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According to Noether's theorem, a symmetry of space-time w.r.t. an observer, will yield a corresponding conservation law for a closed system w.r.t. that observer. Now if our space-time has 3 symmetries w.r.t. an inertial observer namely isotropy and homogeneity of space and homogeneity of time, so only 3 conservation laws are possible in classical mechanics , namely conservation of angular momentum,momentum and energy of a closed system.

Aren't there more conservation laws possible ? What stops me from defining a quantity like $\vec{r} \cdot (\vec{r} $ x $ \vec{f})$ w.r.t. an inertial observer and take its derivative with time, and see when it is conserved w.r.t. time by equaing derivative to be $0$. If yes, how to find the corresponding symmetry ?

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    $\begingroup$ It isn't correct to say that a symmetry of spacetime corresponds to a conserved quantity. Noether's theorem tells us that the symmetry of the action corresponds to a conservation law. $\endgroup$
    – John Rennie
    Commented Jan 16, 2014 at 10:05

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In Lagrangian mechanics, in view of celebrated Noether's theorem, if you have a Lagrangian invariant under the action of a one-parameter group of transformations, or weakly invariant (i.e. $\frac{\partial \cal L_\epsilon}{\partial \epsilon}|_\epsilon = \frac{df}{dt}$, where $\epsilon$ is the parameter of the group), then there exists a conserved quantity along the solutions of the equations of motion. These transformations are called (Lagrangian) symmetries.

I am precisely referring to transformations acting on the coordinates of the points of matter of the physical system. This action is next extended to velocities of these points by formally computing the $d/dt$ derivative (lifting the transformation to the first jet bundle, formally speaking).

The fundamental group of invariance of classical mechanics is Galileo's (Lie) group, $G$, that describes the transformations between coordinate of any pair of inertial systems:

$$t' = t+c\:,\qquad c\in \mathbb R$$ $${\bf x}' = R{\bf x} + t{\bf v} + {\bf c}\:, \qquad R \in O(3)\:, {\bf v}, {\bf c}\in \mathbb R^3\:.$$

$G$ contains $10$ one-parameter (Lie) subgroups that must leave (weakly) invariant every Lagrangian describing an isolated system in an inertial reference system. Correspondingly there must exist 10 scalar conserved quantities. $6$ of them are induced by the Lie group of isometries of the 3D Euclidean space which, in fact has $6$ (Lie) one-parameter subgroups.

All these symmetries are the same for every isolated mechanical system.

Considering specific systems you can have further symmetries and associated conserved quantities. However, in general, these Lagrangian symmetries are not associated with symmetries of the classical spacetime, but are proper of the mechanical system.

When you have a Larangian symmetry it is possible to prove that it transforms solutions of Eulero-Lagrange equations into solutions of Eulero-Lagrange equations. That is another notion of symmetry a dynamical symmetry.

Lagrangian symmetries are dynamical symmetries but the converse is generally false.

Moreover:

Lagrangian symmetries imply the existence of conserved quantities but the converse generally is false.

The correspondence instead is one-to-one if describing all this picture in Hamiltonian formulation and where the Hamiltonian symmetries are assumed to be canonical transformations:

Hamiltonian symmetry $\Leftrightarrow$ Dynamical symmetry $\Leftrightarrow$ conserved quantity

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  • $\begingroup$ Could you explain the sentence: "Lagrangian symmetries imply the existence of conserved quantities but the converse generally is false"? I thought that in general one tries to determine what the conserved quantities of nature are, and then finds an action that has those symmetries as well. In other words, I was under the impression that the conserved quantities observed in nature do constrain the Lagrangian to incorporate the corresponding symmetries. $\endgroup$
    – Hunter
    Commented Jan 16, 2014 at 16:05
  • $\begingroup$ It depends on how you define the notion of symmetry for a Lagrangian. For instance the conservation of Runge vector for the Lagrangian of a point subjected to Coulombian potential cannot be induced by any geometric symmetry in space, extended to velocities by formal time derivation (the standard lift on a jet bundle). A more complicated notion can however be used (as a matter of fact a non-geometrical canonical transformation in Hamiltonian representation). $\endgroup$
    – Valter Moretti
    Commented Jan 16, 2014 at 16:10
  • $\begingroup$ In Hamiltonian formulation (or quantum theories) instead everything is, in a sense, much more trivial: The conserved quantity is the generator of the (canonical) transformation leaving invariant in form the Hamiltonian that, in turn, gives rise to the conservation of that quantity. $\endgroup$
    – Valter Moretti
    Commented Jan 16, 2014 at 16:13
  • $\begingroup$ (+1) Thanks for your answer. I was indeed (subconsciously) thinking about quantum field theories. $\endgroup$
    – Hunter
    Commented Jan 16, 2014 at 16:16
  • $\begingroup$ All this is a bit advanced for me right now. I will revisit this later. So in short the total number of symmetries of the action in classical mechanics is ? $\endgroup$
    – user37026
    Commented Jan 16, 2014 at 17:13

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