One of the main Pilates principles is centring which means activating the core muscles before and during exercises in order to gain stability for quality movement.
❗Core muscles are transversus abdominis (TvA), diaphragm, abdominal oblique muscles, multifidus (deep back extensors) and pelvic floor muscles.

➡️When core muscles are engaged, they provide the lumbopelvic stability by increasing the pressure in abdominal cavity.
➡️ When you try to keep your pelvis from rotating and resist the rotational trunk movements you are engaging multifidus and abdominal obliques.
Obliques work as well when you draw ribs to hips.
➡️ When you inhale, diaphragm slides down and increases the intraabdominal pressure, especially on TvA and pelvic floor. When you exhale your diaphragm relaxes, it slides up but at the same time it activates TvA as it helps with active expiration.

⚠️ But TvA and the pelvic floor muscles also need to be active during inspiration, so how to activate them?
1️⃣ Lie in supine with your knees bent. Do a few pelvic rocks to find a neutral position for your spine. Relax your abs and breathe normally. Place your index fingers on your front hip bones (the pointy part on both sides of the pelvic crest). Ideally, place the fingers directly on the skin. Move the fingers 5cm inwards and then 5 cm downwards. You should feel relaxed abdominals that move only as you breathe. Then as you exhale try to pull front hip bones together or imagine that you have to get into tight jeans. You should feel a very gentle muscle contraction under your fingers. If you feel firm muscles, you have engaged the whole abs. Aim for more subtle, barely noticeable contraction.

2️⃣ Keep your fingers in the same place. Focus on the pelvic floor. Exhale and engage it, pull it inwards, and you are going to feel the same very subtle muscle contraction. TvA and pelvic floor are like best friends, they work together. If you have trouble recognizing the pelvic floor muscles, try to imagine that you need to get to the bathroom but you can still hold it. Those are the muscles. Once you feel them, try to pull them inwards.

NB! In Pilates practice breathing is focused on ribcage not diaphragm.