Editing Momentum (section)

===Lagrangian mechanics===
In [[Lagrangian mechanics]], a Lagrangian is defined as the difference between the kinetic energy {{mvar|T}} and the [[potential energy]] {{mvar|V}}:

<math display="block"> \mathcal{L}  = T-V\,.</math>

If the generalized coordinates are represented as a vector {{math|'''q''' {{=}} ({{var|q}}{{sub|1}}, {{var|q}}{{sub|2}}, ... , {{var|q}}{{sub|{{var|N}}}}) }} and time differentiation is represented by a dot over the variable, then the equations of motion (known as the Lagrange or [[Euler–Lagrange equation]]s) are a set of {{mvar|N}} equations:<ref>{{harvnb|Goldstein|1980|pp=20–21}}</ref>

<math display="block"> \frac{\text{d}}{\text{d}t}\left(\frac{\partial \mathcal{L} }{\partial\dot{q}_j}\right) - \frac{\partial \mathcal{L}}{\partial q_j} = 0\,.</math>

If a coordinate {{math|{{var|q}}{{sub|{{var|i}}}}}} is not a Cartesian coordinate, the associated generalized momentum component {{math|{{var|p}}{{sub|{{var|i}}}}}} does not necessarily have the dimensions of linear momentum. Even if {{math|{{var|q}}{{sub|{{var|i}}}}}} is a Cartesian coordinate, {{math|{{var|p}}{{sub|{{var|i}}}}}} will not be the same as the mechanical momentum if the potential depends on velocity.<ref name=Goldstein54/> Some sources represent the kinematic momentum by the symbol {{math|'''Π'''}}.<ref name=Lerner>{{cite book|editor-last=Lerner|editor-first=Rita G.|editor-link=Rita G. Lerner|title=Encyclopedia of Physics|date=2005|publisher=Wiley-VCH|location=Weinheim|isbn=978-3-527-40554-1|edition=3rd |editor2-last=Trigg |editor2-first=George L.}}</ref>

In this mathematical framework, a generalized momentum is associated with the generalized coordinates. Its components are defined as

<math display="block"> p_j = \frac{\partial \mathcal{L} }{\partial \dot{q}_j}\,.</math>

Each component {{math|{{var|p}}{{sub|{{var|j}}}}}} is said to be the ''conjugate momentum'' for the coordinate {{math|{{var|q}}{{sub|{{var|j}}}}}}.

Now if a given coordinate {{math|{{var|q}}{{sub|{{var|i}}}}}} does not appear in the Lagrangian (although its time derivative might appear), then {{math|{{var|p}}{{sub|{{var|j}}}}}} is constant. This is the generalization of the conservation of momentum.<ref name=Goldstein54/>

Even if the generalized coordinates are just the ordinary spatial coordinates, the conjugate momenta are not necessarily the ordinary momentum coordinates. An example is found in the section on electromagnetism.