Liquidated Damages (LDs) have to be a genuine pre-estimate of the damages caused by the late completion of a project and clearly defined in the agreed contract. LDs limit the damages claimable for late completion to the amount set in the contract, this can save a late contractor from a massive court claim, but can be applied under the contract without going into the courts (or arbitration).
The prerequisites to apply LDs are (a), a properly calculated completion date taking into account all extension of time claims submitted, the claims can all be rejected but must be considered; and (b) work actually continuing past the authorised completion date. LDs can only be claimed for the actual period of delay at the time they are deducted, not a projected period; this means the total LDs deducted are typically updated each month until actual completion.
To enforce LDs, there has to be a proper extension of time (EOT) clause, if there is no EOT clause (or it is defective), any delay by the client converts the enforceable LDs into an unenforceable penalty; but this opens up the option for a court case claiming damages for delay.
In practice, enforcing LDs usually triggers a fight over EOTs and once the actual extended date for completion is determined (usually by an arbitrator or judge), LDs are then assessed on any time between this date and the actual completion date.
While genuine LDs are typically enforceable, this is a complex area of law and varies in detail between jurisdictions, for more on this see: https://mosaicprojects.com.au/PMKI-TPI-080.php#Process1