Key references
- Goldstein, Herbert; "Classical Mechanics"; 1959; Addison-Wesley Publishing Company, Inc..
- Lanczos, Cornelius; "The Variational Principles of Mechanics"; 4th ed.; 1970; Dover Publications, Inc..
- Taylor, John R.; "Classical Mechanics"; 2005; University Science Books.
- Susskind, Leonard & Hrabovsky, George; "The Theoretical Minimum: What You Need to Know to Start Doing Physics"; 2013; Basic Books.
- Coopersmith, Jennifer; "The Lazy Universe"; 2017; Oxford University Press.
Lagrangian mechanics
Hamilton's principle
The actual path which a holonomic system follows between two points and in configuration space in a given time interval, to , is such that the action integral
is stationary when taken along the actual path.
The generalized force and generalized (or conjugate) momentum are respectively given by
The Euler-Lagrange equations can be rewritten as .
Conservation and symmetry
If the Lagrangian does not contain a coordinate (though it may contain ) then the coordinate is said to be ignorable and the corresponding conjugate momentum is conserved.
Conservation of Hamiltonian
If does not depend explicitly on time ( ), then the Hamiltonian
is conserved.
For most systems, is just the total energy and is frequently conserved. However, the identification of as a constant of motion and as the total energy are two separate matters, and the conditions sufficient for one are not enough for the other.
Lagrangian for electromagnetism
The Lagrangian for a charge with mass in an electromagnetic field is
Hamiltonian mechanics
Given the -dimensional phase space of generalised coordinates and conjugate momenta then
are the canonical equations of Hamilton.
If is independent of then the conjugate momentum is conserved and the coordinate is ignorable.
A phase-space orbit is the path traced in phase space by a system as the system evolves in time.
Hamiltonian for electromagnetism
The Hamiltonian for a charge in an electromagnetic field is
Canonical transformations
Coordinate transformations given by
that satisfy
where is some function are said to be canonical (or contact) transformations. The transformation equations are completely specified by the generating function of the transformation which can be written as a function of independent variables in one of only four forms:
For ,
For ,
For ,
For ,
A transformation of time occurs automatically in the -dimensional relativistic Hamiltonian formulation; the invariant parameter of the system is the proper time and the ordinary time appears as one of the coordinates of the particle.
The motion of a system point in time is simply a particular contact transformation of the canonical coordinates in phase space.
The particular form for below generates the identity transformation:
Integral invariants of Poincaré
The integral
over any arbitrary region of phase space is invariant under canonical transformations. The invariance of is equivalent to the statement that volume in phase space is invariant under canonical transformations.
Lagrange brackets
(Note: The bracket notation used in this post is from Goldstein. In particular, the notation for Poisson brackets directly translates to the commutator of Quantum Mechanics for which the Poisson brackets are a direct analogue.)
The Lagrange bracket of and defined by
is invariant under canonical transformations.
The fundamental Lagrange brackets are
Poisson brackets
The Poisson bracket of and is defined by
with the identity
The fundamental Poisson brackets
are canonical invariants.
All Poisson brackets are independent of the set of canonical coordinates they are expressed in.
The canonical equations of motion written using Poisson brackets are
In general,
If is a constant of motion then
Jacobi's identity
If and are two constants of motion and is taken to be then the relation reduces to
Thus, the Poisson bracket of two constants of motion is itself a constant of the motion (Poisson's theorem).
Infinitesimal contact transformations and generators
where is some infinitesimal parameter of the transformation and is given by
It is customary to designate as the generating function even though, strictly speaking, the term refers to .
When and is a small time interval ,
Thus, the Hamiltonian is the generator of the system motion with time.
For any function we have which implies . If is a constant of motion, then and, hence, . Thus, the constants of motion are the generating functions of those infinitesimal canonical transformations which leave the Hamiltonian invariant. One can therefore determine all the constants of motion by an examination of the symmetry properties of the Hamiltonian!
Liouville's theorem
The density of systems in the neighbourhood of some given system in phase space remains constant in time.
Helmholtz circulation theorem
Vortices cannot be created or destroyed in phase space.
Noether's theorem
Every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law.
Hamilton-Jacobi theory
is the Hamilton-Jacobi equation with being the action functional (equation ). , also called Hamilton's principal function, happens to be the (type 2) generating function that characterises a canonical transformation to new coordinates where either all the and are constant or the are cyclic (making the conjugate momenta constant). The corresponding Hamiltonian equals .
The solutions yield and in terms of their starting values and at time .
Geometrical interpretation
Hyper-surfaces of constant represent 'wave-fronts' that evolve in time making the Hamilton-Jacobi theory a direct analogue of the 'matter waves' in Quantum Mechanics. The motion of a single particle can be represented by a wave. This played an important role in the development of Schrödinger's wave mechanics.
Continuous systems (Fields)
<TBD>