madhusūdana (gītā 2.10)

In the gītā, bhagavān śrī kṛṣṇa speaks his first words in the second verse of the second chapter. In the previous verse, vyāsa has introduced this with “viṣīdantam idaṁ vākyam uvāca madhusūdanaḥ”. Of all the synonyms of viṣṇu, vyāsa has chosen the epithet madhusūdana. Why ? ānandagiri quotes the viṣṇusahasranāma bhāṣya by ādi śaṅkarācārya (name #73) --

madhunāmānam asuraṁ sūditavān iti madhusūdanaḥ .

madhusūdana is the one who destroyed (sūdana) the demon named madhu. A slightly more philosophical explanation is given in the gūḍhārtha-dīpikā --

madhusūdana iti svayaṁ duṣṭa-nigraha-kartā .

madhusūdana is that person who destroys all evil. This is fitting because arjuna is described as the despondent one (viṣīdan), and it is this unfitting behaviour that kṛṣṇa seeks to destroy. However, the root sūd does not solely mean destruction. It also means purification. After all, destruction and purification are two sides of the same coin. With this in mind, we can look at the tātparya-bodhinī, which proffers a few different explanations:

(1) madhvākhyam asuraṁ sūdayati iti madhusūdanaḥ, (2) sarveṣāṁ dehe madhuvad iṣṭatvāt madhuḥ ahaṁkāraḥ, tam ātmaprakāśanena sūdayatīti madhusūdanaḥ, (3) madhuḥ svarūpānandaḥ, taṁ svāparokṣavatāṁ sūdayati sphorayati iti madhusūdanaḥ

svāmi śaṅkarānanda first gives the standard explanation – he is the destroyer of the demon named madhu. But, who is this madhu ? madhu is any sweet entity, like honey, which is symbolic of desire. madhu represents our ego (ahaṁkāra), which is the very cause of desire. So, madhusūdana is the person who destroys that ego, through the knowledge of the self. Finally, after showing the destructive aspect, svāmī shows us the constructive aspect of purification. madhu is a sweet entity, representing the true, pure ānanda (bliss). madhusūdana is the person who shows us that true nature of the self, through direct experience.

Not only are these explanations beautiful, they are appropriate to the context. In this drama, kṛṣṇa’s purpose is to destroy the evil despondency, arising out of ego, and bringing about a direct experience of the self, which is of the nature of true ānanda.