By Antonio Romano
This textbook takes a broad yet thorough method of mechanics, aimed toward bridging the distance among classical analytic and modern differential geometric ways to the subject. constructed through the author from 35 years of training adventure, the presentation is designed to offer scholars an outline of the numerous various models used during the background of the field—from Newton to Lagrange—while additionally portray a transparent photo of the main smooth advancements. all through, it makes heavy use of the robust instruments provided by way of Mathematica. the amount is geared up into elements. the 1st makes a speciality of constructing the mathematical framework of linear algebra and differential geometry worthwhile for the rest of the e-book. subject matters coated comprise tensor algebra, Euclidean and symplectic vector areas, differential manifolds, and absolute differential calculus. the second one a part of the booklet applies those subject matters to kinematics, inflexible physique dynamics, Lagrangian and Hamiltonian dynamics, Hamilton–Jacobi conception, thoroughly integrable structures, statistical mechanics of equilibrium, and impulsive dynamics, between others. specific in its scope of assurance and approach to approach, Classical Mechanics should be a really resource for graduate students and complicated undergraduates in utilized arithmetic and physics who desire to realize a deeper figuring out of mechanics.
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Extra info for Classical Mechanics with Mathematica® (Modeling and Simulation in Science, Engineering and Technology)
U1 ; u2 ; u3 / is orthogonal. u1 ; : : : ; un / is determined. Dividing each vector of this system by its length, we obtain an orthonormal system.
1/m. En /. En /. j / D . 1/m. jr / ; < jr are chosen in the set 1; : : : ; n in all possible ways. Aij /. 7. En / given by X X T^L D . 1/m. hrCs /, and m. hrCs /. 2. E3 /; ! E3 / given by T ^ ! 3. E5 / has the component T12 L45 T14 L25 C T15 L24 C T24 L15 along the basis vector Â 1 ^ Â 2 ^ Â 4 ^ Â 5 . 4 Oriented Vector Spaces 37 It is not difficult to verify that T^L is skew-symmetric and independent of the basis. It can also be proved that T ^ L D . 25) We can define the exterior algebra as we did for the tensor algebra.
Instead of Eqs. 5 Tensor Algebra In the previous sections we defined the addition of tensors belonging to the same tensor space. On the other hand, the tensor product of two tensors that might belong to different tensor spaces defines a new tensor that belongs to another tensor space. In conclusion, the tensor product is not an internal operation. However, it is possible to introduce a suitable set that, equipped with the aforementioned operations, becomes an algebra. Let us consider the infinite direct sum (Sect.