Of late I have been intrigued by the idea that all differential equations that make up the bulk of our understanding of physics can be recast as integral equations, subject to some stationary condition. In fact, these all essentially reduce to a condition called Hamilton's Principle, stating that as a system evolves, the "action" is stationary. I will define these ideas in a bit. As I researched the subject, I found that not only classical mechanics could be formulated this way, but also optics, special relativity, general relativity, electromagnetism, quantum mechanics, and even statistical mechanics, essentially all branches of physics. Even string theory can be started from an action principle. The implications are amazing: essentially all of physics can be derived from one principle.

I suppose that now I should formalize: suppose we have an \(n\)-dimensional manifold \(M\) and a target manifold \(T\). Let \(C\) be the set of smooth functions \(\phi:M\to T\). Now consider a functional \(A:C\to F\), where \(F\) is a field. Now we must make some assumptions. First, by one postulate of quantum mechanics, the action must map to the field of real numbers \(F=\mathbb{R}\), since observables (such as action) have real eigenvalues. Also, from relativity, we assume locality, as required for causality. Quantum entanglement does not pose a problem because information cannot be transmitted nonlocally instantaneously. Locality implies that if \(\phi\in C\), we can assume \(A[\phi]\) depends only on a function of \(\phi\) and its derivatives over the manifold \(M\). Hence, \(\forall\phi\in C\):

\[A[\phi]=\int_Md^nx\,\mathcal{L}(\phi(x),\partial\phi(x),\partial^2\phi(x),\dots,x)\]

The function \(\mathcal{L}\) is called the "Lagrangian" function. The Euler-Lagrange equations discussed below can be modified to include higher order derivatives of the Lagrangian*, but through a substitution, the Euler-Lagrange equations can always be reduced to derivatives with respect to the function \(\phi\) and its first derivative, nothing higher (at the expense of additional equations to solve). The Euler-Lagrange equations are derived using the calculus of variations, based on the functional derivative. We can imagine varying the function \(\phi\) until the action integral is maximal or minimal or inflected (ie, "stationary"). Finding this stationary action is the essence of the functional derivative: normally we find a stationary point by setting the derivative of a function with respect to a variable to zero. Here we set the derivative of a functional with respect to a function to zero. There is an issue, in the derivation, of boundary conditions, so that we must specify \(\phi\) on \(\partial M\) if \(M\) is compact, or place some limit on it as \(x\to\infty\). This gives us the Euler-Lagrange equations:

\[\frac{\delta A}{\delta\phi}=-\partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\phi)}\right)+\frac{\partial\mathcal{L}}{\partial\phi}=0\]

where \(\mu=1\dots n\) for each of the dimensions of the manifold. So what is \(\phi\)? These are the physical fields of interest. In classical Lagrangian mechanics, they are the coordinates themselves, expressed as functions of time. In field theory, they are physical fields as functions of spacetime. Hence the target manifold is the set of field values at a given point. For example, in classical mechanics, we might have \(\phi=x(t),\,\partial_{\mu}=d/dt\), so that

\[-\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot{x}}\right)+\frac{\partial\mathcal{L}}{\partial x}=0\]

Hence, with a suitable choice of Lagrangian, all the laws of physics may be derived (though of course in practice this is definitely not the way to always proceed). The question that naturally arises is, "What is the total Lagrangian?" Well, it must be a scalar to preserve isotropy of space and homogeneity of spacetime (that is, empty space looks the same in all directions, and there is no difference between one point and the next). Aside from this, it may be chosen to describe the physics involved. Over a few more posts, I'll derive some important equations in physics from Hamilton's Principle.