# Variational principles in physics

1. Jul 2, 2010

### telegramsam1

I was wondering if anybody could help me crack this one.

Variational principles such as hamilton's principle are used to state the laws of physics. To my knowledge, all of classical theory (including GR) can be stated this way. The resulting DE can then be found using the euler-lagrange equations.

My question is twofold.

1. Can all possible laws of physics be stated this way in principle? Not just all known laws, or all 'true' laws. Can all wild and crazy differential equations for the position of a particle as a function of time be derived from some judicious choice of the Lagrangian.

2. If no. What specific limitations are imposed on the laws of physics if we demand that they are derivable from variational principles?

2. Jul 2, 2010

### lenfromkits

Not sure but couldn't it be argued that evolution/natural selection is just an emergent system from basic physics. So are cognition, emotion, love, fear. Can these be derived from variational principles?

3. Jul 2, 2010

### telegramsam1

Lets spare ourselves the debate over determinism. I'm asking if all laws of physics can be derived from variational principles. I go no further.

4. Jul 2, 2010

### my_wan

So long as a conservation laws applies, ah la "[URL [Broken] theorem[/URL].

Last edited by a moderator: May 4, 2017
5. Jul 2, 2010

### arildno

Hamiltonian formalism (variational formulae) are certainly valid for all complete systems.

However, incomplete systems, notably systems that contain dissipation, sources of friction, or disregard "heat" as a form of the body's total motion will NOT generally be able to be hamiltonially formulated.

At the DEEP theoretical level, therefore, variational principles can always be used, but on the PRACTICAL level, say within mechanics&engineering, those principles can be basically..worthless.

Last edited: Jul 2, 2010
6. Jul 2, 2010

### telegramsam1

Ok I'm actually looking for something else than what I'm getting. I want to know if an arbitrary differential equation of the position of a particle as a function of position,time can be obtained through some variational principle. This means its a classical theory on euclidean 1 dimensional space. An equation of motion of this general form.

f(q,q') = g(t)

From this there exists an L(q,q',t) such that

(doesn't work I know, Euler-Lagrange equation in 1d)
\frac{\partial{L}}{\partial{q}} = \frac{d}{dt} \frac{\partial{L}}{\partial{q'}}

will recover the first equation.
This assumes that q' is the highest derivative of position in the Lagrangian. Thanks in advance.

7. Jul 2, 2010

### alxm

I believe my_wan is correct (and he may know more about it than I do). Any physical system which obeys conservation laws, can probably be expressed in this kind of formalism.

On the other hand, there will also always be mathematically equivalent formalisms.

8. Jul 2, 2010

### telegramsam1

If the lagrangian has some symmetry, Noether's theorem shows that there is a conserved quantity. I'm not sure the converse holds. That is, if there is a conserved quantity, there must be a lagrangian that has that symmetry. If in fact there is a variational principle which determines the equations of motion, then the converse holds. I'm looking for a proof that all DE's of the form I mentioned can be described using a variational principle.