In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics. By contrast, analytical mechanics uses scalar properties of motion representing the system as a whole—usually its total kinetic energy and potential energy—not Newton's vectorial forces of individual particles. A scalar is a quantity, whereas a vector is represented by quantity and direction. The equations of motion are derived from the scalar quantity by some underlying principle about the scalar's variation. Analytical mechanics takes advantage of a system's constraints to solve problems. The constraints limit the degrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as generalized coordinates. The kinetic and potential energies of the system are expressed using these generalized coordinates or momenta, and the equations of motion can be readily set up, thus analytical mechanics allows numerous mechanical problems to be solved with greater efficiency than fully vectorial methods. It does not always work for non-conservative forces or dissipative forces like friction, in which case one may revert to Newtonian mechanics. Two dominant branches of analytical mechanics are Lagrangian mechanics (using generalized coordinates and corresponding generalized velocities in configuration space) and Hamiltonian mechanics (using coordinates and corresponding momenta in phase space). Both formulations are equivalent by a Legendre transformation on the generalized coordinates, velocities and momenta, therefore both contain the same information for describing the dynamics of a system. There are other formulations such as Hamilton–Jacobi theory, Routhian mechanics, and Appell's equation of motion. All equations of motion for particles and fields, in any formalism, can be derived from the widely applicable result called the principle of least action. One result is Noether's theorem, a statement which connects conservation laws to their associated symmetries. Analytical mechanics does not introduce new physics and is not more general than Newtonian mechanics. Rather it is a collection of equivalent formalisms which have broad application. In fact the same principles and formalisms can be used in relativistic mechanics and general relativity, and with some modifications, quantum mechanics and quantum field theory. Analytical mechanics is used widely, from fundamental physics to applied mathematics, particularly chaos theory. The methods of analytical mechanics apply to discrete particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom. The definitions and equations have a close analogy with those of mechanics. (Wikipedia).
B01 An introduction to numerical methods
Most differential equations cannot be solved by the analytical techniques that we have learned up until now. I these cases, we can approximate a solution by a set of points, by using a variety of numerical methods. The first of these is Euler's method.
From playlist A Second Course in Differential Equations
Understand The Work Equation!! (Mechanics)
#Physics #Mechanics #Engineering #TikTok #NicholasGKK #Shorts
From playlist General Mechanics
Linear differential equations: how to solve
Free ebook http://bookboon.com/en/learn-calculus-2-on-your-mobile-device-ebook How to solve linear differential equations. In mathematics, linear differential equations are differential equations having differential equation solutions which can be added together to form other solutions.
From playlist A second course in university calculus.
Fill In The Blank (Dynamics/Friction)
#Physics #Dynamics #Engineering #TikTok #NicholasGKK #shorts
From playlist Mechanical Engineering
In this talk, we will define elliptic curves and, more importantly, we will try to motivate why they are central to modern number theory. Elliptic curves are ubiquitous not only in number theory, but also in algebraic geometry, complex analysis, cryptography, physics, and beyond. They were
From playlist An Introduction to the Arithmetic of Elliptic Curves
A01 Introduction to linear systems
An introduction to linear sets of ordinary differential equations.
From playlist A Second Course in Differential Equations
A05 Explanation of the matrix format of a system of linear differential equations
Explanation of the matrix notation used in systems of linear differential equations.
From playlist A Second Course in Differential Equations
Mission Impossible: Constructing Charles Babbage's Analytical Engine (d)
The Neukom Institute at Dartmouth presents: Mission Impossible: Constructing Charles Babbage's Analytical Engine Given by Doron Swade on May 8th at Dartmouth College Computing is widely viewed as a phenomenon of the electronic age. The mechanical prehistory of computing tends to be seen a
From playlist AnalyticalEngine
Google Workshop on Federated Learning and Analytics: Breakout Session Closing Summaries
A Google TechTalk, 2020/7/30, presented by Peter Kairouz, Marco Gruteser, and all Breakout Session Leaders ABSTRACT:
From playlist 2020 Google Workshop on Federated Learning and Analytics
Some questions around quasi-periodic dynamics – Bassam Fayad & Raphaël Krikorian – ICM2018
Dynamical Systems and Ordinary Differential Equations Invited Lecture 9.2 Some questions around quasi-periodic dynamics Bassam Fayad & Raphaël Krikorian Abstract: We propose in these notes a list of some old and new questions related to quasi-periodic dynamics. A main aspect of quasi-per
From playlist Dynamical Systems and ODE
Marco SERONE - A look at \phi^4_2 using perturbation theory
https://indico.math.cnrs.fr/event/2435/
From playlist Workshop “Hamiltonian methods in strongly coupled Quantum Field Theory”
Classical curves | Differential Geometry 1 | NJ Wildberger
The first lecture of a beginner's course on Differential Geometry! Given by Prof N J Wildberger of the School of Mathematics and Statistics at UNSW. Differential geometry is the application of calculus and analytic geometry to the study of curves and surfaces, and has numerous applications
From playlist Differential Geometry
The greatest machine that never was - John Graham-Cumming
Computer science began in the '30s ... the 1830s. John Graham-Cumming tells the story of Charles Babbage's mechanical, steam-powered "analytical engine" and how Ada Lovelace, mathematician and daughter of Lord Byron, saw beyond its simple computational abilities to imagine the future of co
From playlist Inventions that Shaped History
Digital Fluid Mechanics Laboratory: Sudden Expansion Pressure Loss
This work details the assignment of a digital, fluid mechanics laboratory to a class of undergraduate engineering students at Duke University. In this laboratory, students are asked to numerically simulate flow through a sudden expansion. The primary learning objective of this laboratory i
From playlist Wolfram Technology Conference 2022
Google Keynote: Federated Learning & Federated Analytics-Research, Applications, & System Challenges
A Google TechTalk, presented by Hubert Eichner, Francoise Beaufay, Ravi Kumar & Peter Kairouz, 2021/11/9 ABSTRACT: An overview of federated analytics applications and algorithms, federated learning applications and algorithms, and how we build an infrastructure and scale it. About the S
From playlist 2021 Google Workshop on Federated Learning and Analytics
Semi-Classics, Adiabatic Continuity and Resurgence in Quantum Theories (Lecture 1) by Mithat Unsal
PROGRAM NONPERTURBATIVE AND NUMERICAL APPROACHES TO QUANTUM GRAVITY, STRING THEORY AND HOLOGRAPHY (HYBRID) ORGANIZERS: David Berenstein (University of California, Santa Barbara, USA), Simon Catterall (Syracuse University, USA), Masanori Hanada (University of Surrey, UK), Anosh Joseph (II
From playlist NUMSTRING 2022
The Skellam Mechanism for Differentially Private Federated Learning
A Google TechTalk, presented by Ken Liu (with Naman Agarwal and Peter Kairouz), at the 2021 Google Federated Learning and Analytics Workshop, Nov. 8-10, 2021. For more information about the workshop: https://events.withgoogle.com/2021-workshop-on-federated-learning-and-analytics/#content
From playlist 2021 Google Workshop on Federated Learning and Analytics
B27 Introduction to linear models
Now that we finally now some techniques to solve simple differential equations, let's apply them to some real-world problems.
From playlist Differential Equations
Instabilities in fluid mechanics and convex integration - Francisco Mengual
Short Talks by Postdoctoral Members Topic: Instabilities in fluid mechanics and convex integration Speaker: Francisco Mengual Affiliation: Member, School of Mathematics Date: September 28, 2021
From playlist Mathematics