Why did Sita let Ravana abduct her when she could lift playfully Shiva's bow ?

There are 3 issues mentioned in the question.

1. Sri Rama broke Shiva’s bow, which was playfully lifted by Sita.
2. Ravana could not lift Shiva’s bow.
3. Sita lifts that great bow

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a) According to the narration available in Bala Kanda, it was Shiva’s bow that Shri Rama had broken.  And, as per the narration in Ayodhya Kanda, Shri Rama had broken Varuna’s bow.

Actually, there has not been much difference between what Sita says to Anasuya in Ayodhya Kanda about the great bow that was broken by Shri Rama and that was mentioned in Bala Kanda, except for

• a few shlokas added in the 66th Sarga of Bala Kanda, after shloka 8, and
• the interpolated portion in Bala Kanda (2nd Part of 74th Sarga to 76th Sarga of Bala Kanda) of Parashurama and Shri Rama conflict mentioned while Dasaratha was returning to Ayodhya after the marriage
• This interpolated portion was added to project Parashurama as the incarnation of Vishnu and greatness of Vaishnava Bow

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a) 118th Sarga of Ayodhya Kanda, where Sita mentions that Shri Rama broke a great bow of Varuna

When Anasuya asks Sita to narrate her marraige details, Sita informs her that Shri Rama broke the great bow of Varuna and thus, their marriage was performed. Here, the actual story of Shri Rama breaking the great bow was not narrated in detail.

महा यज्ने तदा तस्य वरुणेन महात्मना |

दत्तम् धनुर् वरम् प्रीत्या तूणी च अक्षय्य सायकौ || (Ayodhya Kanda 118th Sarga 39th Sloka)

"In ancient days, Janaka on the occasion of a great sacrifice, received with affection from Varuna (वरुणेन महात्मना) the rain-god, an excellent bow with two quivers that should never lack arrows."

असंचाल्यम् मनुष्यैः च यत्नेन अपि च गौरवात् |

तन् न शक्ता नमयितुम् स्वप्नेषु अपि नर अधिपाः || (Ayodhya Kanda 118th Sarga 40th Sloka)

"That bow was so heavy in weight that no man could lift it up nor any of the kings were bale to bend it even in their dreams."

b) Bala Kanda Narration of the great bow

The story of King Janaka himself receiving a Great bow from Varuna, the god of Water/rain-god, was altered with the story King Janaka's ancestor Nimi receiving the great bow.

देवरात इति ख्यातो निमेः षष्ठो महीपतिः |
न्यासोऽयं तस्य भगवन् हस्ते दत्तो महात्मना || १-६६-८

Oh, godly sage, there was a king renowned as Devaraata, sixth one from Nimi, [the originator of our lineage,] and this bow was handed down to him for custodial care by the great Soul.

——

Here lies the crux of the issue.

In this shloka it was not stated that it was Shiva who had given that bow to Nimi.

It was only stated that दत्तः तस्य हस्ते न्यासः महात्मना - given as custodial care in his hand by a great soul.

Please note that it did not say Shiva that has given, but only says a great soul - महात्मना

The great soul can be Varuna, as in Ayodhya Kanda also he was referred to as वरुणेन महात्मना .

——

Subsequent 4 shlokas were added by someone else to project this bow as that of Shiva’s :

दक्षयज्ञवधे पूर्वं धनुरायम्य वीर्यवान् |
रुद्रस्तु त्रिदशान् रोषात् सलीलमिदमब्रवीत् || १-६६-९

"Once, during the devastation of the Vedic-ritual of Daksha Prajapati, the mettlesome god Rudra, rancorously outstretching the bowstring of this bow, said this to all gods, superciliously...

यस्माद्भागार्थिनो भागं नाकल्पयत मे सुराः |
वरांगाणि महार्हाणि धनुषा शातयामि वः || १-६६-१०

" 'Oh, gods, whereby you have not apportioned my portion of oblations in Vedic-ritual s, as I am also the desirer of such portion, thereby I will shred the highly revered heads of yours with this bow...' So said Shiva to gods.

ततो विमनसः सर्वे देवा वै मुनिपुंगव |
प्रसादयन्ति देवेशं तेषां प्रीतोऽभवद्भवः || १-६६-११

"Then, oh, best saint Vishvamitra, all gods are truly dismayed, and on their supplicating, Bhava, namely Shiva, the God of Gods is gladdened...

प्रीतियुक्तस्तु सर्वेषां ददौ तेषां महात्मनाम् |
तदेतद्देवदेवस्य धनूरत्नं महात्मनः || १-६६-१२
न्यासभूतं तदा न्यस्तमस्माकं पूर्वजे विभो |

And that Sublime Soul Shiva gladly gave that bow to all of the great souled gods, and oh, godly saint, then those great souled gods gave this gem of a bow of Shiva, the God of Gods, to our ancestor Devaraata, for custodial care...

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If we omit these 4 shlokas,ie., 9th to 12th shlokas in the 66th Sarga of Bala Kanda of Ramayana, then there would be no confusion. Both narrations state that a great bow was kept with Janaka/Nimi by Varuna.

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Further, we have to omit the conflict between Parashurama and Shri Rama, which is an interpolated story added to elevate the status of Vaishnava bow and demeans the Shiva’s bow.

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Hence, Shri Rama had broken the Great bow of VARUNA, but not that of Shiva.

It is to remembered that Varuna was very powerful God in Vedic era, which continued upto Mahabharata period.

Even Arjuna received his Gandiva of Varuna, through Agni only, and after the war it was returned to Varuna.

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b) Seetha further narrates the story of her marriage to Anasuya. Then Seetha relates that story to Anasuya in detail.

पति सम्योग सुलभम् वयो दृष्ट्वा तु मे पिता |

चिन्ताम् अभ्यगमद् दीनो वित्त नाशाद् इव अधनः || २-११८-३४

"Seeing my age to be such, when union with a husband can be easily had, my father (King Janaka) fell a prey to anxiety like an indigent man, feeling miserable by the loss of all his possessions."

तस्य बुद्धिर् इयम् जाता चिन्तयानस्य सम्ततम् |

स्वयम् वरम् तनूजायाः करिष्यामि इति धीमतः || २-११८-३८

"After reflecting thus deeply, the thought came to him, 'I shall inaugurate a Svayamvara, a process of self-choosing marriage, for my daughter."

तद् धनुः प्राप्य मे पित्रा व्याहृतम् सत्य वादिना |

समवाये नर इन्द्राणाम् पूर्वम् आमन्त्र्य पार्थिवान् || २-११८-४१

"My truthful father called all the princes first and informed them in a meeting about the bow to be lifted."

तच् च दृष्ट्वा धनुः श्रेष्ठम् गौरवाद् गिरि सम्निभम् |

अभिवाद्य नृपा जग्मुर् अशक्ताः तस्य तोलने || २-११८-४३

"Seeing that excellent bow, resembling a mountain in weight, and being unable to lift it up, the princes offered salutation to it and went away."

सुदीर्घस्य तु कालस्य राघवो अयम् महा द्युतिः |

विश्वामित्रेण सहितो यज्नम् द्रष्टुम् समागतः || २-११८-४४

लक्ष्मणेन सह भ्रात्रा रामः सत्य पराक्रमः |

"After a very long time, this Rama born in Raghu dynasty with a great splendour possessing a true valour, along with his brother Lakshmana together with a sage Visvamitra, came to witness a sacrifice."

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So there was no mention about Ravana in this narration. So Ravana did not attend to Sita's Swayamvara. And, hence the question of his inability to lift Shiva’s bow does not arise.

c) Finally, Ramayana does not mention about Sita lifting the Great bow. It is an interpolated story, created by film directors of Telugu Film industry.