- #1
Kashmir
- 466
- 74
I dont Understand how we get the final equations relating ##Q_r## with ##\lambda## given the conditions above?
Help in understanding this derivation of Lagrange Equations in Non-Holonomic case
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- Thread starter Kashmir
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SUMMARY
This discussion focuses on the derivation of Lagrange Equations in the context of non-holonomic systems, specifically addressing the relationship between the generalized coordinates ##Q_r## and the constants ##\lambda##. The theorem presented states that if the intersection of the kernels of linear functionals ##f_k## is contained within the kernel of another functional ##f##, then a linear combination of the functionals can express ##f##. This theorem is further generalized to linear operators between vector spaces, establishing a framework for understanding the connections between these mathematical constructs.
PREREQUISITES- Understanding of linear functionals and their properties
- Familiarity with vector spaces and linear operators
- Knowledge of Lagrange Equations in classical mechanics
- Concept of kernels in linear algebra
- Study the implications of the theorem regarding linear functionals in vector spaces
- Explore the derivation of Lagrange Equations in both holonomic and non-holonomic contexts
- Investigate the role of linear operators in functional analysis
- Learn about the application of kernels in solving linear equations
Mathematicians, physicists, and engineering students interested in advanced mechanics, particularly those focusing on non-holonomic systems and the mathematical foundations of Lagrange's equations.
I dont Understand how we get the final equations relating ##Q_r## with ##\lambda## given the conditions above?
- #2
wrobel
Science Advisor
- 1,245
- 1,052
There is a nice theorem.
Let ##f,f_1,\ldots, f_n:X\to\mathbb{R}## be linear functionals defined on a vector space ##X##.
Theorem. Assume that $$\bigcap_{k\in\{1,\ldots,n\}}\ker f_k\subset \ker f.$$
Then there are constants ##\lambda_1,\ldots,\lambda_n## such that
$$f=\sum_{k=1}^n\lambda_k f_k.$$
Moreover this theorem is a special case of the following fact. Let ##X,Y,Z## be vector spaces perhaps infinite dimensional. Let
$$A:X\to Y,\quad B:X\to Z$$ be linear operators such that ##\ker A\subset\ker B##. Then there is a linear operator ##\Lambda:Y\to Z## such that ##B=\Lambda A##.
Let ##f,f_1,\ldots, f_n:X\to\mathbb{R}## be linear functionals defined on a vector space ##X##.
Theorem. Assume that $$\bigcap_{k\in\{1,\ldots,n\}}\ker f_k\subset \ker f.$$
Then there are constants ##\lambda_1,\ldots,\lambda_n## such that
$$f=\sum_{k=1}^n\lambda_k f_k.$$
Moreover this theorem is a special case of the following fact. Let ##X,Y,Z## be vector spaces perhaps infinite dimensional. Let
$$A:X\to Y,\quad B:X\to Z$$ be linear operators such that ##\ker A\subset\ker B##. Then there is a linear operator ##\Lambda:Y\to Z## such that ##B=\Lambda A##.
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