Core exercises whilst cycling

My lower back muscles need building up, I have been advised to do core exercises on and off the bike. The 'off the bike' ones are no problem, but does anyone know of any 'on the bike' core exercises to do whilst cycling?
May the wind always be at your rear!
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I have Arthritis which affects the sacro-illiac area of my back (just above the tailbone) and get the shooting sciatic pain too. I am currently seeing a physio who has given me standard strengthening excercises for off the bike which seem to be working. I am not quite sure of any to do on the bike but would also be interested to know if there are any.
I guess that would fit....hard to keep balance though.... I have a 45 minute commute in the car and try to sit bolt upright for large parts of the drive without supporting myself on the seat. It hurts but I think it is helping as the car creates instability which you have to work against.
Otherwise some weight training might help, but be very careful and get some professional advice before starting.
I do dead lifts and squats with free weights, which builds lower back strength. However, I've been doing this for years and have no previous back trouble; and I avoid doing massive miles on the bike on my weight training days.
Also make sure to even up any lower back exercises with an abb workout.
I commute 21 miles every week day, but again I've been doing this for a while.
- Standing starts
- Sprints from a low speed. Both seated and out of the saddle
- Sprints from speed
Other than that, simply riding harder is usually enough for bike riders