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Lomb Scargle Periodogram

An astronomical time series data is a series of discrete events with either an even or uneven sampling rate. This can be expressed mathematically as a pointwise product between a continuous signal with a rectangular function and a dirac comb.  The Discrete Fourier transform of this results in the convolution of the three functions in the omega space. The square of the Fourier transform is proportional to the power of each periodic signal in the time series or more accurately: Scargle showed that this is equivalent to fitting the time series data with the model  using a least square fitting. The resulting Chi square for each frequency is indicative of the power that the signal contributed to the time series. Essentially the difference between the absolute difference between the chi square assuming non variability and the chi square assuming a sinusoidal variation in the time series is equivalent to the Power, or the classical periodogram given by Lomb. Scargle’s Method of Calcu...
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Virtual Work and d'Alembert Principle

Previously, we learnt the advantage of using generalized coordinates over traditional coordinate spaces while deriving equations of motion. Here, we build on the idea by introducing Virtual Displacements and Virtual Work. We will use these ideas to arrive at d'Alembert Principle, named after the French physicist and mathematician Jean le Rond d'Alembert.

Virtual Displacement

A virtual displacement of a system is an instantaneous, infinitesimal variation in the positions of the constituents of the system in accordance with the constraints. Virtual displacement is denoted by $\delta q_i$. This is by no means a real displacement, but an imaginary one. We "test" the system by nudging it a bit and see how it behaves. Since the constraints and the external forces may change with time, we keep it a constant while we virtually displace it. 

Virtual Displacement has much more nuances to it, but the definition above is good enough for all intents and purposes. But you can learn more about it from [1].

Virtual Work



d'Alembert Principle





References
1. On Virtual Displacement and Virtual Work in Lagrangian Dynamics - Ray, Shamanna 
    arXiv
2. A Student's Guide to Lagrangian and Hamiltonian - Hamill
3. Classical Mechanics - Goldstein, Poole, Safko
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