb{R}^N</math> is ''N''-dimensional [[real number|real]] space (see also [[set-builder notation]]). The particular solution to the Euler–Lagrange equations is called a ''(configuration) path or trajectory'', i.e. one particular '''q'''(''t'') subject to the required [[initial conditions]]. The general solutions form a set of possible configurations as functions of time:

:<math>\{ \mathbf{q}(t) \in \mathbb{R}^N \,:\,t\ge 0,t\in \mathbb{R}\}\subseteq\mathcal{C}\,,</math>

The configuration space can be defined more generally, and indeed more deeply, in terms of [[topology|topological]] [[manifold]]s and the [[tangent bundle]].

==Hamiltonian mechanics==
{{main | Hamiltonian mechanics }}

The [[Legendre transformation]] of the Lagrangian replaces the generalized coordinates and velocities ('''q''', '''q̇''') with ('''q''', '''p'''); the generalized coordinates and the ''[[Canonical coordinates|generalized momenta]]'' conjugate to the generalized coordinates:

:<math>\mathbf{p} = \frac{\partial L}{\partial \mathbf{\dot{q}}} = \left(\frac{\partial L}{\partial \dot{q}_1},\frac{\partial L}{\partial \dot{q}_2},\cdots \frac{\partial L}{\partial \dot{q}_N}\right) = (p_1, p_2\cdots p_N)\,,</math>

and introduces the Hamiltonian (which is in terms of generalized coordinates and momenta):

:<math>H(\mathbf{q},\mathbf{p},t) = \mathbf{p}\cdot\mathbf{\dot{q}} - L(\mathbf{q},\mathbf{\dot{q}},t)</math>

where <math>\cdot</math> denotes the [[dot product]], also leading to [[Hamiltonian mechanics|Hamilton's equations]]:

:<math>\mathbf{\dot{p}} = - \frac{\partial H}{\partial \mathbf{q}}\,,\quad \mathbf{\dot{q}} = + \frac{\partial H}{\partial \mathbf{p}} \,,</math>

which are now a set of 2''N'' first-order ordinary differential equations, one for each ''q<sub>i</sub>''(''t'') and ''p<sub>i</sub>''(''t''). Another result from the Legendre transformation relates the time derivatives of the Lagrangian and Hamiltonian:

:<math>\frac{dH}{dt}=-\frac{\partial L}{\partial t}\,,</math>

which is often considered one of Hamilton's equations of motion additionally to the others. The generalized momenta can be written in terms of the generalized forces in the same way as Newton's second law:

:<math>\mathbf{\dot{p}} = \boldsymbol{\mathcal{Q}}\,.</math>

Analogous to the configuration space, the set of all momenta is the '''generalized [[momentum space]]''':

:<math>\mathcal{M} = \{ \mathbf{p}\in\mathbb{R}^N \}\,.</math>

("Momentum space" also refers to "'''k'''-space"; the set of all [[wave vector]]s (given by [[De Broglie relation]]s) as used in quantum mechanics and theory of [[wave]]s)

The set of all positions and momenta form the '''[[phase space]]''':

:<math>\mathcal{P} = \mathcal{C}\times\mathcal{M} = \{ (\mathbf{q},\mathbf{p})\in\mathbb{R}^{2N} \} \,,</math>

that is, the [[Cartesian product]] of the configuration space and generalized momentum space.

A particular solution to Hamilton's equations is called a ''[[Phase portrait|phase path]]'', a particular curve ('''q'''(''t''),'''p'''(''t'')) subject to the required initial conditions. The set of all phase paths, the general solution to the differential equations, is the ''[[phase portrait]]'':

:<math>\{ (\mathbf{q}(t),\mathbf{p}(t))\in\mathbb{R}^{2N}\,:\,t\ge0, t\in\mathbb{R} \} \subseteq \mathcal{P}\,,</math>

===The Poisson bracket===
{{main |Poisson bracket}}

All dynamical variables can be derived from position '''q''', momentum '''p''', and time ''t'', and written as a function of these: ''A'' = ''A''('''q''', '''p''', ''t''). If ''A''('''q''', '''p''', ''t'') and ''B''('''q''', '''p''', ''t'') are two scalar valued dynamical variables, the ''Poisson bracket'' is defined by the generalized coordinates and momenta:

:<math>
\begin{align}
\{A,B\}  \equiv \{A,B\}_{\mathbf{q},\mathbf{p}} & = \frac{\partial A}{\partial \mathbf{q}}\cdot\frac{\partial B}{\partial \mathbf{p}} - \frac{\partial A}{\partial \mathbf{p}}\cdot\frac{\partial B}{\partial \mathbf{q}}\\
& \equiv \sum_k \frac{\partial A}{\partial q_k}\frac{\partial B}{\partial p_k} - \frac{\partial A}{\partial p_k}\frac{\partial B}{\partial q_k}\,,
\end{align}</math>

Calculating the [[total derivative]] of one of these, say ''A'', and substituting Hamilton's equations into the result leads to the time evolution of ''A'':

:<math> \frac{dA}{dt} = \{A,H\} + \frac{\partial A}{\partial t}\,. </math>

This equation in ''A'' is closely related to the equation of motion in the [[Heisenberg picture]] of [[quantum mechanics]], in which classical dynamical variables become [[operator (physics)|quantum operators]] (indicated by hats (^)), and the Poisson bracket is replaced by the [[commutator]] of operators via Dirac's [[canonical quantization]]:

:<math>\{A,B\} \rightarrow \frac{1}{i\hbar}[\hat{A},\hat{B}]\,.</math>

==Properties of the Lagrangian and the Hamiltonian==
Following are overlapping properties between the Lagrangian and Hamiltonian functions.<ref name="autogenerated1"/><ref>''Classical Mechanics'', T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, {{ISBN|0-07-084018-0}}</ref>
* All the individual generalized coordinates ''q<sub>i</sub>''(''t''), velocities ''q̇<sub>i</sub>''(''t'') and momenta ''p<sub>i</sub>''(''t'') for every degree of freedom are mutually independent. Explicit time-dependence of a function means the function actually includes time ''t'' as a variable in addition to the '''q'''(''t''), '''p'''(''t''), not simply as a parameter through '''q'''(''t'') and '''p'''(''t''), which would mean explicit time-independence.
* The Lagrangian is invariant under addition of the ''[[total derivative|total]]'' [[time derivative]] of any function of '''q''' and ''t'', that is: <math display="block">L' = L +\frac{d}{dt}F(\mathbf{q},t) \,,</math> so each Lagrangian ''L'' and ''L''' describe ''exactly the same motion''. In other words, the Lagrangian of a system is not unique.
* Analogously, the Hamiltonian is invariant under addition of the ''[[partial derivative|partial]]'' time derivative of any function of '''q''', '''p''' and ''t'', that is: <math display="block">K = H + \frac{\partial}{\partial t}G(\mathbf{q},\mathbf{p},t) \,,</math> (''K'' is a frequently used letter in this case). This property is used in [[canonical transformations]] (see below).
*If the Lagrangian is independent of some generalized coordinates, then the generalized momenta conjugate to those coordinates are [[Constant of motion|constants of the motion]], i.e. are [[conserved quantity|conserved]], this immediately follows from Lagrange's equations: <math display="block">\frac{\partial L}{\partial q_j }=0\,\rightarrow \,\frac{dp_j}{dt} = \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_j}=0 </math> Such coordinates are "[[Lagrangian mechanics|cyclic]]" or "ignorable". It can be shown that the Hamiltonian is also cyclic in exactly the same generalized coordinates.
*If the Lagrangian is time-independent the Hamiltonian is also time-independent (i.e. both are constant in time).
*If the kinetic energy is a [[homogeneous function]] of degree 2 of the generalized velocities, ''and'' the Lagrangian is explicitly time-independent, then: <math display="block">T((\lambda \dot{q}_i)^2, (\lambda \dot{q}_j \lambda \dot{q}_k), \mathbf{q}) = \lambda^2 T((\dot{q}_i)^2, \dot{q}_j\dot{q}_k, \mathbf{q})\,,\quad L(\mathbf{q},\mathbf{\dot{q}})\,,</math> where ''λ'' is a constant, then the Hamiltonian will be the ''total conserved energy'', equal to the total kinetic and potential energies of the system: <math display="block">H = T + V = E\,.</math> This is the basis for the [[Schrödinger equation]], inserting [[operators (physics)|quantum operators]] directly obtains it.

==Principle of least action==
[[File:Least action principle.svg|250px|thumb|As the system evolves, '''q''' traces a path through [[configuration space (physics)|configuration space]] (only some are shown). The path taken by the system (red) has a stationary action (δ''S'' = 0) under small changes in the configuration of the system (δ'''q''').<ref>{{cite book |last=Penrose |first=R.| title=The Road to Reality| publisher= Vintage books| year=2007 | page = 474|isbn=978-0-679-77631-4|title-link=The Road to Reality}}</ref>]]

[[Action (physics)|Action]] is another quantity in analytical mechanics defined as a [[Functional (mathematics)|functional]] of the Lagrangian:

:<math>\mathcal{S} = \int_{t_1}^{t_2} L(\mathbf{q},\mathbf{\dot{q}},t) dt \,.</math>

A general way to find the equations of motion from the action is the ''[[principle of least action]]'':<ref>Encyclopaedia of Physics (2nd Edition), [[Rita G. Lerner|R.G. Lerner]], G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3</ref>

:<math>\delta\mathcal{S} = \delta\int_{t_1}^{t_2} L(\mathbf{q},\mathbf{\dot{q}},t) dt = 0\,,</math>

where the departure ''t''<sub>1</sub> and arrival ''t''<sub>2</sub> times are fixed.<ref name=Lanczos/> The term "path" or "trajectory" refers to the [[time evolution]] of the system as a path through configuration space <math>\mathcal{C}</math>, in other words '''q'''(''t'') tracing out a path in <math>\mathcal{C}</math>. The path for which action is least is the path taken by the system.

From this principle, ''all'' [[equations of motion]] in classical mechanics can be derived. This approach can be extended to fields rather than a system of particles (see below), and underlies the [[path integral formulation]] of [[quantum mechanics]],<ref name="autogenerated2004">''Quantum Mechanics'', E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, {{ISBN|978-0-13-146100-0}}</ref><ref name="autogenerated3">Quantum Field Theory, D. McMahon, Mc Graw Hill (US), 2008, {{ISBN|978-0-07-154382-8}}</ref> and is used for calculating [[geodesic]] motion in [[general relativity]].<ref>''Relativity, Gravitation, and Cosmology'', R.J.A. Lambourne, Open University, Cambridge University Press, 2010, {{ISBN|978-0-521-13138-4}}</ref>

==Hamiltonian-Jacobi mechanics==
;[[Canonical transformations]]

The invariance of the Hamiltonian (under addition of the partial time derivative of an arbitrary function of '''p''', '''q''', and ''t'') allows the Hamiltonian in one set of coordinates '''q''' and momenta '''p''' to be transformed into a new set '''Q''' = '''Q'''('''q''', '''p''', ''t'') and '''P''' = '''P'''('''q''', '''p''', ''t''), in four possible ways:

:<math>\begin{align}
& K(\mathbf{Q},\mathbf{P},t) = H(\mathbf{q},\mathbf{p},t) + \frac{\partial }{\partial t}G_1 (\mathbf{q},\mathbf{Q},t)\\
& K(\mathbf{Q},\mathbf{P},t) = H(\mathbf{q},\mathbf{p},t) + \frac{\partial }{\partial t}G_2 (\mathbf{q},\mathbf{P},t)\\
& K(\mathbf{Q},\mathbf{P},t) = H(\mathbf{q},\mathbf{p},t) + \frac{\partial }{\partial t}G_3 (\mathbf{p},\mathbf{Q},t)\\
& K(\mathbf{Q},\mathbf{P},t) = H(\mathbf{q},\mathbf{p},t) + \frac{\partial }{\partial t}G_4 (\mathbf{p},\mathbf{P},t)\\
\end{align}</math>

With the restriction on '''P''' and '''Q''' such that the transformed Hamiltonian system is:

:<math>\mathbf{\dot{P}} = - \frac{\partial K}{\partial \mathbf{Q}}\,,\quad \mathbf{\dot{Q}} = + \frac{\partial K}{\partial \mathbf{P}} \,,</math>

the above transformations are called ''canonical transformations'', each function ''G<sub>n</sub>'' is called a [[Generating function (physics)|generating function]] of the "''n''th kind" or "type-''n''". The transformation of coordinates and momenta can allow simplification for solving Hamilton's equations for a given problem.

The choice of '''Q''' and '''P''' is completely arbitrary, but not every choice leads to a canonical transformation. One simple criterion for a transformation '''q''' → '''Q''' and '''p''' → '''P''' to be canonical is the Poisson bracket be unity,

:<math>\{Q_i,P_i\} = 1</math>

for all ''i'' = 1, 2,...''N''. If this does not hold then the transformation is not canonical.<ref name="autogenerated1"/>

;The [[Hamilton–Jacobi equation]]

By setting the canonically transformed Hamiltonian ''K'' = 0, and the type-2 generating function equal to '''Hamilton's principal function''' (also the action <math>\mathcal{S}</math>) plus an arbitrary constant ''C'':

:<math>G_2(\mathbf{q},t) = \mathcal{S}(\mathbf{q},t) + C\,,</math>

the generalized momenta become:

:<math>\mathbf{p} = \frac{\partial\mathcal{S}}{\partial \mathbf{q}}</math>

and '''P''' is constant, then the Hamiltonian-Jacobi equation (HJE) can be derived from the type-2 canonical transformation:

:<math>H = - \frac{\partial\mathcal{S}}{\partial t}</math>

where ''H'' is the Hamiltonian as before:

:<math>H = H(\mathbf{q},\mathbf{p},t) = H\left(\mathbf{q},\frac{\partial\mathcal{S}}{\partial \mathbf{q}},t\right)</math>

Another related function is '''Hamilton's characteristic function'''

:<math>W(\mathbf{q})=\mathcal{S}(\mathbf{q},t) + Et </math>

used to solve the HJE by [[separation of variables|additive separation of variables]] for a time-independent Hamiltonian ''H''.

The study of the solutions of the Hamilton–Jacobi equations leads naturally to the study of [[symplectic manifold]]s and [[symplectic topology]].<ref name=Arnold>{{cite book |title=Mathematical methods of classical mechanics |last=Arnolʹd |first=VI |year=1989 |publisher=Springer |edition=2nd |page= Chapter 8 |isbn=978-0-387-96890-2 |url=https://books.google.com/books?id=Pd8-s6rOt_cC |no-pp=true}}</ref><ref name=Doran>{{cite book |title=Geometric algebra for physicists |last1=Doran |first1=C |last2=Lasenby |first2=A |publisher=Cambridge University Press |page=§12.3, pp. 432–439 |isbn=978-0-521-71595-9 |year=2003 |url=http://www.worldcat.org/search?q=9780521715959&qt=owc_search}}</ref> In this formulation, the solutions of the Hamilton–Jacobi equations are the [[integral curve]]s of [[Hamiltonian vector field]]s.

==Routhian mechanics==
{{main |Routhian mechanics}}
Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, not often used but especially useful for removing cyclic coordinates.{{cn|date=January 2024}} If the Lagrangian of a system has ''s'' cyclic coordinates '''q''' = ''q''<sub>1</sub>, ''q''<sub>2</sub>, ... ''q<sub>s</sub>'' with conjugate momenta '''p''' = ''p''<sub>1</sub>, ''p''<sub>2</sub>, ... ''p<sub>s</sub>'', with the rest of the coordinates non-cyclic and denoted '''ζ''' = ''ζ''<sub>1</sub>, ''ζ''<sub>1</sub>, ..., ''ζ<sub>N − s</sub>'', they can be removed by introducing the ''Routhian'':

:<math>R=\mathbf{p}\cdot\mathbf{\dot{q}} - L(\mathbf{q}, \mathbf{p}, \boldsymbol{\zeta}, \dot{\boldsymbol{\zeta}})\,,</math>

which leads to a set of 2''s'' Hamiltonian equations for the cyclic coordinates '''q''',

:<math>\dot{\mathbf{q}} = +\frac{\partial R}{\partial \mathbf{p}}\,,\quad \dot{\mathbf{p}} = -\frac{\partial R}{\partial \mathbf{q}}\,,</math>

and ''N'' − ''s'' Lagrangian equations in the non cyclic coordinates '''ζ'''.

:<math>\frac{d}{dt}\frac{\partial R }{\partial\dot{\boldsymbol{\zeta}}} = \frac{\partial R}{\partial \boldsymbol{\zeta}}\,.</math>

Set up in this way, although the Routhian has the form of the Hamiltonian, it can be thought of a Lagrangian with ''N'' − ''s'' degrees of freedom.

The coordinates '''q''' do not have to be cyclic, the partition between which coordinates enter the Hamiltonian equations and those which enter the Lagrangian equations is arbitrary. It is simply convenient to let the Hamiltonian equations remove the cyclic coordinates, leaving the non cyclic coordinates to the Lagrangian equations of motion.

==Appellian mechanics==
{{Main | Appell's equation of motion}}
[[Appell's equation of motion]] involve generalized accelerations, the second time derivatives of the generalized coordinates:

:<math>\alpha_r = \ddot{q}_r = \frac{d^2 q_r}{dt^2}\,,</math>

as well as generalized forces mentioned above in D'Alembert's principle. The equations are

:<math>\mathcal{Q}_{r} = \frac{\partial S}{\partial \alpha_{r}}\,, \quad S = \frac{1}{2} \sum_{k=1}^{N} m_{k} \mathbf{a}_{k}^{2}\,,</math>

where

:<math>\mathbf{a}_k = \ddot{\mathbf{r}}_k = \frac{d^2 \mathbf{r}_k}{dt^2}</math>

is the acceleration of the ''k'' particle, the second time derivative of its position vector. Each acceleration '''a'''<sub>''k''</sub> is expressed in terms of the generalized accelerations ''α<sub>r</sub>'', likewise each '''r'''<sub>k</sub> are expressed in terms the generalized coordinates ''q<sub>r</sub>''.

==Classical field theory==
===[[Lagrangian field theory]]===

Generalized coordinates apply to discrete particles. For ''N'' [[scalar field]]s ''φ<sub>i</sub>''('''r''', ''t'') where ''i'' = 1, 2, ... ''N'', the '''[[Lagrangian density]]''' is a function of these fields and their space and time derivatives, and possibly the space and time coordinates themselves:
<math display="block">\mathcal{L} = \mathcal{L}(\phi_1, \phi_2, \dots, \nabla\phi_1, \nabla\phi_2, \dots, \partial_t \phi_1, \partial_t \phi_2, \ldots, \mathbf{r}, t)\,.</math>
and the Euler–Lagrange equations have an analogue for fields:
<math display="block">\partial_\mu \left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi_i)}\right) = \frac{\partial \mathcal{L}}{\partial \phi_i}\,,</math>
where ''∂<sub>μ</sub>'' denotes the [[4-gradient]] and the [[summation convention]] has been used. For ''N'' scalar fields, these Lagrangian field equations are a set of ''N'' second order partial differential equations in the fields, which in general will be coupled and nonlinear.

This scalar field formulation can be extended to [[vector field]]s, [[tensor field]]s, and [[spinor field]]s.

The Lagrangian is the [[volume integral]] of the Lagrangian density:<ref name="autogenerated3"/><ref>Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, {{ISBN|0-7167-0344-0}}</ref>
<math display="block">L = \int_\mathcal{V} \mathcal{L} \, dV \,.</math>

Originally developed for classical fields, the above formulation is applicable to all physical fields in classical, quantum, and relativistic situations: such as [[Newton's law of universal gravitation|Newtonian gravity]], [[classical electromagnetism]], [[general relativity]], and [[quantum field theory]]. It is a question of determining the correct Lagrangian density to generate the correct field equation.

===[[Hamiltonian field theory]]===

The corresponding "momentum" field densities conjugate to the ''N'' scalar fields ''φ<sub>i</sub>''('''r''', ''t'') are:<ref name="autogenerated3"/>
<math display="block">\pi_i(\mathbf{r},t) = \frac{\partial \mathcal{L}}{\partial \dot{\phi}_i}\,\quad\dot{\phi}_i\equiv \frac{\partial \phi_i}{\partial t}</math>
where in this context the overdot denotes a partial time derivative, not a total time derivative. The '''Hamiltonian density''' <math>\mathcal{H}</math> is defined by analogy with mechanics:
<math display="block">\mathcal{H}(\phi_1, \phi_2,\ldots, \pi_1, \pi_2, \ldots,\mathbf{r},t) = \sum_{i=1}^N \dot{\phi}_i(\mathbf{r},t)\pi_i(\mathbf{r},t) - \mathcal{L}\,.</math>

The equations of motion are:
<math display="block">\dot{\phi}_i = +\frac{\delta\mathcal{H}}{\delta \pi_i}\,,\quad \dot{\pi}_i = - \frac{\delta\mathcal{H}}{\delta \phi_i} \,, </math>
where the [[variational derivative]]
<math display="block">\frac{\delta}{\delta \phi_i} = \frac{\partial}{\partial \phi_i} - \partial_\mu \frac{\partial }{\partial (\partial_\mu \phi_i)} </math>
must be used instead of merely partial derivatives. For ''N'' fields, these Hamiltonian field equations are a set of 2''N'' first order partial differential equations, which in general will be coupled and nonlinear.

Again, the volume integral of the Hamiltonian density is the Hamiltonian
<math display="block">H = \int_\mathcal{V} \mathcal{H} \, dV \,.</math>

==Symmetry, conservation, and Noether's theorem==
;[[symmetry (physics)|Symmetry transformations]] in classical space and time

Each transformation can be described by an operator (i.e. function acting on the position '''r''' or momentum '''p''' variables to change them). The following are the cases when the operator does not change '''r''' or '''p''', i.e. symmetries.<ref name="autogenerated2004"/>
{| class="wikitable"
|-
! Transformation 
! Operator
! Position
! Momentum
|-
| [[Translational symmetry]] 
| <math>X(\mathbf{a})</math>
| <math>\mathbf{r}\rightarrow \mathbf{r} + \mathbf{a}</math>
| <math>\mathbf{p}\rightarrow \mathbf{p}</math>
|-
| [[Time translation]]
| <math>U(t_0)</math>
| <math>\mathbf{r}(t)\rightarrow \mathbf{r}(t+t_0)</math>
| <math>\mathbf{p}(t)\rightarrow \mathbf{p}(t+t_0)</math>
|-
| [[Rotational invariance]] 
| <math>R(\mathbf{\hat{n}},\theta)</math>
| <math>\mathbf{r}\rightarrow R(\mathbf{\hat{n}},\theta)\mathbf{r}</math>
| <math>\mathbf{p}\rightarrow R(\mathbf{\hat{n}},\theta)\mathbf{p}</math>
|- 
| [[Galilean transformation]]s
| <math>G(\mathbf{v})</math>
| <math>\mathbf{r}\rightarrow \mathbf{r} + \mathbf{v}t</math>
| <math>\mathbf{p}\rightarrow \mathbf{p} + m\mathbf{v}</math>
|-
| [[Parity (physics)|Parity]]
| <math>P</math>
| <math>\mathbf{r}\rightarrow -\mathbf{r}</math>
| <math>\mathbf{p}\rightarrow -\mathbf{p}</math>
|-
| [[T-symmetry]]
| <math>T</math>
| <math>\mathbf{r}\rightarrow \mathbf{r}(-t)</math>
| <math>\mathbf{p}\rightarrow -\mathbf{p}(-t)</math>
|}

where ''R''('''n̂''', θ) is the [[rotation matrix]] about an axis defined by the [[unit vector]] '''n̂''' and angle θ.

;[[Noether's theorem]]

Noether's theorem states that a [[continuous variable|continuous]] symmetry transformation of the action corresponds to a [[Conservation law (physics)|conservation law]], i.e. the action (and hence the Lagrangian) does not change under a transformation parameterized by a [[parameter]] ''s'':
<math display="block">L[q(s,t), \dot{q}(s,t)] = L[q(t), \dot{q}(t)] </math>
the Lagrangian describes the same motion independent of ''s'', which can be length, angle of rotation, or time. The corresponding momenta to ''q'' will be conserved.<ref name="autogenerated1"/>

==See also==
*[[Lagrangian mechanics]]
*[[Hamiltonian mechanics]]
*[[Theoretical mechanics]]
*[[Classical mechanics]]
*[[Hamilton–Jacobi equation]]
*[[Hamilton's principle]]
*[[Kinematics]]
*[[Kinetics (physics)]]
*[[Non-autonomous mechanics]]
*[[Udwadia–Kalaba equation]]{{POV statement|1=Reference to article with NPOV issues|date=December 2019}}

==References and notes==
<references/>
{{commons category}}
{{Physics-footer}}
{{Industrial and applied mathematics}}

{{Authority control}}

{{DEFAULTSORT:Analytical Mechanics}}
[[Category:Mathematical physics]]
[[Category:Dynamical systems]]