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There are actually many conditions that can lead to a left hip burning sensation. It is usually with involvement of muscles, nerves,bones, ligaments or joints. With your symptoms, there is possibility that the cause is MUSCULAR STRAIN or KIDNEY STONE. (Note, small kidney stone which has symptoms similar to what you had described, are sometimes cannot be seen with X-ray, ultrasound may catch it. Try to drink plenty of water for now. ) Other probable cause can include spinal stenosis or an infection of the sciatic nerve of the lower leg- however in such situations the symptoms will usually involve the leg. Bone problems as the cause, usually increases the pain with walking or sitting.
You may apply warm compression, and continue your anti-inflammatory drug for few days. Having occasional sharp pain however is quite alarming, so I would highly recommend to have a doctor's check up and have few investigations done.
Good luck and take care!