How does pranayam facilitate activation of the kundalini?

Usually breathing occurs for about an hour through the right nostril. Then for two to four minutes through both the nostrils. Then again through the left nostril for an hour and later for two to four
minutes through both. Then again through the right nostril. This cycle goes on continuously. By practising anulom-vilom pranayam one gradually gets used to breathing through both nostrils.

How to gain control over mind by practicing pranayam?

With practice of pranayam one gets used to concentrating on breathing. Since one gets used to concentrating on breathing, when it becomes rapid one immediately becomes aware of it. This implies that indirectly one becomes aware that either thoughts have increased or emotions are being aroused.

How does tratak help in increasing ones sattvikta?

Muscles of the eyelids and respiration are supplied by both the voluntary and involuntary nervous systems. If one makes efforts to decrease voluntary
movements of these muscles by doing tratak and pranayam then their involuntary movements too can gradually be decreased and brought under control.

What is the motive behind enduring pain in Hathayoga?

Endurance of pain increases mental strength and detachment towards worldly life and therefore the mind turns to Spirituality. By experiencing unhappiness destined actions are completed. However the real motive behind practising Hathayoga is not limited to merely bringing about an increase in the tolerance of the mind but to activating the kundalini.