% ----------------------------------------------------------------------------- %
                               Lagrangians, Variational Forms, and other issues.

1) QUESTION: Given a dynamical system, how do I find what the orbits do?

There is no "universal scheme". Like many other things, you learn and gain
intuition by practice, and build a tool box.    When you face a new system,
maybe the tool box is enough, or maybe it is not. Then you need to develop
new tools, which you may or may not be able to do. Numerical solutions can
be very helpful, but for complicated systems they are often not enough ...
For example:
The parameter space may be too large to explore thoroughly,  and important
    and/or useful behaviors may remain hidden for a long time [or forever].
The dynamical system may be such that  accurate enough numerical solutions
    are not possible.
Even when none of the prior points apply, you may be deluged by the amount
    of data generated by the code.
And so on.  If work-in-all-cases approaches existed, we would already
know and understand everything that, say,  the Navier Stokes equations can
do --- which is obviously not true.    Replace "Navier Stokes" by anything
else [Quantum Electrodynamics, Maxwell Equations, ...], same deal applies.

2) QUESTION: Given a system, how do you find if it has a conserved quantity?

Basically, the answer is the same as for (1). An interesting thing here is
that,  presently,  theoretical physicists propose new models starting from
the need to conserve certain things, such as Energy, as well as have given
symmetries --- these two things are related, by the way, see [NT] below.

This is done using "Lagrangian formulations" -- I may or may not have time
to touch on them.   But "outside of theoretical physics" this is often not
possible, and then a question parallel to "Does my system have a conserved
quantity?" is "Does my system have a Lagrangian?" to which there is, again,
no fool-proof method to get the anwer.

BABY Lagrangian "theory"
History:   in the early times of modern science, researchers often ended up
formulating principles that read like this: "Nature wants to minimize XXX".
Example: Fermat's principle; light rays minimize the optical path.
Later on these principles were justified using the dynamical equations that
governed the phenomena. For example,  in the case of light,  using the wave
equation, Fermat's principle can be justified in the high frequency limit.

Modern view:  at the fundamental level,  physical systems follow extrema of
the action = \int dt L; where L is the Lagrangian.
    e.g.: in particle mechanics L is generally a function of the particle
          positions and velocities, and the integral defining the action is
          an integral over a trajectory, from t = t1 to t = t2.
Defined in this way, the action is a "function" that assigns a number to
every possible trajectory in phase space. It's domain of definition is HUGE!
Then, to find the extrema of the action, one has to compute its "gradient",
and set it to zero. The result is then either an ode or a pde that the
solution has to satisfy.
The branch of mathematics dealing with this is called "Calculus of Variations",
where "variation" is the generalization of the concept of a derivative to the
setting described above. And the gradient is called the variational derivative.

Noether's theorem. ..................................................... [NT]
It can be shown that symmetries in the action are related to conservation.
This is Noether's theorem. For example, action invariant under
(a) time translation yields conservation of energy;
(b) space translation yields conservation of linear momentum;
(c) rotation yields conservation of angular momentum;
plus many more [any continuum symmetry yields a conservation law].

In Mechanics, one of the advantages of Lagrangians is that they make the
formulation of problems with constraints much easier. In terms of forces,
a constraint imposed by some rigid boundary can be resolved by introducing
forces which are "just right" to enforce the constraint. For simple problems
this is doable, but it is not so easy for more complicated ones. As an
example, try to write the equations for a falling flexible but inextensible
rope with a mass rho per unit length. A simpler version of this is: you have
N masses m falling in space, with positions x_n, connected in such a way that
the distance from x_n to x_{n+1} is fixed.
%
% ----------------------------------------------------------------------------- %
EOF