Geometrical formulation of Hamiltonian Mechanics
We consider mechanical systems that are holonomic and conservative. For such a system we can construct a Lagrangian , where which is the difference of the total kinetic T and potential V energies. These mechanical systems evolve according to the n Lagrange equations
(1)
for. These are each second order ordinary differential equations and so the system is determined for all time once initial conditions are specified (or conditions at two different times). The state of the system is represented by a point in configuration space.
2. The passage to Hamiltonian mechanics:
i. Generalized momentum:
We define the generalized momentum for a Lagrangian mechanical system for to be
(2)
Note that we have in general, where and .
In terms of the generalized momenta, Lagrange’s equations become
(3)
For further, in principle, we can solve the relations above which defined the generalized momenta, to find functional expressions for in terms of and , i.e. we can solve the relation defining the generalized momenta to find where and .
ii. The Legendre transformation:
We define the Hamiltonian function as the Legendre transform of the Lagrangian function, i.e. we define it to be
(4)
Where and and we suppose . Note that in this definition we used the form dot product
(5)
From Lagrange’s equation of motion we can deduce Hamilton’s equation of motion using the definitions for the generalized momenta and Hamiltonian
And
and (6)
3. Hamiltonian equations of motion:
Lagrange’s equations of motion imply Hamilton’s canonical equations, for we have
(7)
These consist of first order equations of motion.
Proof:
Using the definition of the Hamiltonian in terms of the Lagrangian and noting for we use chain and product rules, that if while and the definition of the generalized momenta. Directly computing using this sequence or results gives
(8)
For . Again using the definition of Hamiltonian as above, we use the chain and product rule that if , while the definition of generalized momenta and Lagrange’s equations for motion in the form . Computing this equation gives
(9)
For . Then the Lagrange’s equations of motion imply Hamilton’s canonical equations.
If the Lagrangian is independent of explicit, then when we solve the equation that defines the generalized momenta, we find . Hence we see that
(10)
i.e. the Hamiltonian is also independent of explicitly. In general, using the chain rule and Hamilton’s equations we see that
(11)
Hence we have
(12)
Hence if H does not explicitly dependent on t then H is a constant of the motion, or conserved quantity or integral of the motion.
Hence the absence of explicit t dependence in the Hamiltonian H could serve as a more general definition of conservative system, though in general H may not be the total energy. However for simple mechanical systems for which the kinetic energy is a homogeneous quadratic function in , and the potential , then the Hamiltonian H will be the total energy. To see this suppose
(13)
I.e. a homogenous quadratic function in , necessarily , then we have
(14)
which implies
(15)
Thus the Hamiltonian , i.e. the total energy.
Example:(Simple harmonic oscillator).
The Lagrangian for the simple harmonic oscillator, which consists of a mass m moving in quadratic potential field with characteristic coefficient, is:
(16)
The corresponding generalized momentum is:
(17)
Which is the usual momentum This implies and so the Hamiltonian is gives by:
(18)
`note that last expression is the sum of the kinetic and potential energies and so is the total energy. Hamilton’s equations of motion are thus given by:
⬄
⬄
(19)
Note that combining these two equations, we get the usual equation for harmonic oscillator: .
4. Geometrical concepts
i. Differential forms on
A smooth 1-form on is a real valued functional on the set of all tangent vectors to i.e.
(20)
Given a 1-form, for each the map
(21)
Is an element of the dual space . When we extent this notion to all of , we see that the space of 1-form on is dual to the space of vector field on .
In particular, the 1-form are defined by the property that for any vector
(22)
The form a basis for the 1-form on , so any other 1-form may be expressed in the form
(23)
(24)
The 1-form on are part of an algebra, called the algebra of differential forms on . The multiplication in this algebra is called wedge product, and it is skew- symmetric
(25)
One consequence of this is that =0
If each summand of differential form contains k is called k-form. Functions are considered to be 0-form, and any form on of degree must be zero due to the skew-symmetry.
A basis for the k- form on is given by the set
Any k- form may be expressed in the form
(26)
Where I is ranges over all multi- indices of the length k.
Forms act on vector fields to give real –valued functions and k- form act on K- tuples of vector fields to give real - valued function.
Example:
If are 1-forms and are vector fields then
(27)
In general if are 1- forms and are vector fields then
(28)
ii. Differential forms on manifold
Given a smooth manifold M, 1- form on M is real valued function on the set of all tangent vectors to M such that
1- is the linear on the tangent space for each .
2- For any smooth vector v on M, the function is smooth
So for each The map
Is an element of the dual space .
Exterior derivative:
The exterior derivative of differential form of degree k is differential form of degree k+1.
If f is smooth function (a 0- form) then the exterior derivative of f is the differential df is the unique 1- form such that for every smooth vector field X, where is directional derivative of f in direction of X.
In local coordinate, if is a smooth differential k- form on a manifold M its differential or exterior derivative is (k+1) –form
(29)
The differential or exterior derivative d, taking the k- form to (k+1) – form has the following properties:
Linearity for c, constant,
Anti-derivative for a k- form , an 1-form.
5. The Geometrical formulationof Hamilton’s equations
Hamilton’s equations correspond to a construction, by the anti- symmetric 2-form.
(30)
In this case, stating with a Hamiltonian function H, one produces a vector field as follows:
(31)
Hamilton’s equations are then the dynamical system for the vector field .called the Hamiltonian vectors field Here is called a symplectic form and is sometimes called the symplectic gradient of H. While the flow along a gradient vector field of f changes the value of f as fast as possible, flow along keeps the value of H constant since
(32)
Since is anti-symmetric.
One can check that the equation
(33)
Implies Hamilton’s equations for since equating
(34)
And
(35)
Implies
(36)
Another important property of is that
(37)
Since (where is the lie derivative with respect to in general).
i. Hamiltonian vector field:
A vector field that satisfies
(38)
Is called a Hamiltonian vector field and the space of such vector field on will be denoted vect since is non-degenerate, the equation
(39)
Implies that if , then and f = constant as a result, we have an exact sequence of maps.
One can also ask whether all Hamiltonian vector fields (elements of) actually come from a Hamiltonian.
The equation
(40)
Implies
(41)
So is a closed 1-form. Since , this must also be exact, so one can find a Hamiltonian function f .
Just as we saw that along , one can compute the derivative of an arbitrary function g along as
(42)
ii. The contangent Bundle:
Let M be any smooth manifold and let be its contangent bundle. As we saw there is a canonical 2-form on which can be defined as follows: be the base point projection. Then, for every , define
(43)
I claim that is a smooth 1-form on and that is a symplectic form on .
To see this, let us compute in local coordinates.
Let be a local coordinate chart. Since the 10 forms are linearly independent at every point of U, it follows that there are unique functions on so that, for ,
(44)
The functions , then from a smooth coordinate system on in which the projection mapping is given by
(45)
It is then straight forward to compute that, in this coordinate system
(46)
Hence Ω and so is non-degenerate.
A ‘’standard ‘’ structure on . This of as a smooth manifold and let Ω be the form with constant coefficients.
Ω (47)
6.Coadjoint orbits:
Let denote the co adjoint representation of G. This is so called ‘’contragredient’’ representation to the adjoint representation. Thus, for any and , the element is determined by the rule
for all (48)
One must be careful not to confuse with . Insead, as our definition shows, .
Note that the induced homomorphism of lie algebra is given by
(49)
The orbits G. in are called the Coadjoint orbits. Each of them carries a natural symplectic structure to see this is defined, let be fixed, and let be the usual submersion induced by the -action, . . Now let be the left-invariant 1-form on G whose value at e is , thus where is the canonical g-valued 1-form on G.
i. Proposition:
There is unique symplectic form on the orbit satisfying
Proof:
If proposition is true, then must satisfy the rule
for all a symplectic 2-form on .
First, note that, for , we may compute via the structure equations that
(50)
In particular, , if and only if X lies in the null space of the 2-form . In other words, the null space of is , the lie algebra of . Since is . Of course, this is precisely the tangent space at a to the left coset a . Thus for each
(51)
It follows that , is naturally isomorphic to the symplectic quotient space for each . Thus, there is a unique, non-degenerate 2-form on so that
(52)
It remains to show that if however, this letter case occurs if where . Now, for any , we have
(53)
Thus, . Since the following square commutes, it follows that .
All this shows that there is a well-defined, non-degenerate 2-form Ω on which satisfies . Since is a smooth submersion, the equation
(54)
Implies that , as promised.
Note that a consequence of proposition is that all of the Coadjoint orbits are actually even dimensional, as we shall see when we take up the subject of reduction, the Coadjoint orbits are particularly interesting symplectic manifolds.
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