Nostalgia is definitely a modern word. It was first coined by Johannes Hofer in 1688 in his Dissertatio Medica de Nostalgia. About devising the word, he wrote:
(from Nostalgia: Origins and Ends of an Unenlightened Disease, by Helmut Illbruck, p. 5)
However, the construction of desiderium + gen. of the object longed for + a verb that describes how the longing manifested itself is amply attested to in classical Latin.
Parmeno, a slave, runs into Philotis, a courtesan recently returned to Athens after an unhappy sojourn in Corinth:
Edepol, te desiderium Athenarium arbitror, Philotium, cepisse saepe et
tu tuom consilium contempisse
By Pollux, I suppose a longing for Athens often took hold of you, Philotis, and
you regretted your decision [to leave]
Terence, Hecyra (The Mother-in-Law)
In fact, John Barsby (2001) translates this as: I expect you were often homesick …
Cicero, writing to Caelius Rufus in Rome:
Mirum me desiderium tenet urbis, incredibile meorum atque in primus
tui, satietas autem provinciae
An extraordinary longing for the city (Rome) grips me, more than you
would believe, for my family and especially for you, I have had enough
of the provinces
Cicero, Epistulae, 90 (2.11)
Horace speaking to a bailiff who now works and resides in the country but pines for the excitement of the city:
Fornix tibi et uncta popina incutiunt urbis desiderium
The brothel and greasy cook shop excites in you a longing for the city
Horace, Epistulae, 1.14.22
And, of course, Ovid, whose Tristia is nothing if not an extended treatise on the soul-destroying nature of homesickness:
Roma domusque subit desideriumque locorum, quicquid et amissa restat
in urbe mei
Rome, my home, and a longing for its places steals into my thoughts
and whatever remains of me in the city I have lost
Ovid, Tristia, 3.2.21-22
