Editing Generalized coordinates (section)

==Physical quantities in generalized coordinates==

===Kinetic energy===

The total [[kinetic energy]] of the system is the energy of the system's motion, defined as<ref name="Torby 1984 page=269">{{harvnb|Torby|1984|page=269}}</ref>

:<math>T = \frac {1}{2} \sum_{k=1}^N m_k \dot{\mathbf{r}}_k \cdot \dot{\mathbf{r}}_k\,,</math>

in which · is the [[dot product]]. The kinetic energy is a function only of the velocities {{math|'''v'''{{sub|''k''}}}}, not the coordinates {{math|'''r'''{{sub|''k''}}}} themselves. By contrast an important observation is<ref>{{harvnb|Goldstein|Poole|Safko|2002|page=25}}</ref>

:<math>\dot{\mathbf{r}}_k \cdot \dot{\mathbf{r}}_k = \sum_{i,j=1}^n \left(\frac{\partial \mathbf{r}_k}{\partial q_i}\cdot\frac{\partial \mathbf{r}_k}{\partial q_j}\right)\dot{q}_i\dot{q}_j , </math>

which illustrates the kinetic energy is in general a function of the generalized velocities, coordinates, and time if the constraints also vary with time, so {{math|1=''T'' = ''T''('''q''', ''d'''''q'''/''dt'', ''t'')}}.

In the case the constraints on the particles are time-independent, then all partial derivatives with respect to time are zero, and the kinetic energy is a [[homogeneous function]] of degree 2 in the generalized velocities.

Still for the time-independent case, this expression is equivalent to taking the [[line element]] squared of the trajectory for particle {{mvar|k}},

:<math>ds_k^2 = d\mathbf{r}_k\cdot d\mathbf{r}_k = \sum_{i,j=1}^n \left(\frac{\partial \mathbf{r}_k}{\partial q_i}\cdot\frac{\partial \mathbf{r}_k}{\partial q_j}\right) dq_i dq_j \,,</math>

and dividing by the square differential in time, {{math|''dt''{{sup|2}}}}, to obtain the velocity squared of particle {{mvar|k}}. Thus for time-independent constraints it is sufficient to know the line element to quickly obtain the kinetic energy of particles and hence the [[Lagrangian mechanics|Lagrangian]].<ref>{{harvnb|Landau|Lifshitz|edition=3rd|1976|page=8}}</ref>

It is instructive to see the various cases of polar coordinates in 2D and 3D, owing to their frequent appearance.  In 2D [[polar coordinates]] {{math|(''r'', ''θ'')}},

:<math>\left(\frac{ds}{dt}\right)^2 = \dot{r}^2 + r^2\dot{\theta}^2 \,,</math>

in 3D [[cylindrical coordinates]] {{math|(''r'', ''θ'', ''z'')}},

:<math>\left(\frac{ds}{dt}\right)^2 = \dot{r}^2 + r^2\dot{\theta}^2 + \dot{z}^2 \,,</math>

in 3D [[spherical coordinates]] {{math|(''r'', ''θ'', ''φ'')}},

:<math>\left(\frac{ds}{dt}\right)^2 = \dot{r}^2+r^2\dot{\theta}^2 +r^2\sin^2\theta \, \dot{\varphi}^2 \,.</math>

===Generalized momentum===

The ''generalized momentum'' "[[canonical coordinates|canonically conjugate]] to" the coordinate {{mvar|q{{sub|i}}}} is defined by

:<math>p_i =\frac{\partial L}{\partial\dot q_i}.</math>

If the Lagrangian {{mvar|L}} does ''not'' depend on some coordinate {{mvar|q{{sub|i}}}}, then it follows from the Euler–Lagrange equations that the corresponding generalized momentum will be a [[conserved quantity]], because the time derivative is zero implying the momentum is a constant of the motion;

:<math> \frac{\partial L}{\partial q_i} =\frac{d}{dt}\frac{\partial L}{\partial\dot q_i} =\dot{p}_i =0\,.</math>