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.

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

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

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.

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.  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

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.

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