I have heard that Japanese adverbs usually occur along with particles in a
sentence. Also, I perhaps simplistically consider particles to be the analog
of prepositions in English.

In English it seems that there are often two ways to convey the meaning of an
adverb, one with a preposition and one without: For example, "She sang
enthusiastically." or "She sang with enthusiasm."

With that background, my question is whether it is reasonable to say that
Japanese usually uses the "with enthusiasm" form of modifying verbs rather
than the "enthusiastically" form?

This question and hypothesis are mostly for pedagogical reasons, and for the
common case rather than a precise description of the language.
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2014-11-16T19:26:18.083
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grammar particles adverbs
Adverbs in Japanese versus English
There's at least three types of words in Japanese that can be called "adverbs"
in English grammatical terms. There's the 〜く from of what are called
[形容詞]{けいようし} "keiyoshi" or often in teaching "i-keiyoshi" (e.g., [楽]{たの}しく)
and there are [形容動詞]{けいようどうし} "keiyoudoushi" or in teaching it to foreigners
"na-keiyoushi" plus に (e.g., [具体的]{ぐたいてき}に).

The pattern you describe where the "adverb" is made by tacking a particle onto
the word in the manner of a prepositional phrase applies to the "keiyoudoshi"
style of forming "adverbs". Here you put "ni" onto the end of the adjectival
form of the word to get an "adverb".

For example:

> [具体的]{ぐたいてき}に[表]{あらわ}す

gutaiteki-ni arawasu

= to express something concretely.

The pattern for i-keiyoushi is that the "adverb" is a conjugation of the root
adjective. Thus for the keiyoushi "tanoshii" meaning fun, the adverbial form
is "tanoshiku"

> [彼]{かれ}は[楽]{たの}しく[歌]{うた}う。

kare ha tanoshiku utau.

= he sings enjoyably.

There's also words that don't conjugate at all to be adverbs. e.g., ゆっくり
(yukkuri) and ときどき (tokidoki) and さっと (satto).

* * *

There's some disagreement as to whether "adverb" best captures a category of
Japanese words so I've put in quotes.
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answer
Your example may be translated as 彼女は[熱狂的]{ねっきょうてき}に歌った or something similar.
This 熱狂的に with a particle-like ending に is actually an adverbial form of a na-
adjective 熱狂的な, and is very properly translated as "enthusiastically".

Otherwise you can say it like 彼女は熱狂をもって歌った, where 熱狂をもって resembles "with
enthusiasm" in appearance, with "with" being sometimes translated as をもって.
However this kind of wording sounds fairly literary and less versatile, and is
hardly heard in colloquial speech.
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