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theories:classical_theories:classical_mechanics

# Classical Mechanics

## Why is it interesting?

Classical Mechanics is the framework that we use to describe the behaviour of objects we meet in everyday life: from the famous apple that falls from a tree, to the motion of planets in our solar system.

In addition to these “school physics” application, there is a high relevance to almost all topics in modern physics. Namely, it turns out that there is not one unique way to describe classical mechanics but several. Instead, we can describe classical mechanics using:

• Newtonian Mechanics
• Lagrangian Mechanics
• Hamiltonian Mechanics

Each approach has unique strengths and it usually depends on the system that we want to describe which approach is best suited. It is important to note that these three approaches are completely equivalent in what they do. This means, the physics they describe is exactly the same. However, dependending on the system in question, sometimes calculations and the interpretation is easier in Lagrangian Mechanics and sometimes in Newtonian Mechanics or Hamiltonian Mechanics.

It makes sense to study these various approaches instead of just one because:

1. In more modern theories like quantum mechanics or quantum field theory, we are in exactly the same situation: There are several descriptions that all do the same thing. But for specific situations one approach is stronger than the others. To get a full understanding of the theory it is helpful to know all the various approaches. It is thus sensible to get familiar with this “various approaches” idea.
2. The same three approaches are also applicable in quantum field theory and quantum mechanics, which are the best theories of nature that we currently have. In classical mechanics we deal with objects that we are familiar with in everyday life, whereas in quantum field theory and quantum mechanics, things are more abstract. Thus it makes sense to get familiar with the approaches using everyday objects.

Moreover, classical mechanics is an ideal playground to get familiar with many important mathematical tools and concepts.

Nature and Nature’s laws lay hid in night:

God said, “Let Newton be!” and all was light. Alexander Pope

Important Related Concepts:

## Layman

In classical mechanics, we describe what happens in a system in terms of trajectories. For example, the path of a point-like object could look as follows:

## Student

The standard textbook is “Classical Mechanics” by Herbert Goldstein, Charles Poole, and John Safko

## Researcher

A great introduction to high-level concepts in classical mechanics are the lecture notes by David Tong: http://www.damtp.cam.ac.uk/user/tong/dynamics.html Especially part 4 is amazing and explains, for example, nicely how similar classical and quantum mechanics are, if formulated in the same language. (E.g. in classical mechanics, the generators of translations are also given by the momentum, acting via the Poisson bracket.)

a ‘Hamiltonian’ $$H : T^* Q \to \mathbb{R}$$ or a ‘Lagrangian’ $$L : T Q \to \mathbb{R}$$

Instead, we started with Hamilton’s principal function $$S : Q \to \mathbb{R}$$ where $Q$ is not the usual configuration space describing possible positions for a particle, but the ‘extended’ configuration space, which also includes time. Only this way do Hamilton’s equations, like the Maxwell relations, become a trivial consequence of the fact that partial derivatives commute.

Hamilton's principal function is basically just the action

A great book that describes the high-level perspective on classical mechanics is “Introduction to Mechanics and Symmetry” by Jerrold E. Marsden and Tudor S. Ratiu and see also the other books by Marsden, many of which are freely available online here, for example, his “Foundations of Mechanics”.

Classical mechanics can be formulated geometrically using fibre bundles.

The Lagrangian function is defined on the tangent bundle $T(C)$ of the configuration space $C$.

The Hamiltonian function is defined on the cotangent bundle $T^\star(C)$, which is also called phase space.

The map from $T^\star(C) \leftrightarrow T(C)$ is called Legendre transformation.

The phase space is endowed with a symplectic structure, called Poisson Bracket. The Poisson Bracket is an operation that eats two scalar fields $\Phi$, $\Psi$ on the manifold and spits out another scalar field $\theta$:

$$\theta = \{ \Phi,\Psi \}= \frac{\partial \Phi}{\partial p_a}\frac{\partial \Psi}{\partial q^a}-\frac{\partial \Phi}{\partial q_a}\frac{\partial \Psi}{\partial p^a}.$$

If we leave the $\Psi$ slot blank, we can use the Poisson bracket to define a differential operator $\{\Phi,\ \}$. This is a vector field and when in acts on $\Psi$, we get $\{\Phi, \Psi \}$. If we use instead of $\Phi$, the Hamiltonian $H$, we get an differential operator $\{H,\ \}$ that 'points along' the trajectories on in phase space $T^\star(C)$ and describes exactly the evolution that we get from Hamilton's equations.

In this sense, the dynamical evolution of a given system is completely described by the Hamiltonian (= a scalar function).

See page 471 in Road to Reality by R. Penrose, page 167 in Geometric Methods of Mathematical Physics by B. Schutz and http://philsci-archive.pitt.edu/2362/1/Part1ButterfForBub.pdf.

For some more backinfo why there is a symplectic structure in classical mechanics, have a look at https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/

Example1
Example2:

## FAQ

Why don't we use operators in classical mechanics, like we do in quantum mechanics?
Actually, we can do use operators:

As an interesting curiosity, classical mechanics can be formulated using wavefunctions/kets on a complex Hilbert space, with physical observables represented by operators and measurement being probabilistic, just as in quantum mechanics. It's not even all that strange, turning into a complexified reformulation of the classical Liouville equation (which handles probability distributions over the phase space).

The common formalism of QM is quite general and is completely capable of handling classical mechanics; where they disagree is on which operators represent things we actually measure in the world.

What group belongs to the Lie algebra of classical mechanics?
The Lie algebra of classical mechanics is defined through the Poisson bracket. The corresponding Lie group which integrates the Poisson bracket is called the “quantomorphism group”. (Source: https://physics.stackexchange.com/a/75775/37286 )

## History

• A History of Mechanics by René Dugas (book)