Editing Analytical mechanics (section)

==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"/>