Adult horses can have very close bonds and when separated act out like a mare and foal. However, with training and consistency, the horse being ridden and the horse being left at home can learn to relax over time.
By sending your filly to the trainer, you will be "weaning" the pair. This should help when the filly returns. Usually, the horse being handled is more relaxed than the horse being left behind. It is actually a good thing to ride the horses separately. If the horse being ridden is overly excited, you can work the horse at a long trot or canter in a circle. Once the horse relaxes, let the horse stop and rest. However, if the horse's attention goes back to the other horse, continue riding at a long trot or canter. Let the horse rest once its attention is directed to you, not its pasture mate. Remember to stay calm during this process. This is not meant to be punishment, simply redirecting the horse's attention and rewarding the horse for focusing on the ride rather than the other horse.
For the horse left behind, you want to ensure the horse stays safe. Figure out if the horse stays calmer in a stall with hay or a run-in where he can see the other horse. Try to provide an incentive like feed or hay to distract the horse from the missing pasture mate. Make sure you ride the other horse long enough to let the pasture horse settle down.
You may need to reduce the time these two horses spend together, either by dividing your pasture area or separating them with stalls. Finally, consider taking some lessons at home so that a professional can help you through some of the initial retraining of your horses to ride separately. You may ultimately decide you only have time for one horse, and choose to sell the other horse. However, it may be possible to work out the behavioral aspect of owning and riding two horses with time and consistency.
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