Can someone help me to better understand energy

[Gerenuk]

Would it be true to say that energy is the only conserved quantity that transforms as a scalar?

If you consider frame transformations, which is different from the time evolution transformations, then any product of two vectors is a scalar (rest mass of a particle, ...).

Not sure how this refers to time evolution transformation. I think in general energy, momentum and angular momentum are the only integrals of motion unless you deal with a special problem.

Edit: Oh, I see. You meant something that is both
Then in general only energy should be the conserved scalar. Unless the problem is a special one.

 

[jambaugh]

Would it be true to say that energy is the only conserved quantity that transforms as a scalar?

Not at all. Firstly it is only scalar in a non-relativistic setting. In SR energy is one component of the 4-vector energy-momentum. Electric charge and the other gauge charges are space-time scalars (with some slight qualification on weak isospin due to weak parity issues).

The nice part of the link between conserved quantities and symmetry transformations is that how the quantities transform (whether they are scalar, vector, tensor, ...) with respect to the various groups (rotation, Lorentz, Poincare, ...) is made explicit by how the symmetry transformations themselves transform. Consider that translation in the x direction can be rotated to translation in e.g. the y direction. Hence rotations act not only in concert with translations on objects, but also rotations act on translations. This is called the adjoint action of one transformation on another.

Since time translation is "scalar" i.e. invariant under spatial transformations (rotations and spatial translations) Energy is scalar in this non-relativistic setting. Since spatial translation in a given direction gets rotated by a rotation transformation so does momentum. Momentum is a vector quantity.

In the relativistic setting you can think of energy as "momentum in the time direction" and Lorentz transformations act on the four momenta as a 4-vector.

One final note. You can imagine for some system that you can "tweak" the dynamics to have any arbitrary symmetry or that you break a given symmetry. This does not interfere with the link between measurable quantities (which may or may not be conserved) and transformations (which may or may not be symmetries).

EDIT: In short the link is there without worrying about the quantity being conserved or the transformation being a symmetry :END EDIT

This is made most explicit in the formulation of quantum mechanics where the operators generating the transformations are identified mathematically with the operators corresponding to the observables.
The momenta ARE the generator of translations,
Energy (the Hamiltonian) IS the generator of time evolution,
Angular Momenta ARE the generators of rotations.

It is all very elegant and beautiful. That it also yields an accurate description of how nature behaves is gravy on the biscuit!