Lagrange's Equations

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Lagrange's equations describe equations of motion of a system modelled by the action principle .

They are

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

Where q_i, \dot{q}_i are generalised coordinates and velocities, and L is a Lagrangian.

Noting that generalised momenta are equal to \frac{\partial L}{\partial \dot{q}_i}, that is

p_i = \frac{\partial L}{\partial \dot{q}_i}

and generalised forces are equal to \frac{\partial L}{\partial q_i}, that is

F_i = \frac{\partial L}{\partial q_i}

Lagrange's equations may be written in the easy to remember form

\dot{p}_i = F_i

which truly kills many birds with one stone.

We begin with the principle of least action , which asserts that the motion of mechanical systems may be described thusly:

There exists a function L(\mathbf{q},\mathbf{\dot{q}},t), called the Lagrangian, whose integral over each sufficiently small interval of time, tε[t1,t2], takes the least possible value. This integral is called the action . This is described mathematically thusly

For each mechanical system, \exists L: \Re^{2n + 1} \rightarrow \Re, L(\mathbf{q},\mathbf{\dot{q}},t), such that

S = \int_{t_1}^{t_2}{L(\mathbf{q},\mathbf{\dot{q}},t) dt}

is at minimum, for some selection of intervals [t1,i,t2,i] which completely cover the region of interest.

This does not actually impose terribly many restrictions on L. First, clearly, it must be integrable . One notes that if a Lagrangian L describes a system, so does L' = L + \frac{d}{dt} F, where F is any function.

The main point is that, for nearly every system of interest (a proof here, or more insightful description of what actually is described by Lagrangian physics, would actually be very interesting. It should be noted that for every Quantum System, the classical variational principle follows from the use of Feynman Propagator for the time evolution of the system. Are potentially non-quantum justifications for this principle? Are there other systems entirely), mechanical systems may be described by the principle of least action. In other words, this isn't voodoo . The toolset for dealing with mechanical systems, once it is in this form, is very advanced, and has computational and practical advantages over dealing with forces directly -- many of the details are taken care of. (Note: It may be interesting to prove some upper/lower bounds for this).

Now, the requirement that the action S be minimized automatically places some restrictions on the form of the solution of \mathbf{q},\mathbf{\dot{q}} (that's right, after finding a Lagrangian, somehow, after examining systems carefully, we work backwards to impose to find the mechanical path of the system. Luckily, nature isn't terribly malevolent, and the Lagrangians we find are governed by fairly simple laws which we can use outside of the context where we discovered them. It feels backwards, even if we don't think about it much.)

In the language of Calculus of Variations, the First Variation of the Lagrangian must be zero. (This appears to be missed by nearly every classical mechanics text, but additionally, the Lagrangian should either be constant, or the first non-zero nth variation should be positive (otherwise it is actually a maxima.) This step is nearly always dropped, whether sane or not! In fact, in classical mechanics, the fact that there may be multiple solutions to Lagrange's equations seems to be almost universally ignored. I believe that this is due to a false sense of security, brought about by the generally benevolent Lagrangians we face, that do not have multiple local extrema.)

Nonetheless, for any such solution, the First Variation of the action is required to be zero. Lagrange's equations then follow.

W.l.o.g we assume that there is only one degree of freedom, q. Assuming q(t) minimizes S. Consider the small function δq(t), which is zero at t1,t2, (this imposes no loss of generality so long as our chosen intervals can overlap (this allows us to construct, piecewise, a path through phase space), assumed to be universally true (no singularities in time!)). The first variation is then the first order terms (in δq(t)) of

\int_{t_1}^{t_2} L(q + \delta q, \dot{q} + \delta{q}, t) dt - \int_{t_1}^{t_2} L(q, \dot{q}, t) dt

Or, where δ is the functional differential operator,

\delta S = \delta \int_{t_1}^{t_2} L(q, \dot{q}, t)

The functional differential operator works much like a regular differential, so the above becomes

\int_{t_1}^{t_2}\left({\frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot{q}}\delta \dot{q} }\right) dt = 0

Integrating the second term by parts

\delta S = \left[{\frac{\partial L}{\partial \dot{q}}\delta q}\right]_{t_1}^{t_2} + \int_{t_1}^{t_2}\left({\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}}}\right) \delta q dt = 0

This integral must vanish for all possible values of δq. This can only be so if the integrand is zero everywhere (otherwise our nasty δq could be a Dirac Delta Function and pick it out!). This result is known as the Fundamental Lemma of the Calculus of Variations, although it's an altogether haughty name, I think. Anyway, our integrand is

\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} = 0

Repeating the whole shebang for each of the coordinates, we get

\frac{\partial L}{\partial q_i} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_i} = 0

These are Lagrange's equations.

One may ask why the Lagrangian may not be an explict function of acceleration, or any of the other derivatives of the particle motion. The simple and frustrating answer is that such a thing does not exist in nature, as far as we can tell. Even for more complex field theories (as far as I know), a Lagrangian depending on anything other than field density and wave-vector does not exist (can someone who knows what they are talking about enlighten me here?) I suspect something may be said about gauge theories, but really I have no satisfactory reason to believe this (Landau and Lifshitz simply claim that 'we know this due to experiment'.)

One may also say something about Noether's Theorem. The action principle seems to be a guiding light towards exposing symmetries. This can be seen, indirectly, by the proof of Noether's theorem for Lagrangian systems. Precisely why the action principle does this is an interesting question.

This page was recovered in October 2009 from the Plasmagicians page on Lagrange's_Equations dated 04:13, 19 October 2006.

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