I am interested to know where in the Vedas or Upanishad is the philosophy of Nishkama Karma i.e., giving up fruits of actions, mentioned?
What is the origin of Karma Yoga as we know in the Gita?
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Well, I'm aware of at least two instances in Vedas which have the ideas of carrying on doing what you're doing without selfish desire for specific results. In other words, selfless action or Karma Yoga.
Rig Veda 5.46.1:
हयो न विद्वान् अयुजि स्वयं धुरि तां वहामि प्रतरणीमवस्युवम् ।
नास्या वश्मि विमुचं नावृतं पुनर्विद्वान् पथः पुरएत ऋजु नेषति ॥
hayo na vidvān ayuji svayaṃ dhuri tāṃ vahāmi prataraṇīmavasyuvam ।
nāsyā vaśmi vimucaṃ nāvṛtaṃ punarvidvān pathaḥ puraeta ṛju neṣati ॥
"Like a wise horse, the omniscient one joined himself to the "yoke", and I carry the yoke that gives salvation and protection. I do not desire to get rid of it nor do I desire coming back. The Knower of the Path who goes ahead will lead me straight."
Ishavasya Upanishad 1:
ईशावास्यमिदं सर्वं यत्किञ्च जगत्यां जगत् । तेन त्यक्तेन भुञ्जीथाः मा गृधः कस्यस्विद्धनम् ॥
īśāvāsyamidaṃ sarvaṃ yatkiñca jagatyāṃ jagat ।
tena tyaktena bhuñjīthāḥ mā gṛdhaḥ kasyasviddhanam ॥
"The lord dwells within all this, whatever stirs in this universe. Hence, enjoy by giving up and do not covet anybody's wealth."
So this idea of being neutral towards everything in the universe is a very ancient idea in Hinduism.