# Relationship between Principle of Least Action and Continuity Equation

1. Oct 28, 2013

### cryptist

Is there a profound relationship between Principle of Least Action and Continuity Equation? Can we derive one from another?

2. Oct 28, 2013

### UltrafastPED

What is the "Continuity Equation"?

3. Oct 28, 2013

### ModusPwnd

http://en.wikipedia.org/wiki/Continuity_equation

I've never heard of a profound relation between the two nor have I seen one derived from the other.

4. Oct 28, 2013

### AlephZero

Why should you be able to derive one from the other? The continuity equation is about what happens at a single point in space. Least action is about what happens along a path.

If you have done any real analysis, it should be clear you can't deduce what happens in an interval (i.e. a path) from what happens at each individual point. For example think about the difference between "convergence" and "uniform convergence".

5. Oct 29, 2013

### Jano L.

Not from continuity alone, that is too little information. For example, time-asymmetric system of equations can move continuously (system with friction), but it cannot be described by Hamilton's principle.

On the other hand, from Hamilton's principle the continuity does follow.

In my opinion, the most instructive way to arrive at Hamilton's principle is from the Newtonian equations of motion (of certain kind, like particles moving under potential forces).

Although I agree with you answer, I do not understand your argument. It is perfectly possible to derive what happens to integral quantity from the knowledge of what happens at one point.

We can derive the variational condition from the Euler-Lagrange differential equations, for example.

6. Oct 29, 2013

### BruceW

hmm. If we have a Lagrangian, and we have some symmetries, then we can derive continuity equations from them. (This follows from the requirement that the Action is stationary). For example, if our Lagrangian does not change when we make a small spatial translation, then we get a continuity equation for conservation of momentum. And in quantum field theories, if we have some complex field such that our Lagrangian is unchanged by making a small rotation (of the value of the field) in the complex plane, then we get a continuity equation for the conservation of charge. But yeah, if your system does not have a Lagrangian, then I'm pretty sure this method cannot be used to derive continuity equations.