What kind of weapons are the cavalry wielding on the Ludovisi sarcophagus?

The cavalry on the Ludovisi sarcophagus are wielding lances. The Romans are wielding them with an overarm grip (heel of hand to point, thumb to the butt), and a barbarian cavalryman on one end is wielding a lance with an underarm grip (thumb towards point). Images showing the side and both ends are at https://commons.wikimedia.org/wiki/Category:Grande_Ludovisi_sarcophagus

The image you have linked:

https://upload.wikimedia.org/wikipedia/commons/f/ff/Sarcofago_romano_1.jpg

is the Portonaccio sarcophagus (AD190-200) rather than the Ludovisi sarcophagus (which is the latest of the group, c. 250). Still, the cavalry are wielding lances, with an overarm grip. Many of the lances are broken (not as a deliberate part of the art; the sculpture is damaged), which makes them hard to identify.

The lance was the main weapon for hand-to-hand fighting for the Roman cavalry (and for many other cavalry around the world). Similarly lance-wielding cavalry can be seen on Trajan's Column, 100 and 150 years earlier than these sarcophagi:

http://www.trajans-column.org/wp-content/flagallery/scenes126-148/145_cvi_cxlv_8296-composite-web.jpg (note that the lances are missing from the artwork, but the soldier's right hands in the usual overarm lance grip position)

http://www.trajans-column.org/wp-content/flagallery/scenes126-148/142-144_civ-cv_cxlii-cxliii_8290-composite-web.jpg (lances held in the underarm grip)

http://www.trajans-column.org/wp-content/flagallery/scenes22-25/24_6977comp24-web.jpg (underarm and overarm next to each other)

http://www.trajans-column.org/wp-content/flagallery/scenes31-38/38-xxviii-xxxvii_7047-web.jpg (lances gripped underarm, and swords are worn at the right hip)

Modern artwork: https://i.pinimg.com/564x/05/d0/2e/05d02e1636dcf2eb2f1050c6c55f7e5e.jpg

Of course, the spear was a very common weapon for infantry as well. When infantry use spears, the cavalry had better use them too, or they will have no small difficulty when fighting infantry.