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What is the difference between and how can we explain the grammar in the
expressions for "follow" in the examples below?

  1. 子供が母親のあとをついていく (A child is following his/her mother)

  2. 母親の買い物についていく (go shopping with mother)

  3. 彼の考えにはついていけない (I can't go along with his idea)

  4. 彼の後をつける (I follow/tail him-see also * below)

I wondered if this was because:

  * についていく allows the subject to "attach itself to the object" and move after it (grammatically this works well because both are intransitive)?
  * 後をつく and 後をつける are both idioms whereby: 
  * つける can only be used with あと ("mark left behind") and as a transitive verb takes を to mark the object？
  * つく is intransitive so grammatically を is used "spatially" as in 信号を曲がる
  * 〜ついていく can only be used when there is continuous movement/"one journey", as opposed to say a child following its mother 24 hours a day who will start and stop/"make several journeys"?

References/notes on examples 1-4:

  1. All can be found in 総まとめ 語彙 N2-p132 (ex1-3) N1-p100 (ex4); 

  2. *Example 4 also appears in the Apple dictionary as: こっそり人 (の後) を付ける｜follow a person secretly / shadow [stalk / ｟口｠tail] a person

Or (long shot suggestion): \- Is the つく verb the same in all cases? (last week
I noticed that 付く is intransitive but 吐く as in うそをつく is transitive.)

* * *

**Update to reflect two answers received as of 28 Aug 12:**

To summarise answers received, there are two explanations:

**Explanation 1:**

The main difference b/w the examples can be explained by the difference b/w
あとに & あとを:

  * 後に is a temporal/spatial use of を. Essentially the follower is following the same course as the object but not necessarily at the same time. 
  * 跡を is a physical (or if emotionally used, metaphorical) use: the followers moves behind the object of its pursuit in "lockstep", speeding up and slowing down to maintain the same distance. 

Regarding use of つくvsつける： 後をつく is ungrammatical because つく is intransitive but
the ついていく in 後をついていくis thought of as independent transitive verb, taking 後 as
object. (つける is a transitive verb and can take the object あとを）

**Explanation 2 (Revised):**

The main pts of this explanation was proposed after referring to the
explanation 1 (above):

  * （～の後を）ついて行く is defined as "trail", and it classifies this usage with を as being intransitive (【自動】). It might literally be "follow someone's behind".
  * (人）の後について行く is defined as "tread in someone's footsteps"/"tread in the steps of", and ～の後について行く is listed as "follow" and classifies this usage with に as being transitive (【他動】). 
  * 後をつける is used for shadowing/tailing and 跡をつける leaving traces,

**Explanation 3 [Original Explanation 2]:**

  * When the particle に is used the person being followed is the main actor in the sentence.
  * When the particle を is used the main actor in the sentence is unchanged.

This is similar to the difference b/w に伴う and を伴う

(Although this was explanation was replaced, I have kept it here for reference
as I think it still has credibility.)

Any further comments or input are welcome.
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grammar word-choice usage
The difference between "follow" using についていく、「あと？」をつける、「あと？」をついていく
**Edit:** Rewrote as I think user1205935's explanation of them being separate
verbs sounded more plausible than mine.

Looking through Space ALC, there are definitions for
[`後をついて行く`](http://eow.alc.co.jp/%E5%BE%8C%E3%82%92%E3%81%A4%E3%81%84%E3%81%A6%E8%A1%8C%E3%81%8F/UTF-8/)
as well as
[`後について行く`](http://eow.alc.co.jp/%E5%BE%8C%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A1%8C%E3%81%8F/UTF-8/).
One difference does seem to be transitivity:

  * `（～の後を）ついて行く` is defined as "trail", and it classifies this usage with `を` as being intransitive (`【自動】`). It might literally be "follow someone's behind".

  * `（人）の後について行く` is defined as "tread in someone's footsteps"/"tread in the steps of", and `～の後について行く` is listed as "follow" and classifies this usage with `に` as being transitive (`【他動】`). 

Here I do think that `ついて` and `行く` being separate when used with `に` is
likely. In a separate listing, you can see `（人）のすぐ後について行く` as "follow someone
to heel". Here I think the `ついて` means "attach", so maybe in this usage it
might tend to be used to indicate following closely to someone.

* * *

[This page](http://www.practical-japanese.com/2011/06/blog-post_1618.html)
lists `後をつける` and `跡をつける` together, so I think the two might sometimes be
interchangeable ([this page](http://alphatxt.cocolog-
nifty.com/blog/2009/04/post-0fbb.html) says `後をつける` is used for
shadowing/tailing and `跡をつける` leaving traces, so maybe one might be more
appropriate than the other depending on context, though it seems
[Daijirin](http://dic.yahoo.co.jp/dsearch?enc=UTF-8&p=%E3%81%82%E3%81%A8%E3%82%92%E3%81%A4%E3%81%91%E3%82%8B&dtype=0&dname=0ss&stype=0)
and
[Daijisen](http://dic.yahoo.co.jp/dsearch?enc=UTF-8&p=%E3%81%82%E3%81%A8%E3%82%92%E3%81%A4%E3%81%91%E3%82%8B&dtype=0&dname=0na&stype=0)
both include "shadow/tail" as one of the meanings for `跡をつける`.)

That page includes `**こっそりと** 後を追うこと` in the definition so it does seem that
`後をつける`/`跡をつける` also include the nuance of stealthy/secret pursuit. This seems
to also be backed up in the second [Daijirin definition for
`跡をつける`](http://dic.yahoo.co.jp/dsearch?enc=UTF-8&p=%E3%81%82%E3%81%A8%E3%82%92%E3%81%A4%E3%81%91%E3%82%8B&dtype=0&dname=0ss&stype=0):
`後ろからひそかについて行く` ("Stealthily follow from behind").

The fact that definition includes `について行く` also might mean `について行く` doesn't
include the nuance of stealth.

* * *

[This Chiebukuro
page](http://detail.chiebukuro.yahoo.co.jp/qa/question_detail/q119473541) also
seems to back up that the Kanji for `ついていく` is `付いていく`, so I think they both
use `付`, though I think it's frequently written with Hiragana.
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answer
Rather than looking at the verb つく, this may be best explained by focussing on
あと. Compare

```

    母の後についていく
    母の跡をついていく
    
```

The あと from the first example is a spatial or temporal following, whereas the
あと from the second example is more like a metaphor for a physical following,
as in 足跡「あしあと」»footprints«, indicating an emotional following. That the second
あと is sometimes written with 後, rather than 跡, is as subtle a decision as
distinguishing between 診る and 看る (or 見る and 観る), with a preferred 漢字 for a
specific context, but no clear right and wrong. In order to be able to
distinguish between あと as spatial/temporal concept (written 後) and as physical
concept (written 跡), however, the preferred 漢字 for your fourth example would
be 跡.

Lastly, it shouldn't be upsetting that the verb ついていく allows both intransitive
and transitive uses.

The meaning of "follow", is thus spatial/temporal in 後についていく and physical (or
by metaphor emotional) in 後をついていく.

Here the difference between spatial and physical is a matter of distance/time.
Following spatially could mean following the same path, no matter when or at
what speed. Following physically mean as best as one can at the same speed at
the same time, with as little distance as possible.

I hope this answers at least part of your question.

EDIT: In response to your edit, I will write some more about つく, ついていく, and
つける. After some discussion with the locals, I think it is best to think of
ついていく in

```

    後についていく
    後をついていく
    
```

as two different verbs. Let me explain.

The ついていく in

```

    後をついていく
    
```

is thought of as independent transitive verb, taking 後 as object.

The ついていく in

```

    後についていく
    
```

is thought of as a compound of the intransitive verb つく and 行く, where 後に is
the indirect object (is this the correct term?) for つく.

One can say

```

    後をつける
    
```

because つける is a transitive verb, but not

```

    後をつく
    
```

because つく is intransitive...

P.S. The 漢字 for つく is in all cases 付.
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answer