Here is a quote from Vanhees 71 in another thread on Lagrangians. I reposted here as a new thread because I fear going off-topic and redirecting a thread.

In any case, in my study of Lagrangians and Hamiltonians, everywhere I go for tutelage it seems as though everyone is maniacally focused on symmetries. What happens if you move this a little, what happens if you move that a little. Rotational symmetry, translational symmetry, epsilons, deltas, etc. My problem is that I'm willing to buy the fact that it is important, but I'm not getting the intuition as to why. It seems as though this fixation on what happens when you change something a little bit just seems to be a preoccupation that each next generation of physicists feel they need to be neurotic about because their mentors were.

Of course I jest a little, but the fact is that it is not getting through to me on an intuitive basis what these symmetry properties are or why they are so important. Perhaps someone can provide some intuition?

Intuition is something very subjective and for each person it means something different. Of course, intuition is nothing you just have but what you've trained in your thinking about problems.

Symmetry principles are so important, because they allow to find the fundamental laws of nature in a very systematic way. The real difficulty is to find the symmetries. Before Einstein this ideas vere only very hidden in physics. Einstein was the one who started his famous paper "On Electrodynamics of Moving Bodies" with the very programmatic words (translation mine):

"It is well known that Maxwell's electrodynamics, as it is interpreted today, leads to asymmetries that seem not to be observed in the phenomena, whenever one applies it to moving bodies."

This sets the theme of the whole endeavor to formulate what's nowadays known as the special theory of relativity. The conclusion has been quite drastic: While hin contemporaries, like Poincare, Lorentz, FitzGerald, and others thought the problem with the electromagnetics of moving bodies is either due to some mistake in Maxwell's equations, i.e., in the dynamics of electromagnetic fields in interaction with matter or that there must be a preferred frame of reference, defined by a socalled ether (or aether, depending on the author's spelling), which was meant to be a substance which enables the propagation of electromagnetic wave fields. The former idea was abandoned quite soon since many predecessors of Maxwell's theory like the most popular one by Weber with instantaneous actions at a distance, have been formulated in a Galilei invariant way but were clearly falsified by observation. After the direct discovery of electromagnetic waves by H. Hertz in 1894 Maxwell's theory has been favorized by almost any physicist of the time.

Einstein took up all these ideas (where it is still not clear which publications he has read before his groundbreaking work, since his paper doesn't contain any citations) but interpreted them in a radically new way, namely that it is the very foundation of whole physics which as to be adapted to the electromagnetic phenomena, i.e., the mathematical description of space and time.

As ofthen the mathematicians were much earlier in developing very clever and farreaching general ways to characterize "geometries". In his habiltation thesis (a German specialty which is necessary to become a professor) Bernhard Riemann formulated already in 1854 the general symmetry principles that allow to characterize any differentiable manifold (to put it in modern terms). The theory of invariants has been a lively topic in the 2nd half of the 19th century, mostly in GĂ¶ttingne, where eminent mathematicians as Hilbert, Klein, and finally Emmy Noether and Minkowski worked in this field.

Einstein's analysis was more motivated by physics, and he didn't use the formal theory of invariants at this time, but he was the one who interpreted what was known as Lorentz transformations, which give the mathematical relations between quantities measured in one inertial frame in terms of another. This transformation has been found before by Wolemar Voigt and other to keep Maxwell's equations invariant, but they thought of it only as a mathematical trick to make some calculations concerning electromagnetic fields simpler but not a real physically relevant thing. Einstein came to the conclusion that this is not true but that it's the very structure of space and time which has to admit Lorentz transformations instead of Galilei transformations to switch from one inertial frame to another. Not long thereafter the complete mathematical theory has been worked out by Minkowski in terms of the formal theory of invariants, and he was the one who could use this theory to work out all the known laws of (macroscopic) electrodynamics in a relativistically covariant way, and this theory stands still today (despite the fact that we know nowadays that it is an effective theory for macroscopic setups but that the underlying fundamental theory is quantum electrodynamics, but that's another story).

Although this was a very "revolutionary" thought in these days, nearly all physicists at the time were immediately convinced of this work, and among the early "fans" of Einstein's theory were very conservative guys like Planck. I guess this was not the least due to the very "intuitive" formulation in terms of a symmetry principle, based on a subtle analysis what's measureable concerning lengths and time intervals when one assumes a finite universal speed of light independent of the velocity of the light source, which after all was a known fact at the time (although even Einstein himself was uncertain in his later days whether he has known the negative result to measure the "ether wind" in the famous experiment by Michelson and Moreley).

Today, our theories and models on the fundamental level are all based on symmetry principles, which have been found by many experiments. The best theory of space and time is still Einstein's general relativity, which came out after 10 years of hard work attempting to describe also gravitation (the second fundamental interaction known in the early 20th century) in a relativistic way. It turned out in this work that space and time itself is part of the dynamical action. The best fundamental theory of elementary particles and matter composed of them is (relativistic) quantum field theory, which is completely based on (special) relativity and the underlying symmetry principles. Only the symmetry principles admit an "intuitive" derivation of what is measurable within this theory and thus only the symmetry principles admit an interpretation of the theory in a way that it can be applied to the experiments.

I'd like to add that the search for symmetry and invariance is not a modern tendency, but it was the matter even for the first seeds of physics (Ionian school). As refered in wikipedia (http://en.wikipedia.org/wiki/Ionian_School_(philosophy)) : "Most cosmologists thought that although matter can change from one form to another, all matter has something in common which does not change." . So the main object of their philosophy was to find what is that invariant quality.

Symmetry is important in physics because of mathematics.
We have Noether's Theorem, the reason for the conservation of the Hamiltonian (a.k.a. energy), angular momentum, etc. This theorem states that every time the Lagrangian of a system is invariant upon small perturbations in a quantity (local transformations - this is a symmetry), then there is a conserved quantity.
This is why group theory is important in physics. Generally, groups arise from symmetries of mathematical objects. The transformations between redundant degrees of freedom ('gauge') generate Lie groups. This connects symmetries with vector fields, and thus field theories. This is where the term 'gauge theory' comes from. For quantized theories, there are gauge bosons that arise from these gauge fields (e.g. photon, W+,W-,Z_0 in the std. model).

Thank you Chown for that nugget of insight and thank you samalkhaiat for that much larger chunk. It looks tailor made for my question and plan to spend some time going over it.

I guess now looking at it my confusion here lied with just assuming that the equations of motion and in physics in general HAD to be symmetrical. What good would a model be if it gave different laws at different orientations of the way you viewed the problem? Anyway, hopefully I will be fully enlightened after reviewing samalkhaiatâ€™s paper!