A strong core is fundamental for good skiing at any ability level. Using a stability ball, also called a Swiss ball or balance ball, is one way to work to improve core strength that doesn't require that much upper body strength. I've used a stability ball in hotel and resort fitness rooms. When I started ski conditioning 7+ years ago, we had a ball at home but I'd never used it for exercise since it was bought at a yard sale as a toy when my daughter was young. It was the second training aid I used at home, after a BOSU.
Regardless of what other exercises I do with a stability ball, I usually do hip curls or hip lifts to get in a little hamstring work. It doesn't matter too much what size ball is available.
BEGINNER CORE | 15 Minute STABILITY BALL Workout for BEGINNERS, 15 min
https://youtu.be/GKa1Kx0uuMg
CORE & MORE Stability Ball WORKOUT - Intermediate, 14 min
This 12-min TRX HIIT workout is pretty advanced. Meaning it's better for someone who already has a fairly strong upper body. By the time my personal trainer introduced this routine, I'd been working with her and the TRX for over five years. Of course, the advantage of using TRX is that it's possible to start at an easier level and increase the amount of body weight being applied as strength increases simply by placing the feet a slightly different distance from the anchor point.
There are 12 exercises, each done for 45 seconds, with 15 second rest periods The video provides timing for three sets, but I normally just do one set and then move on to different exercises. Keeps me from getting bored.
36 Minute TRX Full Body Workout #9, 45 min (12 x 3)
https://youtu.be/4_ntfCMCRFI
Face anchor, 2 straps: squat, or squat with arm raise, or squat jump; can mix
Feet in straps, chest towards mat: body saw, or crunch, or alternate
Heels in straps, back towards mat: pull thru
Face anchor, 1 strap: single arm row
Face anchor, 1 strap: single arm row, other arm
Face anchor, 2 straps: single leg squat
Face anchor, 2 straps: single leg squat, other leg
Face anchor, 1 strap: torso rotation, both hands on handle
Away from anchor, 1 strap: single arm push up
Away from anchor, 1 strap: single arm push up, other arm
Face sideways, 1 strap: hip drop, foot closer to anchor in front
Face sideways, 1 strap: hip drop, foot closer to anchor in front, other side