Conserved quantities in mechanics

[aaaa202]

So having been through translation and rotation I can conclude that my book has found 3 conserved quantities in classical dynamics:
Energy
Angular momentum
Momentum
That is 7 separate quantities which are conserved E,Lx,Ly,Lz,px,py,pz
But this question is bothering me: How do we know that there are not more conserved quantities?

 

[AJ Bentley]

Conservation laws come from invariance of the Lagrangian with each degree of freedom.
If you can think of another degree of freedom you can have another conservation law.

 

[Studiot]

Is momentum always conserved?

Suppose I direct a hosepipe horizontally at a wall.

Please explain conservation of horizontal momentum at the wall face in this case.

 

[mfb]

Is momentum always conserved?

In closed systems.

Some systems have additional conserved quantities, for example the Kepler problem or harmonic oscillators. But in the general case, you just have those 7 conserved quantities.

 

[Studiot]

@mfb
Yes it is good to specify the conditions.
These were not originally specified and mechanics is not exclusively about closed systems, and should not be taught or thought of as such.

In my career I have seen spectacular failures of mechanical systems due to failure to appreciate this.

 

[D H]

Is momentum always conserved?

Suppose I direct a hosepipe horizontally at a wall.

Please explain conservation of horizontal momentum at the wall face in this case.

Simple. The (effectively) infinitely massive wall moves at an (effectively) infinitesimally small rate. The product mass*velocity is some finite, non-zero quantity that exactly obeys conservation of momentum.

 

[Studiot]

So where does the perpendicular (to the flow) momentum come from at the wall face?