Preposition: Under

1. Under means in a lower position than something else.
Pattern 1: verb + under + noun
We sat under the tree and had a picnic.
Pattern 2:
verb + noun + under + noun
Let's put the desk under the window; then wel'll have a great view.

2. Under means covered by something else; underneath.

Pattern 1: verb + under + noun
The children hid under the table, thinking we couldn't see them.
   

Typical verbs before under:

be, hide, lie, rest, sit, sleep, stand, wait, walk  

Pattern 2: verb + noun + under + noun
She stores all her boxes under the bed.
 

Typical verbs:
bury, find, hide, place, push, put, store, wear              

3. Under means less than.                       

Pattern: under + noun
I'm sure she was driving under the speed limit.
He has three children under age ten.
                           

Nouns commonly used after under:
any number, age, average, height, limit, maximum, minimum, norm weight               

Expression:
to be under age—to not be old enough to do something
She can't vote because she is under age.                                

 4. Under can indicate control.            

 Pattern 1: under + noun
Under this boss we have little freedom to express our own ideas.
You have a lot more benefits under the new insurance policy.
                        

Typical nouns used after under:
boss, coach, contract, dictator, doctor, general, king, mayor, policy, principal, president, professor, supervisor, teacher

 Pattern 2: under + the + noun + of + noun
She is under the care of a doctor.
                                

Typical nouns after under the:

administration, care, control, dictatorship, direction, eye, management, presidency               

Pattern 3: under + ø + noun
The children are under supervision at all times.
They were arrested under orders of the chief.
              

 Typical nouns:
control, orders, supervision, surveillance      

 5. Under can indicate a current situation or state:                     

Pattern 1: be + under + ø noun
Those two thugs are under investigation by the police.
That problem is still under discussion by the board.
              

 Typical nouns used after under:
consideration, construction, discussion, investigation, suspicion                  

Pattern 2: be + under + noun
They are under the influence of their new friends.
                   

Typical nouns used after under:
circumstances, conditions, impression, influence                  

 6. Under can identify the category of a noun in written reference material.                                                                                 

 Pattern: look up/find + noun + under + name of category
You can find my name under ''Y" in the telephone book.
I looked up butterflies under "Insects" in my encyclopedia.