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Can all differential equations be derived from a variational principle?

  1. Jul 3, 2006 #1
    In classical mechanics, for conservative systems, it well knows that the differential laws of motion can be derived from a variational principle called "least action principle".

    I know also that some non-conservative systems can be derived from a variational principle: the damped harmonic oscillator has a time-dependent Lagragian and an associated least action principle.

    Physically, I wanted to know if:

    the least action is something special happening in special conditions, and which conditions
    or ​
    if it is a general rule that applies to (nearly) all differential systems of equations​

    Can all differential equations be derived from a variational principle?
    I would greatly enjoy your ideas, comments suggestion or any track.
    Examples could be very useful too.

    Last edited: Jul 3, 2006
  2. jcsd
  3. Jul 5, 2006 #2
    That is a very interesting question. My initial opinion would be that "the least action is something special happening in special conditions, and which conditions," since I think (and this is opinion only, of course) that the variational principle [itex]\delta S = 0[/itex] has some physical content.

    But someone may just come and prove me wrong. Maybe it is the second one, and that the real trick is to find which quantity we want to minimize in order to get some diff. equations out. It just so happens that for physical situations, it's generally a simple quantity that we want to minimize.
  4. Jul 6, 2006 #3

    That's indeed the reason why I asked the question here after a short discussion in the CP section. The limit from CM to QM is a well known physical interpretation. The damped oscillator admits a variational principle too, but I don't see the physical interpretation. The Schrödinger equation can also be derived from a variational principle, but what is the physical meaning then?

    Actually I don't know about some differential equation that could not be derived from a variational principle, at least in physics. (I will ask that in the CP section)

    If only a small class of DE could be derived from a variational principle, then it would mean that physical laws are really special and this would deserve a further study.

    If variational principles are (nearly) the rule, they would be no less interresting, but their status would be more comparable to differential equations.

    Last edited: Jul 6, 2006
  5. Jul 6, 2006 #4
    Mathematically, I believe that any second order differential equation could be represented as the minimization (or maximization, as is sometimes the case) of an appropriate integral, but I haven't seen a proof of that so I'm not 100%.

    Whether this is physically significant.....I'm not sure. All differential equations can be represented as integral equations (I'm 100% on that), but humans don't tend to set them up that way. Any given problem can usually be represented a number of different ways, but certain methods tend to take precendence. I've always thought this said more about people than the physical universe.
  6. Jul 6, 2006 #5
    inverse problem of the calculus of variations

    I browsed on the net to find some track.
    I guess the problem I submitted is known and called

    inverse problem of the calculus of variations ​

    Quickly reading here and there suggested me that it is solved only for second-order differential equations (one variable) and maybe some even-order systems (one variable). I don't understand that quite well, since I am thinking essentially first-order and any number of variables.

    Here is one of the links I found: Towards a solution of the inverse problem ... .

    Any suggestion, comment, reading, ... ?

  7. Jul 6, 2006 #6


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    You need to separate the mathematics from the physics. When you say "all differential equations" you are asking a mathematics question. As a pure mathematical concept DEQs have nothing to do with physics or any principle of physics. It is a very happy coincidence that physical properties can be expressed in terms of a well understood mathematical language.

    It is observed, that in the physical world, systems involving changes in energy evolve in such a way that energy is minimized. This may be a major factor in why the concept of energy is so useful.

    When asking this type of question you must remember that ALL DQs can be, and are, studied completely independent of Physics. You do not need ANY physical principles to discuss the mathematics of Differential Equations.
  8. Jul 7, 2006 #7

    I don't agree with your philosophy, at least as I understand it from your post.
    (I would better assume that mathematics is physics !)

    Assume it could be proved that all systems of differential equations can be derived from a variational principle, and eventually how to construct a Lagrangian for it. Then the consequence for the interpretation of physics is very important. This would mean the equivalence of these two questions:
    Why a least action principle, why a variational principle for CM or QM ... ?
    Why are differential equations at the root of physics ?

    It is well known that the transition from quantum mechanics to classical mechanics gives a physical interpretation for the least action principle: "the waves do really have the ability to sense all possible trajectories and concentrate on stationary phase path" -so to speak physically- . But is this interpretation really useful if it could be proved that all DEqs could be derived from a variational principle? Actually, would that not indicate somehow the reverse route from CM to QM !

    Now consider the Schrödinger equation. It can also be derived from a variational principle. Same for the Maxwell's equations. Many other examples can be considered. Actually, I don't know one physical theory that cannot be derived from a variational principle. Do you? Therefore my question. Maybe all differential equations can be derived from a variational principle!

    And there is more: the Lagrangian may even not be unique. (see this paper). Again, then a mathematical problem questions the physics. If the relation between two possible Lagrangian is simply related to a symmetry, this already points to some (known) physics. But could the possibilities be broader?

    Sorry for the philosophy. This was simply to explain my motivation.
    The question remains: can all DE be derived from a VP ?

    Last edited: Jul 7, 2006
  9. Jul 8, 2006 #8
    As a physics-trained engineer, specialising in CFD research, perhaps I could add in a perspective:

    1. Physicists use Conservation Laws to derive relationships between physical quantities.

    2. A simple method is to use the Continuum Principle, which lends itself to differential equations applying at each point in the continuum. This can be converted to an integral form if so desired.

    3. The usual rules of mathematics are then used to analyse these equations.

    4. In many cases, numeric methods need to be employed eg. for Navier-Stokes.

    So, if I were to address the OP's original question, could this be re-couched in terms of an apparent relationship between the Conservation Laws & the Variational Principle? In other words, "Do the Conservation Laws always express a minimisation of 'some (conserved) quantity'?"
  10. Jul 9, 2006 #9


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    Pretty big assumption isn't it? How could you possibly make this sort of assumption then carry on to a conclusion that means anything.

    Rather then assuming this, please show us how to prove it.
  11. Jul 9, 2006 #10

    jim mcnamara

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    IMO - arguing assumptions for math motivation like this one isn't very productive.

    There has been a dichotomy in Mathematics for a long time. In general, maths evolved to answer physical questions (fluxions, for example), but as early as Euclid, "pure" math also became defined, probably as an exercise in logic.

    And this is the point of contention here - call it pure vs applied.
  12. Jul 9, 2006 #11
    Pure vs. applied... That's an excellent way to put things.

    As a physicist-type, I tend to see Mathematics as a useful toolbox of tricks I can use to go where I need to go along my research trail. When matters of Mathematical purity emerge, then I conveniently find time to move onto the next tool. :shy:
  13. Jul 9, 2006 #12


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    we are getting off the topic, towards personal biases. it was interesting until then.

    there is to me no reason that avariational principle could not have a purely mathematical formulation, and thus make sense in either context.

    look for example at the proof of existence of solutions of odes by eulers or picards methods. these rewrite it as an integral equation, or integral operator, and take limits.

    picards point of view is that of an integral operator transforming one functions into another, and the solution is always a function which is "fixed" or transformed into itself. this is surely some kind of minimization principle.

    think of the gradient principle in calculus, of loking at the gradient of the function and always moving along the gradient. a minimum is a point where this leaves the point fixed, since the gradient there is zero and the point is "stable" for the transformation. think of a marble at the bottom of a bowl.

    just brainstorming on this very intersting question. i of course am totally ignorant even if what most of the words mean in this topic, but that does not stop me from babbling analogies. thats how i work.
  14. Jul 9, 2006 #13

    This is also one of the reason I asked for opinions here.
    Apparently non conservative systems can also derive from a variational principle.
    For example, the damped oscillator can be derived from a variational principle. But its lagrangian is time dependent (contains an exponential decay).

  15. Jun 15, 2007 #14
    Why isn´t it as simple as taking for the general PDE

    F(x,y,z,f_x,f_y,f_z,f_xy,...) = 0

    the functional G(F) := Integral(F^2(f))

    This is =0 for the solution f of the PDE, >=0 for all other functions (not solving the PDE) and if there were some notion of continuity this could be interpreted as a minimum problem? Of course it would be a trivial solution and not very useful...
  16. Jun 15, 2007 #15
    lalbatros: I believe you are using logic in a wrong way. As most DE's in physics are derived on the assumption of energy conservation, they admit a variational formulation, but as others pointed out before, the set of DE's you can write down in a piece of paper is far more extense than the set of DE's that have physicall meaning.

    I am almost sure that not all DE's admit an energy functional, and I am pretty sure that variational methods do not work on all DE's. Let me do a little research with my variational friends and I'll get back to you.

    As Integral said, if you have a proof of your statemen, by all means show it, as it should be an award winning paper.


    This post reminded me of a problem that a brilliant professor of mine once talked about:

    All ODE's can be written as difference equations by means of Fourier methods. Now, do all difference equations come from a ODE?
    Last edited: Jun 15, 2007
  17. Jun 15, 2007 #16
    The question posed by lalbatros is actually extremely important. It basically boils down to wether the diff. equations or the action principle are primordial. The question is very important because the fancy theoretical physicists trying to quantize a system usually start by writing an action, implicitly assuming the system has one. What if it doesn't ?

    I've heard not all differential equations can be derived by varying some action.

    Suppose you are given an action S. You derive equations from that action by setting the first functional derivatives to zero. The question is: given the equations, is it always possible to find an action which has those equations as functional derivatives.

    I make the following loose analogy with ordinary calculus. An analogous question in calculus would be: given some functions, is it always possible to find a function with such partial derivatives. The answer in calculus is NO - the given functions must obey certain consistency conditions (the cross derivatives must be equal) to be partials of the same function.

    By analogy, the answer is NO in the action case. I wonder if anyone has derived the conditions a set of equations has to satisfy to be derivable by an action (maybe the cross functional derivatives must be equal) analogously to the conditions a set of functions have to satisfy to be partials of a single function.

    I'm reading a book in classical mechanics and there they say that a system with non-holonomic type of constraints can't be described by action cause the constraints can't be integrated to a function and incorporated in the action by lagrange multiplier.

    It's another story MOST of the systems in physics we KNOW DO have actions. That doesn't mean the systems we don't know will continue to be derivable by actions.
    Last edited: Jun 15, 2007
  18. Jun 15, 2007 #17
    I found an article,
    which states:

    "...A strict mathematical theorem tells us the
    condition of the existence of a variational principle for a given differential (or almost
    any kind of) equation (see for example in [?, ?])..."

    Unfortunately the authors don´t give the source of this assertion. Probably it is an unfinished pre-print, but I didn´t find a final version with a bibliography. Maybe one of you has?
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