Euler-Lagrange equations

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Given a system described by one $n\,$ generalized coordinates $q_i\,$ , their velocities $\dot{q}_i\,$ , along with purely holonomic constraints, d'Alembert's principle yields $n\,$ equations of motion

$\frac{d}{dt} \left( \frac{\partial T}{\partial \dot q_j}\right) - \frac{\partial T}{\partial q_j} - Q_j = 0\,$ ,

where $T = \textstyle\sum_i \frac{1}{2}m_i v_i^2\,$ is the kinetic energy and

$Q_j = \sum_i\mathbf{F}_i \cdot\frac{\partial \mathbf{r}_i }{\partial q_j}\,$ .

is the generalized force corresponding to $q_j\,$ . For a conservative system, the forces $\mathbf{F}_i\,$ may be written in terms of a potential function $V(\mathbf{r}_1, \mathbf{r}_2, ...)\,$ , such that

$\mathbf{F}_i = - \boldsymbol{\nabla}_i V\,$ .

Therefore

$Q_j = - \sum_i \frac{\partial V}{\partial \mathbf{r}_i} \cdot\frac{\partial \mathbf{r}_i }{\partial q_j} = -\frac{\partial V}{\partial q_j}\,$ .

The equations of motion become

$\frac{d}{dt} \left( \frac{\partial (T-V)}{\partial \dot q_j}\right) - \frac{\partial (T-V)}{\partial q_j}= 0\,$ ,

where we have made use of the fact that $V\,$ depends only on the generalized coordinates $q\,$ and not their velocities. This motivates the definition of the Lagrangian

$L = T - V\,$ ,

from which the Euler-Lagrange equations follow:

$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0\,$ .

The utility of the Lagrangian approach is that, by virtue of d'Alembert's use of generalized coordinates, (holonomic) constraint forces do not appear explicitly.

Non-conservative forces

More generally, we may suppose that some of the generalized forces are non-conservative, and write them as

$Q_i = Q_i^{(c)} + Q_i^{(nc)}\,$ ,

where $Q_i^{(c)}\,$ may be written as

$Q_i^{(c)} = -\frac{\partial V}{\partial q_j}\,$ .

We may still construct the Lagrangian, and the Euler-Lagrange equations are recast in the following form

$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = Q_i^{(nc)}\,$ .

It is sometimes advantageous to treat one or more conservative forces in this way as well; that is, to exclude them from the potential $V\,$ , and to write them on the right hand side of the equations of motion.

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Euler-Lagrange equations

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Non-conservative forces

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