What is Analytical mechanics: Definition and 66 Discussions

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.

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  1. R

    Help with this integral on page 34 of Analytical Mechanics by John Bohn

    TL;DR Summary: Am stuck on an integral at the bottom of page 34 Hi - I am working thru (by myself) the small textbook by Bohn on Analytical Mechanics. Its very good but am stuck on Page 34, at the bottom. It concerns the "action" of a simple pendulum - I understand the math concept of action...
  2. D

    I Why L=L(v^2) in inertial reference system?

    Why he said that beacause space's propertiy is the same in both direction, so L=L(v^2), or do I misunderstand him incorrectly? btw this conclusion appears in somewhere like page 5 and its about Galilean principle of relativity.
  3. ronniegertman

    Analytical mechanics- working with Lagrangian and holonomic constraints

    The tutor solved the problem using kinetic spinning energy though I find it very difficult and confusing to do so, therefore, I would like to know if there is a way to solve the problem using effective potential energy, Veff = J2/(2mr2 below is a sketch of the problem
  4. N

    Oscillation with friction - Analytical mechanics

    Hi, I had those exercises and want to know if they're correct. Also, feedback/tips would be great from you, professionals. $$A$$ 1. Let's consider the oscillator with a friction parameter... \begin{equation} m \ddot{x}+\alpha \dot{x}=-\kappa x \end{equation} but with \begin{equation}...
  5. C

    Classical How is the book "Structure and Interpretation of Classical Mechanics"?

    I learned some computer science basics from the book SICP ( Structure and Interpretation of Computer Programs, Authors: Gerald Jay Sussman, Hal Abelson, Julie Sussman ) and I've witnessed a book about mechanics from the same author called Structure and Interpretation of Classical Mechanics...
  6. V

    I Non-Newtonian description of an accelerated object

    Hello, everyone! In the first few chapters of Physics 1, there is the description of motion, and the computation of the acceleration vector as being the sum of the rate of change of velocity, with the division of linear velocity squared by the instant radius of curvature: The above equation...
  7. L

    Relativity Special relativity in Lagrangian and Hamiltonian language

    Some introduction books on Lagrangian and Hamiltonian mechanics use classical mechanics as the theoretical framework, and when it come to special relativity it goes back to the basics and force language again. I would like to ask for some recommendations on good books that introduces Lagrangian...
  8. patric44

    How does Generalized Work have dimensions of Work?

    hi guys my analytical mechanics professor asked a question the other day about, how come the generalized forces##Q_{\alpha}## doesn't need to have a dimension of force, and the generalized coordinated ##q_{\alpha}##as well doesn't need to have a dimension of length, but the generalized work...
  9. A

    Analytical Mechanics: Regularity Conditions on Constraint Surface

    Hi, In my course in analytical mechanics, it is said that for a system of n particles subjected to r constraint equations, it is necessary to impose regularity conditions on the constraint surface defined by G = 0 where G is a function of the position of the position of the particles and time...
  10. A

    Particles on Deforming Surfaces: Theory & Analysis

    All books in analytical mechanics explain the case of a particle moving on a given static surface. But what happen if, for example, the surface is having some deformation?. I imagine that the principle of virtual work, and hence, D'Alembert are no longer valid since the normal force by the...
  11. dRic2

    Can analytical mechanics be deduced from Newtonian mechanics?

    As you can see from the very last line of my post, this whole post may come from the fact that I don't get sarcasm o0)Hi, reading the above mentioned book I ran into the following footnote: Postulate A was earlier stated as: An alternative, but equal, version of Postulate A is given the page...
  12. sams

    A The δ Notation in Calculus of Variations

    On page 224 of the 5th edition of Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion, the authors introduced the ##δ## notation (in section 6.7). This notation is given by Equations (6.88) which are as follows: $$\delta J = \frac{\partial J}{\partial...
  13. Gbox

    Prove: angular momentum is preserved

    3. Find the hamilton equations 4. using 3. prove the the angular momentum in the z axis ##L_z=m(x\dot y-xy\dot)## is preserved. I got in ##3##: How can I prove 4?
  14. Celso

    Differential equation for the simple pendulum

    How do I start this? I plugged the differential equation at wolfram alpha and it semmed too complicated for such an exercise. I've also looked at a sample of an answer on cheeg where the initial approach is to rewrite the equation as ##\frac{d}{dt} (\frac{\dot\theta^2}{2}-cos(\theta)) = 0## How...
  15. T

    How come "terminal velocity" and "final velocity" are different?

    Homework Statement Someone shot the bullet perpendicular to the ground. And there is air resistance. Velocity of bullet is v^2 = Ae^(-2kx)-g/k upward v^2 = g/k-Be^(2kx) downward A,B is constant, g is a gravitational acceleration, k = c2/m , c2 is a resistance constant, m is a mass...
  16. K

    Virtual work in Atwood's machine

    The first chapter in Goldstein's Classical Mechanics ends with 3 examples about how to apply Lagrange's eqs. to simple problems. The second example is about the Atwood's machine. The book says that the tension of the rope can be ignored, but I don't understand why. The two masses can move...
  17. sams

    A Summation Index Notation in the Transformation Equations

    In Chapter 7: Hamilton's Principle, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 258-259, we have the following equations: 1. Upon squaring Equation (7.117), why did the authors in the first term of Equation (7.118) are summing over two...
  18. F

    Energy and motion from ##R^6 \rightarrow R##

    We know that energy is a function of space and velocity and it’s constant (in ideal case) though time. So ## E(\vec{x}(t) , \vec{\dot{x}}(t)) = E_0## where ##\vec{x} , \vec{\dot{x}} \in \mathbb{R}^3##. So my function is ##E : \mathbb{R}^6 \rightarrow \mathbb{R}##. Then there is my question...
  19. sams

    Scleronomic or Rheonomic Mechanical System?

    I would really appreciate if someone could advise me whether the system below is a scleronomic or a rheonomic mechanical system, or a mix of both. If we consider the first pendulum, the constraint is fixed which leads to a scleronomous case while the constraint of the second pendulum is not...
  20. sams

    I A Question about the Lagrangian

    Is there any proof for the Lagrangian: $$L = T - U $$ And why L = T - U ? Any help is much appreciated. Thank you.
  21. T

    Hamilton-Jacobi theory problem

    Homework Statement A particle moves on the ##xy## plane having it's trajectory described by the Hamiltonian $$ H = p_{x}p_{y}cos(\omega t) + \frac{1}{2}(p_{x}^{2}+p_{y}^{2})sin(\omega t) $$ a) Find a complete integral for the Hamilton-Jacobi Equation b) Solve for ##x(t)## and ##y(t)## with...
  22. J

    Proving stable equilibrium: Rotating circular hoop

    Homework Statement A circular hoop of radius R rotates with angular frequency ω about a vertical axis coincident with its diameter. A bead of mass m slides frictionlessly under gravity on the hoop. Let θ be the bead’s angular position relative to the vertical (so that θ = 0 corresponds to the...
  23. Pushoam

    Obtaining the equation of motion using analytical mechanics

    Let's consider a particle moving along x – axis, its position at t = 1s is 1m and speed is 1 m/s. How can one calculate acceleration on the basis of this information?
  24. Jianphys17

    Is understanding analytical mechanics essential for studying QM?

    Hi at all. I'm wondered is necessary learn well analytical mechanic ( eg learning The Goldstein's book), for understand and study QM ?
  25. Jianphys17

    Better textbook for analytical mechanics

    Hi at all. According to you which of the two texts, between Landau-Lifshitz (mechanics) and the Goldberg (classical mech) is better for study Analytical Mech ? Or there are other better ones ?
  26. S

    Rocket Launch: Achieving Escape Velocity w/ Fuel-Mass Ratio of 300

    Homework Statement Rockets are propelled by the momentum of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that for a Rocket starting...
  27. S

    How do you solve the problem of a rocket with variable mass?

    Homework Statement Rockets are propelled by the momentum of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that for a Rocket starting...
  28. TheCapacitor

    Classical Best analytical mechanics textbook recommandation

    Hello, I'm a second year physics student. We are going to use "hand and finch analytical mechanics", however the reviews I saw about this book are bad. I've already taken calculus for mathematicians, linear algebra, classical mechanics, special relativity, and electromagnetism. The topics it...
  29. J

    Very simple Lagrangian mechanics problem

    Homework Statement [/B] Consider a mass m moving in a frictionless plane that slopes at an angle \alpha with the horizontal. Write down the Lagrangian \mathcal{L} in terms of coordinates x measured horizontally across the slope, and y, measured down the slope. (Treat the system as...
  30. amjad-sh

    Prerequisites needed to grasp analytical mechanics.

    1.What are the mathematical prerequisites needed to grasp analytical mechanics conceptually and technically? 2.To understand the very concept of analytical mechanics, do I need to train my self in Euclidean and Riemannian manifolds, and do I need to have good knowledge in topology and...
  31. davidbenari

    Classical Studying Analytical Mechanics this July

    So I'm taking analytical mechanics next semester. The textbook they use is Thornton and Marion's "Classical Dynamics of Particles and Systems". I want to learn it all this month. It is feasible, but extenuating. But there is another option. I could read Robert Fitzpatrick's Newtonian Dynamics...
  32. Frimus

    Classical Analytical Mechanics by Johns

    Is there someone in PF community who knows "Analytical Mechanics for Relativity and Quantum Mechanics" by Johns? What is your opinion on the book?
  33. A

    Time derivatives of sin and cos phi

    Homework Statement By using chain rule of differentiation, show that: $$ \frac{\mathrm{d} sin\phi }{\mathrm{d} t} = \dot{\phi} cos\phi , \frac{\mathrm{d} cos\phi }{\mathrm{d} t} = -\dot{\phi} sin\phi , $$ Homework EquationsThe Attempt at a Solution I got this right for a homework problem...
  34. A

    Simple question on proving polar coordinates

    Homework Statement Prove: $$\frac{d\hat{r}}{dt} = \dot{\phi} \hat{\phi }$$ and $$\frac{d\hat{\phi}}{dt} = -\dot{\phi} \hat{r }$$Homework EquationsThe Attempt at a Solution I solved this for an Analytical Mechanics assignment a month ago, and completely forgot how it goes.. $$\hat{r} ⊥...
  35. M

    Analytical Mechanics and Waves

    Hello everyone. I'm taking two physics courses this semester. Analytical Mechanics and Waves. I was hoping if you guys could help me out a little and recommend some good books for both courses. Also, if you know good lectures that were recorded and I have access too (YouTube for example) I...
  36. N

    Analytical Mechanics: Proving Equilibrium at Bottom of Bowl

    Homework Statement A point particle of mass m is confined to the frictionless surface of a spherical bowl. There are 2 degrees of freedom. 1. Prove that the equilibrium point is the bottom of the bowl. , 2.Does the bowl need to be exactly spherical for this to be true ? 3. Near the...
  37. P

    Analytical mechanics question.

    Hi, Homework Statement I was given the setup in the attachment and was asked to find the angular frequency of small oscillations around the equilibrium. m1=m; m2=√3m Homework Equations The Attempt at a Solution I have found L = 1/2*(3+√3)*mR2\dot{θ}2 + mgRcosθ+√3mgRsinθ and the...
  38. jtbell

    Analytical Mechanics by Fowles and Cassiday

    Author: Grant Fowles and George Cassiday Title: Analytical Mechanics Amazon Link: https://www.amazon.com/dp/0534494927/?tag=pfamazon01-20 Prerequisities: Calculus-based intro physics and a year of calculus; differential equations and matrix algebra highly recommended before or concurrently...
  39. I

    Analytical Mechanics Lectures: Get Help Now

    i need lectures in analytical mechanics, please help
  40. T

    Some questions about Analytical Mechanics

    Hi, I have some questions about some fundamental things of analytical mechanics... The Lagrangian contains all the physical information concerning the system and the forces acting on it. Is that correct? If it is, it only applies to systems where the applied forces are conservative, correct...
  41. F

    Analytical Mechanics 6th Ed. by Fowles & Cassiday

    This is the book I used for classical mechanics in College. I'm looking through it again, trying to study and really deeply learn the things I winged in my undergrad years, hopefully to take a GRE and go into graduate school. I'm having some issues with this text. Often, it will use things...
  42. K

    Analytical Mechanics question

    A cannon launches a 10cm diameter cannonball w/ a muzzle velocity of 100 m/s horizontally from the top of a 40m high wall under STP. The cannon has a mas of 40 kg and is free to recoil. a. What is the initial v of the cannonball wrt the ground b.calculate the landing position neglecting air...
  43. J

    Hey, so I am just working on a second year Analytical Mechanics

    Hey, so I am just working on a second year Analytical Mechanics assignment, and right now dealing with oscillations. I have two questions I am stumped on and don't know if I have it right. It is probably basic, but just checking. 6. The frequency fd of a damped oscillator is 100 Hz, and the...
  44. E

    Analytical Mechanics - rolling wheel (accelerating)

    Homework Statement A wheel of radius b rolls along the ground with constant forward acceleration a0. Show that at any given instant, the magnitude of the acceleration of any point on the wheel is (a0^2 + (v^4 / b^2))^(1/2) relative to the center of the wheel. Here v is the...
  45. Y

    Analytical Mechanics by Hand,Finch

    Hello, Has anyone used Analytical Mechanics by Hand,Finch for their study in Classical Mechanics? Did the authors publish the solution for the questions given in the book? I can't seem to find them any where. Pardon me if i post this question in the wrong section. thanks, yinx
  46. L

    Analytical mechanics for dummies ?

    Analytical mechanics "for dummies"? I'm starting my 2nd year course in analytical mechanics tomorrow and am quite worried that I'll get thrown off very early on, as I didn't fully grasp all of the concepts in lagrangian mechanics in my previous mechanics course (only studied up to lagrangian...
  47. S

    Analytical mechanics book to go along with meriam-kraige dynamics course

    We will be doing a course based entirely on Engineering Mechanics Vol. 2: Dynamics by Meriam and Kraige in the semester coming up. I want to supplement it with some good theoretical grounding in Lagrangian and Hamiltonian mechanics. Now, I cannot handle an entire separate course for those topics...
  48. H

    Analytical Mechanics in Mechanical Engineering

    How important is Analytical Mechanics in Mechanical Engineering? I'm asking that because my regular course does not offer Analytical Mechanics in its original curriculum, so I would have to get it as an elective from Physics course (curriculums are pretty much pre-defined in here). Thing is...
  49. T

    Analytical Mechanics: Solving Equations of Motion for Block Sliding Down Ramp

    Homework Statement a block(mass m), is sliding down a long straight ramp with angle theta, this block is undergoing frictional forces(mu) and air resistance proportional to the speed of the block. In its sliding the block reaches a terminal velocity. Determine the equation of motion and...
  50. F

    Projectile Range & Max Range: Analytical Mechanics

    Homework Statement A projectile is launched from a platform that is moving horizontally with velocity v=(v,0,0). The initial velocity of the projectile is u with a launch angle theta relative to the platform. (a) Show that the range of he projectile is R= u^2*sin(2*theta)/g + 2*v*u*sin(theta)...
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