This dissertation details our experiments on studying the Brownian motion of an optically trapped microsphere with ultrahigh resolution, and cooling of its motion towards the quantum ground state. We have trapped glass microspheres in water, air and vacuum with optical tweezers. We developed a detection system that can monitor the position of a trapped microsphere with Angstrom spatial resolution and microsecond temporal resolution. We studied the Brownian motion of a trapped microsphere in air over a wide range of pressures. We measured the instantaneous velocity of a Brownian particle. Our results provide direct verification of the Maxwell-Boltzmann velocity distribution and the energy equipartition theorem for a Brownian particle. For short time scales, the ballistic regime of Brownian motion is observed, in contrast to the usual diffusive regime. We are currently developing a new detection system to measure the instantaneous velocity of a Brownian particle in water. In vacuum, we have used active feedback to cool the three center-of-mass vibration modes of a trapped microsphere from room temperature to millikelvin temperatures with a minimum mode temperature of 1.5 mK, which corresponds to the reduction of the root mean square (rms) amplitude of the microsphere from 6.7 nm to 15 pm for that mode. The mean thermal occupation number of that mode is reduced from about 6.8$\times 10^8$ at 297 K to about 3400 at 1.5 mK. / text

Η παρούσα εργασία αφορά στη μελέτη Χαμιλτώνιων συστημάτων Ν μη γραμμικών ταλαντωτών, όπως είναι αυτό των Fermi Pasta και Ulam (FPU), με στόχο την βαθύτερη κατανόηση της δυναμικής των σχεδόν-περιοδικών τροχιών και του ρόλου των αντίστοιχων τόρων στο χώρο φάσεων, καθώς αυξάνουμε την ενέργεια Ε και τον αριθμό βαθμών ελευθερίας Ν του συστήματος. Το βασικό μας αποτέλεσμα είναι ότι υπάρχουν τόροι χαμηλής διάστασης, που προκύπτουν από τη συνέχεια των αντίστοιχων του γραμμικού συστήματος, οι οποίοι ευθύνονται για τις FPU επαναλήψεις και εμποδίζουν την ισοκατανομή της ενέργειας μεταξύ όλων των κανονικών τρόπων ταλάντωσης. Αναλύοντας ευστάθεια αυτών των τόρων, μπορέσαμε να δώσουμε μια πληρέστερη ερμηνεία στο Παράδοξο των FPU, συνδέοντας και συμπληρώνοντας έτσι δύο από τις επικρατέστερες ερμηνείες του εν λόγω φαινομένου. / The present work concerns the study of Hamiltonian systems of N nonlinear coupled oscillators, as it is the one by Fermi Pasta and Ulam (FPU), in order to understand the dynamics of quasi-periodic orbits and the role of their corresponding tori in phase space, as we increase the energy E and the number N of the degrees of freedom. Our fundamental result is that there exist tori of low dimension, that come from the continuation of the corresponding tori of the linear system, which are responsible for the FPU recurrences and prevent the system from equipartition of the energy among all normal modes. By investigating the stability of these tori, we achieved to provide a more complete explanation for the FPU paradox, connecting and supplementing in this way two of the most dominant approaches for this paradox.