Hamiltonian Principle: Uncovering Physics Beyond Euler-Lagrange Eq.s

[ledamage]

Hi there!

After some years of physics studies I'm accustomed to the Hamiltonian principle but I sometimes still wonder why physicists tacitly assume that the eq.s of motion of any physical theory (no matter if quantized or not, relativistic or not, strings etc.) can be obtained as Euler-Lagrange equations of some variational problem which severely restricts the possible eq.s of motion. Did I overlook something obvious? Even Ramond (in Field Theory - A Modern Primer) says

It is a most beautiful and awe-inspiring fact that all fundamental laws of Classical Physics can be understood in terms of one mathematical construct called the Action.

How do we know that maybe important new physics don't lie beyond the realm of Euler-Lagrange eq.s?

 

[Cantab Morgan]

It's worth pointing out that Hamilton didn't develop his ideas to explain mechanics, but to explain optics. It was a decade or so before he realized that the very same mathematics that could explain the angles of rays of light in a telescope could also explain stuff like momentum in a mechanical system. Of course, this recognition kicked off the long chain of events that led to quantum mechanics.

I wonder if the successor to Euler-Lagrange is already out there somewhere, but not yet applied to an area of physics where it will be revolutionary.

 

[sir-pinski]

The Hamiltonian principle, also known as the principle of least action, is a powerful tool in physics that allows us to describe the motion of particles and systems in terms of a single mathematical construct called the action. This principle is based on the idea that the actual path taken by a system is the one that minimizes the action, which is defined as the integral of the Lagrangian over time. This approach has been incredibly successful in explaining and predicting the behavior of physical systems, leading many to wonder why we rely so heavily on the Euler-Lagrange equations.

It is important to note that the Hamiltonian principle is not just a mathematical trick, but it has a deep physical significance. The fact that all fundamental laws of classical physics can be derived from the principle of least action is not a coincidence, but rather a reflection of the underlying symmetries and principles that govern our universe. For example, the principle of least action is intimately connected to the conservation of energy and momentum, which are fundamental principles in physics.

So, why do we assume that all physical theories can be described by the Euler-Lagrange equations? The reason is that these equations arise naturally from the principle of least action, which we have seen is a fundamental principle in physics. This means that any theory that can be described in terms of the action can also be described by the Euler-Lagrange equations. It is not a limitation of the Hamiltonian principle, but rather a consequence of its universality.

However, it is possible that there may be new physics beyond the realm of Euler-Lagrange equations. In fact, many physicists are actively searching for new theories that may go beyond the standard model of particle physics or general relativity. These theories may require new mathematical tools and principles to fully describe them, but it is unlikely that they will contradict the fundamental principles of the Hamiltonian principle.

In summary, the Hamiltonian principle is a powerful and elegant tool that has been incredibly successful in describing the laws of physics. While it is possible that new physics may lie beyond the realm of the Euler-Lagrange equations, it is unlikely that they will contradict the fundamental principles that underlie the Hamiltonian principle. So, we can continue to rely on this principle with confidence as we explore and uncover the mysteries of the universe.