#### Generating Functions for Transformations of Coordinates

In order to describe the motion of a system mathematically we need to be able to specify the instantaneous configuration of the system - the position. For example the motion of a projectile can be specified by the horizontal distance of the projectile...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1522-generating-functions-for-transformations-of-coordinates.html#### Infinitesimal Canonical Transformations

Perturbed and unperturbed phase curves of a Hamiltonian system can be connected via a canonical transformation. This suggest that perturbation theory for Hamiltonian systems is really the study of canonical transformation which depend upon a parameter...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1523-infinitesimal-canonical-transformations.html#### Introduction to Perturbation Theory

Real problems rarely have Hamiltonians with equations of motion having simple solutions depending upon elementary functions, so some method of approximating the solutions is needed. Typically we start with a Hamiltonian of the form (1) where the...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1525-introduction-to-perturbation-theory.html#### Making Equations Dimentionless

It is often useful to remove the units from an equation. We can see which physical mechanisms are more important and the equation to be solved is simpler. We scale all variables so that the variables become dimensionless. We can solve the equation...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1526-making-equations-dimentionless.html#### Moments of Inertia of Some Basic Shapes

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1527-moments-of-inertia-of-some-basic-shapes.html#### Possible Forms of Generating Function

Given an arbitrary canonical transformation it may not be possible to treat and as independent variables since the condition for this to be possible is that the equation can be solved to give in terms of and so so this transformation cannot be applied...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1528-possible-forms-of-generating-function.html#### Preservation of Area by Time Independent Systems

A system influenced by time dependent forces or which is represented in a rotating or non inertial reference frame has a Hamiltonian which depends explicitly on time, The rate of change of the Hamiltonian is given by On using Hamilton's equations of...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1529-preservation-of-area-by-time-independent-systems.html#### Proof of Conservation of Angular Momentum

Theorem Angular momentum is conserved for a body subject to no net external torque. Proof Let a body consist of particles at position vectors The motion of each particle is determined by Newton's second law or Taking the cross product with and summing...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1531-proof-of-conservation-of-angular-momentum.html#### Proof of Newton's First Law of Motion

Theorem A body continues in a state of uniform motion unless acted on by an external force. Proof Let a body consist of particles of mass at position vectors The equation of motion for particle is The force acting on the body is The position vector of...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1532-proof-of-newton-s-first-law-of-motion.html#### Proof of the Law of Conservation of Momentum

It is not obvious why momentum should be conserved. Momentum is avery abstract concept, and the principle of conservation of momentummay be seen as a consequence of other, fundamental Laws of physics: The principle of Relativity The Principle of...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1533-proof-of-the-law-of-conservation-of-momentum.html#### Proof That the Moment of a Force Acting on a Body Equals Rate of Change of Angular Momentum of the Body

It is not obvious why momentum should be conserved. Momentum is avery abstract concept, and the principle of conservation of momentummay be seen as a consequence of other, fundamental Laws of physics: The principle of Relativity The Principle of...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1534-proof-that-the-moment-of-a-force-acting-on-a-body-equals-rate-of-change-of-angular-momentum-of-the-body.html#### Proof That Trajectory of BodyAbout Another Body is a Conic Section

Suppose a body moves under the influence of a single mass producing a gravitational field. The equation of motion of the body is Taking the cross product of this equation with gives Now since angular mometum is conserved. Hence a constant vector. We...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1535-proof-that-trajectory-of-bodyabout-another-body-is-a-conic-section.html#### Condition for a Transformation to Preserve Hamiltonian Form

The area of a region of the phase space diagram at time is given in terms of the area at some earlier time by Differentiating this expression gives or is a solution curve of the system so then for a transformation The second bracket equals 0 for...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1536-condition-for-a-transformation-to-preserve-hamiltonian-form.html#### The Hamiltonian

The Hamiltonian represents the energy of the system which is the sum of kinetic and potential energy, labelled and respectively. For a one dimensional system, we may write so where Note that is a function of only and is a function of only. In general...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1537-the-hamiltonian.html#### The Lagrangian Equation of Motion

The Lagrangian is defined as where and is a function of Hence The first of Hamilton's equations gives so the bracketed term vanishes and leaves The other of Hamilton's equations is and we can use to give This is Lagrange's equation of motion. It is a...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1538-the-lagrangian-equation-of-motion.html#### The Relationships Between Different Generating Functions

There may be more than one generating function for the same transformation, so it seems logical that relationships exist between each generating functions. These do in fact exist. Example: Find the relationship between and Hence The set of...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1539-the-relationships-between-different-generating-functions.html#### The Rocket Equation

Without gravity With Gravity:

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1540-the-rocket-equation.html#### Time Dependent Canonical Transformations

Time dependent canonical transformations are very similar to the time independent case, and as with the time independent case, the preservation of area is a part of the analysis. Consider the transformation The condition for such a transformation to...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1541-time-dependent-canonical-transformations.html#### Damped and Forced Vibrations

Free undamped, unforced vibrations in simple harmonic motion obey the equation or Typically the vibration will be subject to a resistive term R, which for low speeds is proportional to velocity and in the opposite direction hence The equation for...

https://astarmathsandphysics.com/university-physics-notes/classical-mechanics/1542-damped-and-forced-vibrations.html#### Alternative Forms of The Continuity Equation

The continuity equation is usually written (1) where is the density of the fluid at a point. < > It may also be written (2) or (3) where is the Stokes derivative. < > (2) follows from (1) on using the identity < > To derive (3) write

https://astarmathsandphysics.com/university-physics-notes/fluid-mechanics/1561-alternative-forms-of-the-continuity-equation.html