Lagrangian sought for given conservation law

[birulami]

Lagrangian sought for given conservation "law"

Reading about the Lagrangian and conservation laws, I was wondering if, given the conservation law

$$|\dot{x(t)}| = const$$

where $x$ is an n-dimensional vector, we can find the Lagrangian $L(t, x(t), \dot{x(t)})$ that produces this conservation law via the action integral's minimum or maximum?

Note that a constant absolute velocity does not mean the particle has to go in a straight line. It only means that any acceleration is always strictly orthogonal to $\dot{x}$.

 

[birulami]

Another way to put it, would be to ask for the Lagrangian which leads to the conservation of momentum absolute value, i.e.
$$(1)\qquad |p| = const$$
which is different to the usual law of conservation of momentum coordinate-wise:
$$(2)\qquad p_\mu = const \; \forall \mu = 1..N$$
(2) requires that the Lagrangian $L$ is independent of spatial translation, and then (1) of course follows as a consequence.

I would expect that for (1) to hold but not (2) we would have a less strict independence rule.

 

[vanhees71]

You could use Noether's theorem to think further about this. The conserved quantity is the generator of the one-parameter subgroup of the system's symmetry group that leads to this conservation law. Thus, you should check, which transformations your conserved quantitity generates and then think about the most general Hamiltonian (or Lagrangian) that preserves this symmetry.

 

[medthehatta]

EDIT: Forgive me: the constraint you've listed is (second-order! ) nonholonomic. The techniques I pointed out, of course, won't work for a nonholonomic constraint.

My erroneous response was:
I hope this isn't an overly flippant reply, but here are some suggestions, both of which essentially reduce to thinking about the "conservation law" as a "kinematical constraint" and then using the standard techniques:

1. Use Lagrange multipliers to enforce the constraint

2. Use generalized coordinates that implicitly respect the constraint (like, say, have two of the coordinates be the direction of the vector x-dot, and don't have a coordinate for its length)

You'll certainly find that there will not be a unique Lagrangian respecting that conservation law, of course.

 

[birulami]

Well, actually I was hoping that a langrangian for $|v|=const$ is (well) known. and was just curious to see it. Working with Noether's theorem will be tough, since just know I try to work my way through Neuenschwander's book "Emmy Noether's wonderful theorem".

Nevertheless thanks for the possiblel recipes.