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#### Introduction to Newtonian Mechanics

#### Vectors in Multiple Dimensions

#### Newton's Laws of Motion

#### Newton's Laws (cont.) and Inclined Planes

#### Work-Energy Theorem and Law of Conservation of Energy

- Introduction to Newtonian Mechanics
Thanks for watching introduction to Newtonian Mechanics on www.raptvlive.com

- Vectors in Multiple Dimensions
Thanks for watching Vectors in Multiple Dimensions on www.raptvlive.com

- Newton's Laws of Motion
Thanks for watching Newton's Laws of Motion on www.raptvlive.com

- Newton's Laws (cont.) and Inclined Planes
Thanks for watching Newton's Laws (cont.) and Inclined Planes on www.raptvlive.com

- Work-Energy Theorem and Law of Conservation of Energy
Thanks for watching Work-Energy Theorem and Law of Conservation of Energy on www.raptvlive.com

#### Introduction to Newtonian Mechanics

#### Vectors in Multiple Dimensions

#### Newton's Laws of Motion

#### Newton's Laws (cont.) and Inclined Planes

#### Work-Energy Theorem and Law of Conservation of Energy

Math help

Who is William Rowan Hamilton?

A Irish mathematician from the 1800's.

Who is by Joseph Louis Lagrange?

A mathematician and astronomer born in Turin, Piedmont, who lived part of his life in Prussia and part in France in the 1700's.

What do Hamilton and Lagrangian have in common?

A new and equivalent way of looking at Newtonian physics. Generally, these equations do not provide a more convenient way of solving a particular problem in classical mechanics. Rather, they provide deeper insights into both the general structure of classical mechanics and its connection to quantum mechanics as understood through Hamiltonian mechanics, as well as its connection to other areas of science.

What is symplectic manifold?

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ?, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds, e.g., in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field: The set of all possible configurations of a system is modelled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

What is mathematical formalism?

Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system. The function H is known as the Hamiltonian or the energy function. The symplectic manifold is then called the phase space. The Hamiltonian induces a special vector field on the symplectic manifold, known as the Hamiltonian vector field.

What is phase space?

A space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables i.e. the cotangent space of configuration space.

What is Poisson algebra?

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson-Lie groups are a special case. The algebra is named in honour of Siméon-Denis Poisson.

What are Moyal brackets?

The Moyal bracket is a way of describing the commutator of observables in quantum mechanics when these observables are described as functions on phase space. It relies on schemes for identifying functions on phase space with quantum observables, the most famous of these schemes being Weyl quantization. It underlies Moyal’s dynamical equation, an equivalent formulation of Heisenberg’s quantum equation of motion, thereby providing the quantum generalization of Hamilton’s equations.

What is Lie algebra?

an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used.

What is vector space?

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms.

What is a Euclidean vector?

a geometric object that has a magnitude (or length) and direction and can be added according to the parallelogram law of addition. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B.

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