To really feel a sense of lightness in Ardha Chandrasana (balancing half moon,) you need to engage a lot of muscles. In my own practice, I like to start cuing this pose from the ground up. (And I'm actually serious–you know how in your yoga class teachers say things like "pull your low belly in and up," "press into the mat," "spread your fingers wide," etc.? When I practice by myself I give myself directions like this inside of my head!) 

Give it a try!

1. Press down into the standing leg foot to lift up out of the hip. 
This is the concept of rooting down to rise up. Every action creates and equal and opposite reaction. When you press down into the the earth through your foot, you lift up and out of the hip. This takes pressure out of the hip join, and the hip is instead supported by the muscles of the leg! 

3. Engage the standing leg quadricep to lift the kneecap. 
This is a cue that I also use in Tadasana (mountain pose). This quadricep engagement helps with this lifting out of the hip, while also helping to lengthen the hamstring in the back of the leg (as the agonist muscle). 

3. Pull the low belly in and up. 
This is the cue that is generally used to refer to uddiyana bandha. This is one of the energy locks in yoga that helps us feel light in standing poses, and helps us fly in arm balances and inversions. Think of it as the deep deep core a few inches below the navel. 

4. Squeeze the glutes, especially of the lifted leg!
The lift in the back leg happens primarily from the strong muscles in the center of the body. That means primarily the glutes and the core. Sure, other muscles of the legs can help like the the quads, hamstrings, and calves, but we start from the big central muscles and work outwards. 

5. Flex the lifted foot
In the video I wrote "flex the foot to engage the leg." You can technically engage the leg whether the foot is pointed or flexed. Specifically, if your foot is pointed it encourages engagement in the muscles in the front of the body, which is why you might choose to point the toes in Navasana (boat pose). If your foot is flexed it encourages engagement in the back of the body, which allows greater access in this case to the glute muscles which in my experience are the prime "movers" in lifting the leg in Ardha Chandrasana. 

Try it out and let me know how it goes!