Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor Course website:

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Presentation transcript:

Analytical mechanics Fall 2015 S. Barraza-Lopez. Instructor Course website:

Syllabus (available at Chapters 1 to 12. Marion and Thornton, Fifth Edition. Three cumulative tests: -Monday September 28 (50 minutes) -Monday November 2 (50 minutes) -Monday December 14 (2 hours) About 14 homeworks to be assigned on Monday, and due Monday of next week

Grading Policy: Each exam: 20% of your grade HW: 35% of your grade Assistance: 5% Grading scores: : A71-85: B56-70: C40-55: D

HW 1 assigned (12 problems: quick review of vector analysis and matrix alegebra) Available at: Due on Friday August 28.

Vectors A scalar is a single quantity (a number or a single-valued function), but a vector is a collection of numbers or/and functions. Notation: A vector will usually be represented by a bold letter: (e.g., r), by a regular letter with a “vector sign” on top (e.g., Ā). (this vector has three entries)

Dot product and vector length Given two vectors a and b with the same number of entries, we define a “dot product” as: Similarly, we define the length of a vector v as.

A vector with n-entries

“Unit” vectors A unit vector is usually written with a “hat” symbol, and it is defined as follows: An important example of unit vectors are those that indicate orthogonal vectors…

“Unit” vectors A unit vector is usually written with a “hat” symbol, and it is defined as follows: An important example of unit vectors are those that indicate orthogonal vectors…

“Unit” vectors So in three-dimensions, these special unit vectors are: Question: express r in terms of i, j, and k.

Vector algebra r=x1i+x2j+x3k, is an example of vector algebra. What is  r where a is a scalar? What is  r when  =0? A+b = b+A? (why? Or why not?)

Towards matrix algebra: Rotation of coordinates Galileo once stated that the laws of mechanics should not depend on the choice you make of your coordinate axes (this is known as Galilean relativity). This means that the (right-handed) set of vectors on this figure are not “special:” others sets of vectors will describe mechanical phenomena just as well.

So we have that k’=k (the axis that is kept unchanged is the axis of rotation) We also see that i’, j’ are linear combinations of i, j.

We will address the conditions this matrix must satisfy in order to be a rotation matrix next time.

As a way to conclude: Most difficulties faced will be related to learning the notation. Please make a few notes with the definitions introduced today. Always seek clarification, if in doubt.

Next time: Basic calculus on vector quantities